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Multi-Objective Optimization:Algorithm Development and Applications in

Rubber Technology

A Thesis submitted to Gujarat Technological University

for the Award of

Doctor of Philosophy

in

Chemical Engineering

byDESAI RUPANDE NITINBHAI(Enrolment No. 139997105010)

under supervision of

Dr. S A Puranik

Gujarat Technological UniversityAhmedabad

December-2019

DECLARATION

I hereby declare that the thesis entitled “Multi-Objective Optimization: Algorithm De-

velopment and Applications in Rubber Technology" submitted by me for the degree of

Doctor of Philosophy is the record of research work carried out by me during the period from

January 2014 to March 2019 under the supervision of Dr. S. A. Puranik and this has not

formed the basis for the award of any degree, diploma, associate ship, fellowship, titles in this

or any other University or other institution of higher learning.

I further declare that the material obtained from other sources has been duly acknowledged in

the thesis. I shall be solely responsible for any plagiarism or other irregularities, if notices in

the thesis.

Signature of Research scholar: ............................................... Date:December ,2019.

Name of Research Scholar: Desai Rupande Nitinbhai

Place: Ahmedabad

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Development and Applications in Rubber Technology" submitted by Ms. Desai Rupande

Nitinbhai was carried out by the candidate under my supervision. To the best of my knowl-

edge:(i) The Candidate has not submitted the same research work to any other institution for

any degree/diploma, associate ship, fellowship or other similar titles. (ii) The thesis submitted

is a record of original research work done by the Research Scholar during the period of study

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PRIMARY SOURCES

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www.cs.uoi.grInternet Source

Narendra Patel, Nitin Padhiyar. "Fast Mesh-Sorting in Multi-objective Optimization∗∗IITGandhinagar, Ahmedabad, Gujarat, India",IFAC-PapersOnLine, 2015Publicat ion

sop.tik.ee.ethz.chInternet Source

Narendra Patel, Nitin Padhiyar. "Modif iedgenetic algorithm using Box Complex method:Application to optimal control problems",Journal of Process Control, 2015Publicat ion

www.gtu.ac.inInternet Source

Maria Gordina, Michael Röckner, Feng-Yu

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Wang. "Dimension-Independent HarnackInequalities for Subordinated Semigroups",Potential Analysis, 2010Publicat ion

www.iitk.ac.inInternet Source

Kasat, R.B.. "Multi-objective optimization of anindustrial f luidized-bed catalytic cracking unit(FCCU) using genetic algorithm (GA) with thejumping genes operator", Computers andChemical Engineering, 20031215Publicat ion

www.hanserpublications.comInternet Source

edoc.siteInternet Source

Narendra Patel, Nitin Padhiyar. "Multi-objectivedynamic optimization study of fed-batch bio-reactor", Chemical Engineering Research andDesign, 2017Publicat ion

Abeykoon, Chamil, Adrian L. Kelly, Elaine C.Brown, Javier Vera-Sorroche, Phil D. Coates,Eileen Harkin-Jones, Ken B. Howell, Jing Deng,Kang Li, and Mark Price. "Investigation of theprocess energy demand in polymer extrusion:

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A brief review and an experimental study",Applied Energy, 2014.Publicat ion

R. Saravanan, S. Ramabalan, C. Balamurugan."Evolutionary multi-criteria trajectory modelingof industrial robots in the presence ofobstacles", Engineering Applications ofArtif icial Intelligence, 2009Publicat ion

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xix

Synopsis

0.1 Synopsis Abstract

Evolutionary computation is becoming the most proven method for Global optimization of

complex problems. Among them, Genetic Algorithm (GA) has become more popular being

robust, flexible and relatively efficient. However, GAs are computationally more expensive

than the classical methods and hence suitable mainly for the off-line applications. Moreover,

the GAs are naturally designed for unconstrained problems and hence require additional

mechanism for constraint handling. GAs can handle single and multi-objective optimization

problems. Even with the developments in the computational powers of computers, solving

the complex multi-objective problems requires very long time. There is always a need for

development of robust and computationally efficient algorithms for large and complex prob-

lems. This research focuses on upgrading the GA to enhance the convergence and constraint

handling capabilities for multi-objective optimization. The proposed approaches are tested by

benchmark test functions and further validated using rubber extruder screw design application.

A mathematical model for rubber extrusion is developed using finite difference technique

considering temperature dependent viscosity modelled using Carreau-Yasuda model. This

model is used for optimization of screw design parameters and temperature profile simulta-

neously to maximize throughput minimizing power consumption. The temperatures of the

material under process within the extruder and residence time distribution of product are

also tracked for assured quality of product. The screw helix angle, channel depth, and screw

xx

speed are used as manipulated design parameters along with barrel temperature profile. Best

screw geometry, screw speed and barrel temperature profile are obtained using proposed

multi-objective optimization algorithm. These multiple optimum solutions assist the decision

maker in selecting an appropriate design which is the best according to his needs.

0.2 Brief description on the state of the art of the research topic

Multi-Objective Optimization(MOO) is a class, which deals with multiple conflicting objec-

tives simultaneously. MOO problems with conflicting objectives will have a set of solutions

(representing trade-offs among the objectives), which are called pareto optimal solutions, of

which none can be said to be better than the others with respect to all objectives. Usually the

decision makers want a small set of solutions to make a choice among them. The challenge is

to provide them with a set, as small as possible, that represents the whole set of choices, but to

compute this set in an efficient way.

Rubber extrusion process consists of pushing blend by means of screw extruder through

feeding channels and extrusion dies of relatively complex geometry. The channels are used to

condition the rubber flow parameters (velocity, temperature) and to distribute the flow rate of

different blends in the case of co-extrusion. The extrusion die orifice has to be designed to

produce profile with the required geometry. The process involves several complex phenomena:

complex rheological behaviour, fluid flow with free surfaces, etc. The critical part of extruder is

a screw. The extruder screw lies at the heart of many processing methods. It is obviously one of

the most crucial parts of a single screw extruder. Optimization of extrusion includes selection

of the operating and geometrical variables that maximize mass output maintaining quality

and minimize the remaining (in order to save energy, increase efficiency and avoid polymer

degradation, respectively). Rauwendaal (2014a). The objective of screw design is to deliver

the largest amount of output at minimum energy needs. Unfortunately high output & mixing

quality are, to some extent, conflicting requirements. The Helix angle is the most important

Ph.D. thesis Desai Rupande Nitinbhai

0.3 Definition of the Problem xxi

parameter effecting a performance of screw. It affects throughput and power consumption.

Discharge pressure also depends upon helix angle. Increasing the screw speed throughput

increases. Too high speed will result in greater temperature variation and poor mixing and thus

deteriorated quality of products. Effect of different design parameters, operating parameters

and material properties on conflicting performance parameters: throughput and power for a

single screw extruder is investigated formulating multi-objective optimization problem.

0.3 Definition of the Problem

Genetic Algorithm (GA) is a widely accepted population based stochastic optimization

technique for single and multi-objective optimization problems. Though, it is more computa-

tionally expensive algorithm compared to the gradient based algorithms, it is preferred tool

for complex functions and off-line analysis because of its capability of providing potentially

global optimum solution. GA is either binary coded, wherein the real values of population

members are encoded in binaries (0, 1), or real value coded, wherein the population members

are represented as real values. There are four steps in a GA for both real coded or binary coded

namely initiation, selection, mutation, and crossover. GA starts with a randomly generated

population of initial guess values of all the decision variables uniformly spread over the entire

solution space. This population of multiple members is then processed through recombination

and/or mutation to add diversity in the population to obtain better offspring. In every genera-

tion of GA, mutation and crossover operations are performed to add diversity in the population

members Deb (2001). This new population has potential to provide better fitness value com-

pared to the parent generation. The fitter members of parent and offspring populations survive

to the next generations, which then go through the crossover and mutation process once again

after selection step depending upon their fitness values. Thus, while crossover and mutation

operators add diversity in the population leading to high probability of convergence to global

optimum solution, selection method guides the GA to achieve appropriate convergence. GAs

are criticised for their slow convergence rate but are appreciated for their capacity to handle

Desai Rupande Nitinbhai Ph.D. thesis

xxii

complex problems.

Though there are many tuning parameters in GAs, determining proper values of parameters

is crucial. Choosing a population size too small increases the risk of converging to a local

minimum, on the other hand a larger population has a greater chance of finding the global min-

imum at the expense of more CPU time. Same way accuracy of encoding decision variables

plays crucial role in binary coded GA. Choosing shorter chromosomes has more probability

of exploring search space during initialization, and evolution (crossover and mutation) stages.

We in this work propose to use two parallel populations, one binary coded and one real coded.

Binary coded population is exploited to scan search space using shorter binary chromosome

lengths and real population is explored for convergence with desired accuracy. The binary

population takes care of global searching and supports the real population to escape any local

optima. The proposed binary real coded hybrid algorithm to explore search space is presented

as Parallel Universe Alien GA.

Rubber extrusion process consists of pushing blend by means of screw extruder through

feeding channels and extrusion dies of relatively complex geometry. The channels are used to

condition the rubber flow parameters (velocity, temperature) and to distribute the flow rate of

different blends in the case of co-extrusion. The extrusion die orifice has to be designed to

produce profile with the required geometry. The process involves several complex phenomena:

complex rheological behaviour, fluid flow with free surfaces, etc. The critical part of extruder

is designing a screw. The extruder screw lies at the heart of many processing methods. It is

obviously one of the most crucial parts of a single screw extruder. Optimization of extrusion

includes selection of the operating and geometrical variables that maximize mass output

maintaining quality and minimize the remaining (in order to save energy, increase efficiency

and avoid polymer degradation, respectively). Effect of different design parameters, operating

parameters and material properties on performance of a single screw extruder can be simulated

and optimized.


0.4 Objective and Scope of work xxiii

0.4 Objective and Scope of work

Considering the literature gap identified the current research focuses on the following three

objectives:

• Generating GA for multi-objective optimization problem, which has high probability of

providing global optimum solution at the same time in less computational time.

• The above developed algorithm will be tested using benchmark multi-objective opti-

mization problems.

• The algorithm will be tested on Extruder design optimization for maximization of

throughput and minimization of power consumption and betterment of selected proper-

ties, all conflicting objectives.

As a part of our PhD study we propose to develop GA program with modifications in the

existing algorithm to make it more robust and efficient. We will test the developed program

with benchmark test functions. The multi-objective optimization application for throughput

maximization and power minimization will be developed and solved using proposed algorithm.

0.5 Original contribution by the thesis

We developed hypothesis to modify GA using two sub-populations, one real coded and an-

other, binary coded. We call the concept of Parallel Universe having different encoding. Best

members from binary coded population known as Alien members will go to real coded popu-

lation and take part in evolution. Alien will transfer the information from one sub-population

(universe) to another; we call this concept as Parallel Universe Alien GA (PUALGA). This

approach increases robustness without any additional computational burden by combining the

capacity of both, binary and real coded GAs. In fact, dividing the population in sub-population


xxiv

will reduce the calculations needed for sorting and selection and hence will increase the overall

efficiency of the algorithm. Though, the proposed algorithm can be used with any population

based evolutionary optimization, we choose to use GA to demonstrate the clear benefits of

the proposed concept of hybridization. The algorithm flowchart for the proposed Parallel

Universe Alien GA (PUALGA) is presented in Fig. (0.1).

Figure 0.1. Parallel Universe Alien GA Evolution Scheme

We have also developed a generalized constraint handling approach for population based

EAs using Boundary Inspection (BI) approach. The BI approach converts every infeasible

member to a feasible one during the evolution process. The algorithm attempts to move

infeasible point in a direction joining an infeasible point and a feasible point such that we

reach within feasible area. At every generation using this approach all infeasible members

are converted to feasible members by moving towards randomly selected feasible point. The

parameter deciding the location of the new point is used from a predefined pool of values

based on its success history.


0.5 Original contribution by the thesis xxv

A randomly created population is classified in two groups, namely feasible and infeasible

ones. For every member from the infeasible group, one member from feasible group is

selected randomly. The BI approach can be applied using half moves as demonstrated in

the Fig. (0.2 a). Point R is the worst point selected from infeasible group and point S is the

corresponding point selected from feasible group. Point N1 is located moving R towards S in

the direction joining R and S, half the distance between point R and S. The point N1 is not

feasible, hence further half distance move from N1 is carried out, reaching to N2. That point

is also not feasible hence we move to point N3 moving half distance towards S, which is a

feasible point. We apply this procedure to all infeasible point and convert them to feasible

point at every generation of evolution.

Figure 0.2. Boundary Inspection Approach

We propose to use an predefined ensemble of parameter λ to locate the new point on the

line joining an infeasible point and the corresponding feasible point selected as shown in Fig.

(0.2 b). Each value in the ensemble is given equal opportunity during initial learning period.

The success count by each value in the learning period is converted to success probability,

which is used in the next learning period. During the learning period the success probability

is kept constant. Value of parameter λ to locate the new point is selected based on its suc-

cess probability. Thus the value of parameter λ generating feasible point will automatically

preferred over the value failing. This will avoid the parameter tuning during evolution and

problem specific tuning to the algorithm.


xxvi

For each infeasible member R, one member, S from feasible population is selected ran-

domly. A new point, N dividing the line joining point S and infeasible point, R in the λ : 1

ratio is obtained such that it is feasible. The division ratio is selected from a predefined pool

of λ values based of past performance history. An ensemble of possible values of ratio λ used

are [-0.6, -0.3, 0.3, 0.6, 1, 1.5, 2].


considering temperature dependent viscosity modelled using Carreau-Yasuda model. The

model solution algorithm is also proposed and tested to converge velocity and temperature

profiles within extruder channel. This validated model is used for optimization of screw design

parameters and temperature profile simultaneously to maximize throughput minimizing power

consumption. The temperatures of the material under process within the extruder and residence

time distribution of product are also tracked for assured quality of product. The screw helix

angle, channel depth, and screw speed are used as manipulated design parameters along

with barrel temperature profile. Best screw geometry, screw speed and barrel temperature

profile are obtained using the proposed multi-objective optimization algorithm. These multiple

optimum solutions assist the decision maker in selecting an appropriate design which is the

best according to his needs.

0.6 Methodology of Research, Results and discussions

The proposed unconstrained Parallel Universe Alien GA (PUALGA) algorithm and Boundary

Inspection (BI) approach for constraint handlingis are implemented in Matlab R2018a. we

have used ZDT (from Zitzler-Deb-Thiele’s study Zitzler et al. (2000)) test problems [ZDT 1,

ZDT2, ZDT3, ZDT4, and ZDT6] to test performance of the proposed PUALGA algorithm.

All the problems have two objective functions, which are to be minimized. Each test function

presents certain difficulties for multi-objective optimisation. For testing the efficiency and


0.6 Methodology of Research, Results and discussions xxvii

effectiveness of the proposed BI approach for constraint handling with EAs, we use three

two-objective constrained optimization test problems with known pareto optimal solutions.

The three test problems are namely, Constr-Ex , BNH (Binh and Korn 1997) , OSY (Osyczka

and Kundu 1995). All the problems have two objective functions, which are to be minimized.

Each test function presents certain difficulties for constrained multi-objective optimisation.

The detailed discussion of the problem and its solution are available in Deb (2001).

The general performance criteria for the multi-objective optimization algorithms are: (1)

Accuracy - how close the generated non-dominated solutions are to the best known predic-

tion. (2) Coverage - how many different non-dominated solutions are generated and how

well they are distributed. (3) Variance for every objective - which is the maximum range of

non-dominated front, covered by the generated solutions. Performance metrics are important

performance assessment measure, which also allow us to compare algorithms and to adjust

their parameters for better results. They are classified in three categories, metrics evaluating

closeness to the pareto optimal front (convergence), metrics evaluating distribution (diversity)

amongst non-dominated solutions and metrics evaluating convergence and diversity Deb

(2001). Two critical issues normally taken into consideration while evaluating performance of

multi objective optimization algorithms are: distance between obtained solutions and, spread

and uniformity among the obtained solutions. We use generational distance (GD) metric as

measure for convergence to true pareto front and the spread metric to represent the distribution

of solutions in the pareto front.

Generational distance is an average distance of the solutions to the true pareto front. For a

set Q of N solutions from a known set of the pareto optimal set P∗, the Generational Distance

(GD), γ is defined as follows,

γ =

(∑|Q|i=1 dp

i

)1/p

|Q|(0.1)


xxviii

where Q represents solution set having |Q| members. we use p=2 and di is minimum

distance between the member in solution set and nearest member is true pareto set, which is

defined as.

di = min

√M

∑m=1

( f (i)m − f ∗(k)m )2

(0.2)

where M represents number of objectives, i and k represent member index in solution set

and true pareto set respectively.

f ∗(k)m is the mth objective function value of the kth member of P∗ and f (i)m is the correspond-

ing objective function value from the true pareto front. When the objective function values are

of different order or magnitudes, they should be normalized by an appropriate weighing factor

in defining the distance, di.

The spread matrix is defined as follows,

∆ =∑

Mm=1 de

m +∑|Q|i=1 |di− d|

∑Mm=i de

m + |Q|d(0.3)

where, dem is the distances between the extreme solutions and the boundary solutions of

the obtained non-dominated solution set Q from the known end solutions of known solution

set P∗. The parameter di is the distance measured between the neighbouring solutions and d is

the mean value of this distance measure. Note that the maximum value of ∆ can be greater

than one. Though, a good distribution would make all distances di equal to d and would make

dem = 0. Thus, the most widely and uniformly spread of the non-dominated solutions result to

the zero value of ∆. For any other distribution, the value of the metric would be greater than

zero.

There are some metrics which evaluates closeness and diversity. They are Hypervolume,

attainable surface based statistical metric, weighted metric, non-dominated evaluation met-


0.6 Methodology of Research, Results and discussions xxix

ric, and Inverted Generational Distance (IGD). IGD is a well known and widely accepted

performance measure, which accounts convergence and distribution both. Let P∗ be a set of

uniformly distributed true pareto optimal solutions and A is the obtained solution set, then

IGD value is the average distance from P∗ to A. Note that the smaller the IGD value, better is

the performance of the MOO algorithm. We use IGD metric for performance comparison of

results obtained using different constrained MOO algorithms.

Genetic Algorithm Program developed in MATLAB is used in this work. It uses single

point crossover and binary mutation for binary population evolution. It uses tournament

selection, simulated binary crossover, SBX (with ηc = 20, crossover probability 0.90) and

non-uniform mutation (with b = 4, mutation probability 1/n) for real population evolution.

It uses non-dominated sorting along with elitism survival selection operators for both binary

and real coded GA. The PUALGA uses same binary and real coded GA operators along with

alien operator to exchange information between populations. All MOO programs use non-

dominated sorting, crowding distance calculation and binary tournament selection operators

as recommended in the NSGA-II Deb et al. (2002). The jumping gene GA uses randomly

created five bit chromosome with probability of 0.2 Guria et al. (2005). The decision vari-

ables, their upper and lower limits for all the problems are taken as used in Deb et al. (2002).

Population size is 100 for all test problems. Since techniques used are stochastic optimization

technique, it does not converge to the same solution every time even with the same initial

population. Hence, we carried out twenty simulation runs for every test problem with dif-

ferent initial population and average results are presented for the comparison of the algorithms.

Since the selected test problems have known true pareto fronts, it is possible to evaluate

convergence. Key result plots for critical test functions are only presented here. Convergence

metric γ for ZDT4 test functions are presented in Fig. (0.3). Three to ten times faster

convergence is observed for PUALGA for all the test functions. The statistical analysis in

Table 0.1 also conforms consistent better performance of PUALGA compared to other all


xxx

algorithms. The statistical analysis presented is at the end of 250 generations for ZDT1, ZDT2,

ZDT3 and 250 generations for ZDT4 and ZDT6 test functions.

0 50 100 150 200 250

0

2

4

6

8

10

12

14

16

18

20

Generation No

mean γ

rNSGA−II

bNSGA−II

aJGGA

PUALGA

Figure 0.3. Generation wise convergence metric γ (average of 20 runs) for ZDT4 test function

Table 0.1. Statastical analysis of convergence metric γ for 20 simulation runs

Problem binGA realGA jgGA PUALGA

ZDT1mean 0.0021 0.1148 0.0105 0.0010std 0.0005 0.0323 0.0033 0.0001

ZDT2mean 0.0016 0.2342 0.0136 0.0009std 0.0003 0.0683 0.0039 0.0001

ZDT3mean 0.0026 0.0525 0.0042 0.0025std 0.0002 0.0228 0.0006 0.0002

ZDT4mean 0.0025 4.8790 0.1295 0.0007std 0.0004 2.3012 0.1144 0.0001

ZDT6mean 0.0029 0.0027 0.0027 0.0068std 0.0002 0.0001 0.0001 0.0097

The diversity metric, ∆ represents spread of solutions. It is a measure of distribution

of solution along Pareto front. Zero value of the diversity metric indicates solutions are

uniformly distributed covering full range of true front; smaller the value, better the spread.

The generation wise progress of diversity metric, ∆ is presented in Fig. 0.4. The figure clearly

indicates that distribution is also observed to be the best for PUALGA compared to all other

algorithms.

The statistical analysis of distribution and coverage of pareto front are presented in Table


0.6 Methodology of Research, Results and discussions xxxi

0 50 100 150 200 250

0.6

0.8

1

1.2

1.4

1.6

Generation No

mean ∆

rNSGA−II

bNSGA−II

aJGGA

PUALGA

Figure 0.4. Distribution and coverage of pareto front as spread metric ∆ (average of 20 runs) for ZDT3 testfunction

0.2. The statistical analysis presented is at the end of 250 generations for ZDT1, ZDT2, ZDT3

and 250 generations for ZDT4 and ZDT6 test functions. The results indicate that PUALGA

performance is observed to be better in terms of convergence, distribution and coverage all

the three criteria. To represent the same information on pareto front an intermediate pareto

front for one of the run is presented for ZDT4 test function in Fig. 0.5. The figure clearly

shows that PUALGA population has already converged at 200 generations, where as other

algorithms are away from true pareto front.

Table 0.2. Statasical analysis of distribution and coverage as spread metric, ∆ for 20 simulation runs


ZDT1mean 0.3733 1.0443 0.5670 0.3827std 0.0329 0.1360 0.0807 0.0262

ZDT2mean 0.3599 1.3100 0.8082 0.3766std 0.0322 0.1246 0.1110 0.0277

ZDT3mean 0.5516 1.1660 0.7838 0.5514std 0.0231 0.1118 0.1053 0.0335

ZDT4mean 0.3845 0.7346 0.5856 0.3892std 0.0328 0.0467 0.3691 0.0380

ZDT6mean 0.3492 0.2878 0.2817 0.4836std 0.0318 0.0300 0.0330 0.2748

The PUALGA algorithm implemented in MATLAB using non-dominated sorting and elite

survival selection operator for MOO is used to evaluate three constraint handling approaches.


xxxii

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

f1

f 2

rNSGA−II

bNSGA−II

aJGGA

PUALGA

Figure 0.5. Pareto front for ZDT4 test function at the end of 200 generations

The population size is kept as 100 for all the test problems. We carried out twenty simulation

runs for every test problem with distinct initial populations and a statistical analysis is pre-

sented for the comparison study of various algorithms. Number of function evaluations (NFEs)

and number of constraint evaluations (NCEs) are the two important measures for evaluating

the computational expense of any constrained optimization algorithm. Performance metric

IGD values are presented as the functions of NFEs and NCEs for the augmented penalty,

ignore infeasible and boundary inspection to compare the computational performance. The

method proposed converts all infeasible solutions in to feasible solutions and consistently

showed better performance for all the three test functions.

Modelling and simulation is exhaustively explored to assist engineers in enhancing designs,

but complex processing requirements along with complex behaviours of polymers has limited

its application in polymer processing area Li and Hsieh (1996); Ghoreishy et al. (2000); Wood

and Rasid (2003); Vera-Sorroche et al. (2013); Chaturvedi et al. (2017). With recent advance-

ments in computational powers and modelling simulation tools, computations for polymer

processing has become feasible. Extrusion is an important polymer processing equipment for

rubber, plastic and food industries. A mathematical model for rubber extrusion is developed

using finite difference technique considering temperature dependent viscosity modelled using


0.6 Methodology of Research, Results and discussions xxxiii

Carreau-Yasuda model. The model solution algorithm is also proposed and tested to converge

velocity and temperature profiles within extruder channel. This validated model is used for

optimization of screw design parameters and temperature profile simultaneously to maximize

throughput minimizing power consumption. The temperatures of the material under process

within the extruder and residence time distribution of product are also tracked for assured

quality of product. The screw helix angle, channel depth, and screw speed are used as manipu-

lated design parameters along with barrel temperature profile. Best screw geometry, screw

speed and barrel temperature profile are obtained using multi-objective optimization algorithm.

These multiple optimum solutions assist the decision maker in selecting an appropriate design

which is the best according to his needs.

Rubber extrusion process consists of pushing compound by means of screw through feeding

channels and die. The channels are used to condition the rubber flow parameters (velocity,

temperature) and to distribute the flow rate of different blends in the case of co-extrusion. The

critical part of extruder is designing a screw, which lies at the heart of extruder. Optimization

of extrusion includes selection of the operating and geometrical variables that maximize mass

output maintaining quality with minimum energy demand. All these objectives are conflicting

with each other hence it is a good MOO problem to investigate. We review different modelling

approaches to develop an extruder model to optimize rubber extruder screw design Azhari

et al. (1998); Desai and Patel (2005); Ghoreishy et al. (2005); Ha et al. (2008); Trifkovic

et al. (2012); Rauwendaal (2014b). We formulate MOO problem considering, throughput

maximization, energy consumption minimization and residence time distribution as three

objectives. The design parameters considered are the screw helix angle φ , screw channel

height H, screw rotation speed N and barrel temperature profile T b. The three objective MOO

problem formulated is represented as follows:


xxxiv

max f1 = Q

min f2 = E

min f3 = IRTDdev

φ ,H,N,Tb

(0.4)

The fully developed velocity profile at entrance of metering section is shown in Fig. (0.6).

The velocity component in x and z directions along channel height H are shown at the figure.

The velocity in the x direction clearly reflects that the net flow in x direction is zero. The

distribution of velocity in the z direction shows the contribution to the net flow in z direction.

The total net flow at any location along z direction is always equal to Q.

0

40

2

100

Channel H

eig

ht (H

) m

m

20

4

u mm/sec w mm/sec

50

6

00

Velocity in x direction (u)

Velocity in z direction (w)

Figure 0.6. Fully developed Velocity profile

Temperature profile along extruder metering section length and channel height is plotted

as surface plot in Fig. (0.7). The temperature profile values are used to detect local heating.

Analytical solutions for RTD calculation considering non-Newtonian flow are not feasible.

We use of the tanks-in-series (TIS) model to analyse non-ideal flow in extruder. The TIS

model is a one parameter model used for reactor analysis and modelling. We analyse the RTD

to determine the number of ideal tanks, n, in series that will give approximately the same RTD.

We get n=1 for perfect mixing and very large value of n indicate ideal plug flow. Residence

time distribution function is plotted at Fig. (0.8). The distribution of residence time is one of


0.6 Methodology of Research, Results and discussions xxxv

0100

20

40

40

Tem

pera

ture

, deg C

30

60

Channel Lenght

50

Channel Height

80

2010

0 0

Figure 0.7. Temperature profile

the key parameter in evaluating performance of extruder screw design.

0 2 4 6 8 10 12 14

Residence time (ti) sec 105

0

1

2

3

4

5

6

7

8

Channel H

eig

ht (H

) m

m

0 2 4 6 8 10

Residence time (t) sec 105

0

0.5

1

1.5

2

E(t

)

10-5

Figure 0.8. Residence time distribution

The objective of screw design is to deliver the largest amount of output of acceptable

quality. The helix angle is the most important parameter affecting the performance of screw.

It affects throughput, power consumption, mixing and discharge pressure. Throughput can be

calculated by empirical equation (0.5) suggested by rauwendaal Rauwendaal (2014b) which

is used to calculate estimated u,w in iterative numerical solution procedure.

Q =

(4+n

10

)WHπDN cosφ −

(1

1+2n

)WH3

4η

(d pdz

)(0.5)

The throughput is calculated using velocity profile obtained at exit of the extruder channel


xxxvi

using Eq. (0.6) for MOO solution in this work.

f1 = Q = f (φ ,H,N,Tb) =W∫ H

0wdy (0.6)

The energy consumed by the extruder screw is subtotal of energy consumed for viscous

heating, for increase in pressure and kinetic energy Zuilichem et al. (2011). Kinetic energy is

very very small compared to the other energy components, hence the total energy consumed E

is considered as sum total of energy consume for viscous energy dissipation in screw channel

Evsc, in Screw tip Evst , and increasing pressure E p. The multi-objective optimization problem

formulated is solved using PUALGA algorithm to get pareto solutions for maximization of

throughput minimizing energy demand. The resultant pareto front obtained at the end of 300

generation for a population size of 100 is plotted at Fig. 0.9. The conflicting nature of two

objectives, throughput and power requirement for single screw extruder are clearly reflected

in the pareto plot. Point C is the Eutopia point for throughput-power Pareto front. Eutopia

point is the best point, which is at minimum distance from reference point R. The helix angle

corresponding to Eutopia point is 35deg. A screw was tested with help of Pioneer Rubber

Industries for this configuration. The experimental results were very close to the simulation

results as shown at Fig. 0.9, marked as point E.

36 38 40 42 44 46

Throughput (m3/h)

7.5

8

8.5

9

9.5

10

Pow

er

(kW

)

10-4

A

B

R

CE

Figure 0.9. Throughput-Power pareto front for single screw extruder


0.7 Achievements with respect to objectives xxxvii

The effect of helix angle on throughput and power requirement is presented in Fig. 0.10.

As helix angle increases from 25 degree, initially throughput increases at a fast rate, but near

45 degree the influence of helix angle on throughput reduces. The reverse phenomena is

observed in case of power requirements. This two plots clearly show the conflicting nature

of throughput and power requirements and presents influence of helix angle as manipulated

variable.

25 30 35 40 45

Helix Angle (degree)

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

Pow

er

(kW

)

10-3

25 30 35 40 45


34

36

38

40

42

44

46

Thro

ughput (m

3/h

)

Figure 0.10. Influence of Helix Angle on Throughput and Power for single screw extruder

0.7 Achievements with respect to objectives

Objective AchievementGenerating GA for multi-objective optimiza-tion problem, which has high probability ofproviding global optimum solution at the sametime in less computational time.

Developed PUALGA algorithm DevelopedBoundary Inspection approach for constrainthandling in EAs.

The above developed algorithm will be testedusing benchmark multi-objective optimizationproblems.

Validated the proposed concept using bench-mark unconstrained and constrained test func-tions for MOO.

The algorithm will be tested on Extruder de-sign optimization for maximization of through-put and minimization of power consumptionand betterment of selected properties, all con-flicting objectives.

The Rubber Extruder Screw design MOO prob-lem formulated and solved using new devel-oped algorithm.


xxxviii

0.8 Synopsis Conclusion

Hybridization of binary and real coded GA is explored to enhanced convergence rate. The

focus of the hybridization is to combine the strengths of both algorithms. Binary encoding

has flexibility of adjusting accuracy of decision variables by adjusting binary chromosome

size. The mechanism of binary encoding gives better exploration of search space using small

chromosome size. Use of small chromosome size supports very good initial convergence

but can not converge to true solutions at later stage of evolution. Real coded GA takes up

that responsibility of convergence at that stage. The algorithm uses the concept of parallel

population and combined binary and real GA. Non-dominated sorting is used in all algorithms

for survival selection. The advantage of using two sub populations reduces the complexity

of sorting and achieves better results with same computational efforts (Number of function

Evaluations). Though, this concept can be applied for any population based MOEAs, it has

been used to obtain the results under GA framework in this study. The proposed PUALGA

has been compared with its native binary and real coded GAs and Jumping Gene Adaptation

of GA. The proposed PUALGA algorithm drastically enhances the initial convergence rate

for all bench mark MOO test problems taking the benefit of exploration capacity of binary

encoding.

Constraint handling is always a critical part in performance of optimization method. Multi-

objective constrained optimization problems are typically very difficult to solve. A new

generalized Boundary Inspection (BI) approach based constraint handling mechanism for pop-

ulation based evolutionary algorithms(EAs)has been proposed . The concept is general and can

be used with any population based EAs, we demonstrate its implemented for Multi-Objective

Optimization (MOO) in this work. In the proposed algorithm, every infeasible member is

projected through the randomly selected feasible member. The selection of parameter which

locates the new point on the line joining infeasible and feasible point is based on success

probability history, hence it is automated avoiding adaptive tuning during the evolution process.

The efficacy of the BI approach is presented using multi-objective PUALGA algorithm and


0.8 Synopsis Conclusion xxxix

has been tested with three bench mark test functions and the performance is compared with

two popular constraint handling algorithms, namely augmented penalty function and ignore

infeasible.


considering temperature dependent viscosity modelled using Carreau-Yasuda model. This

model is used for optimization of screw design parameters and temperature profile simulta-

neously to maximize throughput minimizing power consumption. The temperatures of the

material under process within the extruder and residence time distribution of product are also

tracked for assured quality of product. The screw helix angle, channel depth, and screw speed

are used as manipulated design parameters along with barrel temperature profile. Best screw

geometry, screw speed and barrel temperature profile are obtained using proposed PUALGA

algorithm for multi-objective optimization.


Dedicated to

Pradip Mukherjee

My Life Mentor

and

Devindra Desai and Nitin Desai

My Beloved Mother Father

xliii

Abstract

Evolutionary computation is becoming the most proven method for global optimization of

complex problems. Amongst them, Genetic Algorithm (GA) has become more popular, being

robust, flexible and relatively efficient. GAs can handle single and multi-objective optimiza-

tion problems. Evolutionary optimization algorithms are computationally more expensive

compared to traditional optimization methods, but their flexibility and robustness attribute to

their importance. The global optimization of complex problem can be covered successfully

in most cases. However, it does not give guarantee being stochastic in nature. Exploration

capabilities are excellent for GA, but Lack of convergence appears to be a drawback. Par-

ticularly convergence becomes much slower near the optimal solution. Hybridization is one

of the approach used to overcome this convergence problem. Hybridization of binary coded

and real coded GA is explored in this work. Exploration capabilities of binary encoding

are exploited to enhance convergence. Alien transport information between binary and real

encoded population. The concept is presented as Parallel Universe Alien Genetic Algorithm

(PUALGA).

Moreover, the GAs are naturally designed for unconstrained problems, and hence, require

additional mechanism for constraint handling. Boundary Inspection (BI) approach is presented

for constraint handling under PUALGA framework. It converts all infeasible members at every

generation of evolution to feasible members. Infeasible member is moved using randomly

selected feasible member to cross the boundary separating feasible and infeasible region.

Even with the developments in the computational powers of computers, solving the complex

multi-objective problems requires very long time. There is always a need for development

of robust and computationally efficient algorithms for large and complex problems. This

present research work focuses on upgrading the GA to enhance the convergence and constraint

handling capabilities for multi-objective optimization. The proposed approaches are tested by

benchmark test functions and further validated using rubber extruder screw design application.

A model for rubber extruder is developed using finite element analysis. The extruder model

considers temperature dependent viscosity using Carreau-Yasuda model. The FEA model and

xliv

solution algorithm developed is used for extruder parameter analysis. The model solutions

are validated with analytical and empirical model results. Multi-objective optimization is

carried out for maximization of throughput and minimization of power consumption. Helix

angle, channel height and, rotational speed of extruder screw are considered as manipulated

design variables along with temperature profile across the barrel. The temperatures of the

material under process within the extruder and residence time distribution of product are

also tracked for the assured quality of product. Best screw geometry, screw speed and barrel

temperature profile are obtained using multi-objective optimization algorithm : PUALGA

with BI approach. These multiple optimum solutions assist the decision maker in selecting an

appropriate design which is the best according to the needs.


xlv

Acknowledgements

I am deeply thankful to my supervisor Dr. S. A. Puranik for his unconditional support dur-

ing each and every stage across the tenure of this research work. The research would not

have come to this shape without his deep involvement. I thank him for his critical obser-

vations about my capabilities and limitations, which brought a significant change in my

development during this research. I am very much thankful to my DPC members, Dr. R.

Sengupta and Dr. A. P. Vyas for their guidance and focused reviews across the tenure of the

research, which helped me understand my work in great depths. I especially thank Dr. N. M

Patel, for his inputs at every stage of the research, which brought significant clarity in my work.

I am very much thankful to my collogues Prof. B H Shah and Prof. S. R. Shah, who stood

by me wherever I needed their support. I am extremely thankful to my life mentor Pradip

Mukerjee and friends Narendra Patel, Binita Vyas and Falguni Pathak for their motivation

and unconditional support to overcome frustrating stages during the tenure of this research. I

am deeply thankful to my mother Devindraben, father Nitinbhai, brother Jay, sister-in-law

Jigna and nephews Jagravi, Shardul and Rucha for their support to manage life along with

this research.

I am very much thankful to Dr. G. P. Vadodaria, principal of my parent institute and all my

collogues Prof. S. J. Padhiyar, Prof. B. D. Patel, Prof. G. G. Bhatt, Prof. P. N. Chavda, Prof.

R. Y. Modan, Prof. A. D. Bhatt, Prof. H. C.Shah and Prof. N. D. Solanki for their motivation

and support at all the crucial stages of this work. I express my sincere thanks to Dr. Sachin

Parikh, Prof. C. G. Bhagchandani, Dr. D. D. Mandaliya, Prof. Sahil Prajapati and Prof. R. P.

Bhatt for their support. I will always remain indebted to the wonderful group of friends who

always stood by me whenever I needed them.

Desai Rupande Nitinbhai

xlvii

Contents

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiCertificate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vCourse-work Completion Certificate . . . . . . . . . . . . . . . . . . . . . . . . . viiOriginality Report Certificate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixPhD THESIS Non-Exclusive License . . . . . . . . . . . . . . . . . . . . . . . . . xvThesis Approval Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Synopsis xix0.1 Synopsis Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix0.2 Brief description on the state of the art of the research topic . . . . . . . . . . xx0.3 Definition of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi0.4 Objective and Scope of work . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii0.5 Original contribution by the thesis . . . . . . . . . . . . . . . . . . . . . . . xxiii0.6 Methodology of Research, Results and discussions . . . . . . . . . . . . . . xxvi0.7 Achievements with respect to objectives . . . . . . . . . . . . . . . . . . . . xxxvii0.8 Synopsis Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii

Abstract xliii

Acknowledgements xlv

Contents xlix

List of Abbreviations li

List of Figures lvii

List of Tables lx

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Key Issues in Multi-Objective Search . . . . . . . . . . . . . . . . . . . . . 51.3 Major Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Parallel Universe Alien Genetic Algorithm (PUALGA) . . . . . . . . 61.3.2 Boundary Inspection Approach for Constrained handling . . . . . . . 71.3.3 Multi-Objective Optimization Problem Formulations for Rubber Ex-

truder Screw Design . . . . . . . . . . . . . . . . . . . . . . . . . . 8

xlviii Contents

1.4 Objectives of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Literature Review 112.1 Multi-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Classical Methods for MOO . . . . . . . . . . . . . . . . . . . . . . 172.2 Evolutionary Multi-Objective Optimization Algorithms . . . . . . . . . . . . 202.3 Evolutionary Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Hybrid Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . 262.3.2 Constraint Handling in Evolutionary Algorithms . . . . . . . . . . . 29

2.4 Rubber Extruder Design Optimization . . . . . . . . . . . . . . . . . . . . . 332.4.1 Modelling of Rubber Extruder . . . . . . . . . . . . . . . . . . . . . 362.4.2 Throughput Power relations for Rubber Extruder . . . . . . . . . . . 42

2.5 Rheology of Rubber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5.1 Effect of Temperature on Rheology of Rubber . . . . . . . . . . . . . 522.5.2 Effect of Pressure on Rheology of Rubber . . . . . . . . . . . . . . . 53

2.6 Summery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Op-timization 573.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2 Binary and Real coded GA . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3 Non-dominated Sorting GA . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4 Jumping Gene GA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5 Proposed Parallel Universe Alien GA . . . . . . . . . . . . . . . . . . . . . 66

3.5.1 Proposed PUALGA algorithm . . . . . . . . . . . . . . . . . . . . . 683.6 MOO test problems and Performance measures . . . . . . . . . . . . . . . . 693.7 Sensitivity Analysis of Proposed Algorithm . . . . . . . . . . . . . . . . . . 723.8 Performance Evaluation Results and Discussion . . . . . . . . . . . . . . . . 94

3.8.1 Convergence to true pareto front . . . . . . . . . . . . . . . . . . . . 953.8.2 Distribution of solutions within pareto front . . . . . . . . . . . . . 98

3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4 Boundary Inspection Approach for Constrained handling in Evolutionary Opti-mization Algorithms 1054.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.2 Boundary Inspection Approach for Constraint Handling . . . . . . . . . . . . 107

4.2.1 Ensemble of the projection parameter λ in BI approach . . . . . . . . 1104.3 Parallel Universe Alien Genetic Algorithm (PUALGA) with BI Approach . . 1124.4 Sensitivity analysis of propose BI approach . . . . . . . . . . . . . . . . . . 114

4.4.1 Feasible circular area inside square . . . . . . . . . . . . . . . . . . 1154.4.2 Feasible circular area outside the circle within a square . . . . . . . . 1194.4.3 Feasible square area inside a square . . . . . . . . . . . . . . . . . . 1214.4.4 Feasible area outside the small square within a square . . . . . . . . . 123

4.5 Performance Measure for MOO . . . . . . . . . . . . . . . . . . . . . . . . 1264.5.1 Convergence to true pareto front . . . . . . . . . . . . . . . . . . . . 127


Contents xlix

4.5.2 Matrix to measure distribution of solutions . . . . . . . . . . . . . . 1284.5.3 Matrix evaluating closeness and diversity . . . . . . . . . . . . . . . 129

4.6 Test Problems and Engineering Design Applications . . . . . . . . . . . . . 1294.6.1 Test problem-1: Constr-Ex . . . . . . . . . . . . . . . . . . . . . . . 1304.6.2 Test problem-2: BNH . . . . . . . . . . . . . . . . . . . . . . . . . 1304.6.3 Test problem -3: OSY . . . . . . . . . . . . . . . . . . . . . . . . . 1314.6.4 Engineering application-1: Design of welded beam . . . . . . . . . . 1324.6.5 Engineering application-2: Design of disk brake . . . . . . . . . . . 133

4.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.7.1 Test problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.7.2 Design applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5 Rubber Extruder Modelling and Simulation 1475.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.2 Mathematical modelling of single screw extruder . . . . . . . . . . . . . . . 1505.3 FEA Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.3.1 Finite Difference implementation for FEA model . . . . . . . . . . . 1565.3.2 Numerical Solution Algorithm . . . . . . . . . . . . . . . . . . . . 158

5.4 Sensitivity of parameters influencing Extruder Throughput . . . . . . . . . . 1595.5 Simulation using the FEA model . . . . . . . . . . . . . . . . . . . . . . . . 1665.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6 Multi-Objective Optimization: Application to Rubber Extruder Screw Design 1716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.2 Multi-Objective optimization of extruder screw design . . . . . . . . . . . . 174

6.2.1 Throughput Maximization . . . . . . . . . . . . . . . . . . . . . . . 1766.2.2 Energy Consumption Minimization . . . . . . . . . . . . . . . . . . 177

6.3 Residence Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7 Conclusions and Scope of Future Work 1917.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917.2 Scope of Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Bibliography 197

A Non-dominated sorting Genetic Algorithm (NSGA)-II 217

B NSGA-II-JG 221

C List of Publications 225


li

List of Abbreviations

AC Attempt CountBI Boundary InspectionCSTR Continuous Sterred Tank ReactorDE Differential EvolutionDM Decision MakerEAs Evolutionery AlgorithmsFEA Finite Element AnalysisGA Genetic AlgorithmGD Generational DistanceHDPE High Density Poly EthyleneIGD Inverted Generational DistanceJG Jumping GeneLP Learning PeriodMAs Memetic AlgorithmsMODE Multi Objective Differential EvolutionMOEAD Multi-Objective Evolutionary Algorithm based on DecompositionMOEAs Multi-Objective Evolutionary AlgorithmsMOEP Multi Objective Evolutionary ProgrammingMOGA Multi-Objective Genetic AlgorithmMOO Multi-Objective OptimzationNCEs Number of Constraint EvaluationsNFEs Number of Function EvaluationsNPGA Niched Pareto Genetic AlgorithmNR Natural RubberNSGA Non-Dominates Sorting Genetic AlgorithmPFR Plug Flow ReactrorPSO Particle Swam OptimizationPUALGA Parallel Universe Alien Genetic AlgorithmRDGA Rank Density based Genetic AlgorithmRTD Residence Time DistributionSC Success CountSOO Single Objective OptimzationSR Stochastic RankingTIS Tank In SeriesVEGA Vector Evaluated Genetic Algorithm

liii

List of Figures

0.1 Parallel Universe Alien GA Evolution Scheme . . . . . . . . . . . . . . . . . xxiv0.2 Boundary Inspection Approach . . . . . . . . . . . . . . . . . . . . . . . . . xxv0.3 Generation wise convergence metric γ (average of 20 runs) for ZDT4 test

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx0.4 Distribution and coverage of pareto front as spread metric ∆ (average of 20

runs) for ZDT3 test function . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi0.5 Pareto front for ZDT4 test function at the end of 200 generations . . . . . . . xxxii0.6 Fully developed Velocity profile . . . . . . . . . . . . . . . . . . . . . . . . xxxiv0.7 Temperature profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv0.8 Residence time distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv0.9 Throughput-Power pareto front for single screw extruder . . . . . . . . . . . xxxvi0.10 Influence of Helix Angle on Throughput and Power for single screw extruder xxxvii

2.1 Pareto front for maximum throughput and minimum power for Extruder . . . 142.2 Classification of Multi-objective Optimization Methods (source: Rangaiah

(2017) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Graphical interpretation of the weighting method (left) and the constraint

method (right). (Source: Zitzler (1999)) . . . . . . . . . . . . . . . . . . . . 192.4 General Scheme of an Evolutionary Algorithm . . . . . . . . . . . . . . . . 25

3.1 Ranking of population using non-dominated sorting . . . . . . . . . . . . . . 643.2 Crowding distance calculation for population member in a pareto front (Source:

Deb (2001)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3 Schematics of replacement and reversion JG adaptations for GA (Source:

Kasat and Gupta (2003)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4 Parallel Universe Alien GA Evolution Scheme . . . . . . . . . . . . . . . . . 693.5 Generational Distance matrix for an obtained MOO solution set (Source: Deb

et al. (2002)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.6 Spread matrix for an obtained MOO solution set (Source: Deb et al. (2002)) . 733.7 Effect of binary fraction on generation wise performance (Generational Dis-

tance metric) for SCH1 test function . . . . . . . . . . . . . . . . . . . . . . 743.8 Effect of binary fraction on generation wise performance (Spread metric) for

SCH1 test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.9 Sensitivity of binary fraction on performance metric at 50th Generation for

SCH1 test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

liv List of Figures

3.10 Effect of number of alien transfer on generation wise performance (Genera-tional Distance metric) for SCH1 test function . . . . . . . . . . . . . . . . . 77

3.11 Effect of number of alien transfer on generation wise performance (Spreadmetric) for SCH1 test function . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.12 Sensitivity for number of alien transfer on performance metric at 50th Genera-tion for SCH1 test function . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.13 Effect of binary fraction on generation wise performance (Generational Dis-tance metric) for FON test function . . . . . . . . . . . . . . . . . . . . . . . 80

3.14 Effect of binary fraction on generation wise performance (Spread metric) forFON test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.15 Sensitivity of binary fraction on performance metric at 70th Generation forFON test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.16 Effect of number of alien transfer on generation wise performance (Genera-tional Distance metric) for FON test function . . . . . . . . . . . . . . . . . 82

3.17 Effect of number of alien transfer on generation wise performance (Spreadmetric) for FON test function . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.18 Sensitivity for number of alien transfer on performance metric at 70th Genera-tion for FON test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.19 Effect of binary fraction on generation wise performance (Generational Dis-tance metric) for POL test function . . . . . . . . . . . . . . . . . . . . . . . 85

3.20 Effect of binary fraction on generation wise performance (Spread metric) forPOL test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.21 Sensitivity of binary fraction on performance metric at 100th Generation forPOL test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.22 Effect of number of alien transfer on generation wise performance (Genera-tional Distance metric) for POL test function . . . . . . . . . . . . . . . . . . 87

3.23 Effect of number of alien transfer on generation wise performance (Spreadmetric) for POL test function . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.24 Sensitivity for number of alien transfer on performance metric at 100th Gen-eration for POL test function . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.25 Effect of binary fraction on generation wise performance (Generational Dis-tance metric) for KUR test function . . . . . . . . . . . . . . . . . . . . . . 90

3.26 Effect of binary fraction on generation wise performance (Spread metric) forKUR test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.27 Sensitivity of binary fraction on performance metric at 250th Generation forKUR test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.28 Effect of number of alien transfer on generation wise performance (Genera-tional Distance metric) for KUR test function . . . . . . . . . . . . . . . . . 92

3.29 Effect of number of alien transfer on generation wise performance (Spreadmetric) for KUR test function . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.30 Sensitivity for number of alien transfer on performance metric at 250th Gen-eration for KUR test function . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.31 Generation wise convergence metric γ (average of 20 runs) for ZDT1 testfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


List of Figures lv





3.36 Distribution and coverage of pareto front as spread metric ∆ (average of 20runs) for ZDT1 test function . . . . . . . . . . . . . . . . . . . . . . . . . . 99



3.39 Pareto front for ZDT4 test function at the end of 200 generations . . . . . . . 1013.40 Pareto front for ZDT6 test function at the end of 200 generations . . . . . . . 102

4.1 Boundary Inspection Approach for Constraint Handling . . . . . . . . . . . . 1084.2 Boundary Inspection Approach for Constraint Handling . . . . . . . . . . . . 1094.3 Different test cases of feasible regions for study in two dimensional space,

feasible area inside or our side of circle or square . . . . . . . . . . . . . . . 1144.4 Effect of % feasible are on BI treatment count and NCEs for FIcircle case . . 1174.5 Generation wise infeasible members requiring BI treatment FIcircle case . . . 1174.6 Generation wise NCEs required in BI treatment for FIcircle case . . . . . . . 1184.7 Effect of feasible area on adoptive learning probability distribution for FIcircle

case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.8 Effect of % feasible are on BI treatment count and NCEs for FOcircle case . . 1204.9 Generation wise infeasible members requiring BI treatment and NCEs re-

quired for FOcircle case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.10 Effect of feasible area on adoptive learning probability distribution of selecting

a division ration value from an Ensemble pool . . . . . . . . . . . . . . . . . 1214.11 Effect of % feasible are on BI treatment count and NCEs for FIsquare case . . 1224.12 Generation wise infeasible members requiring BI treatment and NCEs re-

quired for FIsquare case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.13 Effect of feasible area on adoptive learning probability distribution of selecting

a division ration value from an Ensemble pool . . . . . . . . . . . . . . . . . 1244.14 Effect of % feasible are on BI treatment count and NCEs . . . . . . . . . . . 1244.15 Generation wise infeasible members requiring BI treatment and NCEs re-

quired for FOsquare case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.16 Effect of feasible area on adoptive learning probability distribution of selecting

a division ration value from an Ensemble pool . . . . . . . . . . . . . . . . . 1264.17 Average IGD values against Run time for ConstrEx test function . . . . . . . 1354.18 Average IGD values against NFEs for ConstrEx test function . . . . . . . . . 1364.19 Average IGD values against NCEs for ConstrEx test function . . . . . . . . . 1374.20 Pareto Front for ConstrEx test function . . . . . . . . . . . . . . . . . . . . . 137


lvi List of Figures

4.21 Average IGD values against Run time for BNH test function . . . . . . . . . 1384.22 Average IGD values against NFEs for BNH test function . . . . . . . . . . . 1384.23 Average IGD values against NCEs for BNH test function . . . . . . . . . . . 1394.24 Pareto Front for BNH test function at 50 Generations . . . . . . . . . . . . . 1394.25 Average IGD values against Run time for OSY test function . . . . . . . . . 1404.26 Average IGD values against NFEs for OSY test function . . . . . . . . . . . 1404.27 Average IGD values against NCEs for OSY test function . . . . . . . . . . . 1414.28 Pareto Front for OSY test function . . . . . . . . . . . . . . . . . . . . . . . 1414.29 Average IGD values against Run time for Welded Beam design application . 1424.30 IGD convergence profiles for Disk Welded Beam design application . . . . . 1434.31 Pareto Front for Welded Beam design design application . . . . . . . . . . . 1434.32 Average IGD values against Run time for Disk Break design application . . . 1444.33 IGD convergence profiles for Disk Break design application . . . . . . . . . 1454.34 Pareto Front for Disk Break design application . . . . . . . . . . . . . . . . 145

5.1 Rubber Extruder schematic diagram . . . . . . . . . . . . . . . . . . . . . . 1515.2 Rubber Extruder screw and barrel . . . . . . . . . . . . . . . . . . . . . . . 1515.3 Rubber Extruder screw channel . . . . . . . . . . . . . . . . . . . . . . . . . 1525.4 Computational Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.5 Effect of helix angle and channel height on throughput . . . . . . . . . . . . 1615.6 Effect of helix angle on throughput for different channel height . . . . . . . . 1635.7 Effect of channel height on throughput for different helix angle . . . . . . . . 1635.8 Effect of channel height on throughput for different viscosity index . . . . . . 1645.9 Effect of helix angle on throughput for different viscosity index . . . . . . . . 1655.10 Effect of polymer viscosity on throughput for different channel height and

helix angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.11 Velocity Profile at exit along x and z direction . . . . . . . . . . . . . . . . . 1675.12 Three dimensional view of u and w velocity profile at screw channel exit . . . 1675.13 Pressure profile along extruder screw channel length . . . . . . . . . . . . . 1685.14 Pressure gradient profile along extruder screw channel length . . . . . . . . . 1685.15 Three dimensional view of Temperature profile along screw channel height

and length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.1 Throughput-Power pareto front for single screw extruder . . . . . . . . . . . 1826.2 Influence of Helix Angle on optimum Throughput and Power consumption

for single screw extruder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.3 Influence of Channel Height on optimum Throughput and Power consumption

for single screw extruder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.4 Velocity profile in x direction . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.5 Velocity profile in z direction . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.6 Temperature profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.7 Time distribution for RTD calculation . . . . . . . . . . . . . . . . . . . . . 1866.8 The E(t) curve presentation of RTD . . . . . . . . . . . . . . . . . . . . . . 1876.9 The F(t) curve presentation of RTD . . . . . . . . . . . . . . . . . . . . . . . 188

A.1 Non-dominated sorting algorithm (NSGA) pseudo code . . . . . . . . . . . . 217


List of Figures lvii

A.2 Non-dominated sorting Genetic Algorithm-I (NSGA-I) . . . . . . . . . . . . 218A.3 Non-dominated sorting Genetic Algorithm-II (NSGA-II) . . . . . . . . . . . 218A.4 Illustrative example of Pareto optimality in objective space (left) and the

possible relations of solutions in objective space (right). . . . . . . . . . . . . 219

B.1 Flowchart of NSGA-II-JG . . . . . . . . . . . . . . . . . . . . . . . . . . . 224


lix

List of Tables

0.1 Statastical analysis of convergence metric γ for 20 simulation runs . . . . . . xxx0.2 Statasical analysis of distribution and coverage as spread metric, ∆ for 20

simulation runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi

2.1 Main Features, Merits and Limitations of MOO Methods (Source: Rangaiah(2017)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Literature search count on various scientific databases with the key words"hybrid Evolutionary Algorithms" . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Details of MOO test functions (Source: Deb (2001)) . . . . . . . . . . . . . 703.2 Sensitivity analysis for binary fraction on performance metrics for SCH1 test

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3 Sensitivity analysis for number of alien tranfer on performance metrics for

SCH1 test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.4 Sensitivity analysis for binary fraction on performance metrics for FON test

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.5 Sensitivity analysis for number of alien transfer on performance metrics for

FON test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.6 Sensitivity analysis for binary fraction on performance metrics for POL test


POL test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.8 Sensitivity analysis for binary fraction on performance metrics for KUR test


KUR test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.10 Statastical analysis of convergence metric γ for 20 simulation runs . . . . . . 983.11 Statasical analysis of distribution and coverage as spread metric, ∆ for 20

simulation runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.1 Details of test cases of feasible regions for study in two dimensional space,feasible area inside and our side of circle or square . . . . . . . . . . . . . . 116

4.2 Pareto optimal solutions for the OSY problem . . . . . . . . . . . . . . . . . 132

5.1 Parameteres of single screw extruded used in simulation . . . . . . . . . . . 1625.2 Matrial Properties used for simulation of single screw extruded . . . . . . . . 166

lx List of Tables

6.1 Parameteres and matreial properties of single screw extruded used in optimiz-ing screw design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181


1

Chapter 1

Introduction

Optimization is the process of finding the best. The best is decided by the criteria of selecting

the best, which in the terminology of optimization is known as objective function. The

best value of the objective functions is to be obtained satisfying the conditions (constraints)

imposed. Optimization problem can be stated in a generic form as follows:

Minimize/Maximize f (x) = { f1(x), f2(x), .. fm(x)}

Subject to g j(x)≤ 0, j = 1,2,3....J;

hk(x) = 0, k = 1,2,3....K;

li ≤ xi ≤ ui, i = 1,2,3....n.

(1.1)

Here, f1 to fm represents m objective functions. Thus, for a single and bi-objective op-

timization problem, the value of m becomes one and two, respectively. The n dimensional

decision vector x have upper and lower limits, li, and ui respectively. The functions g and h

denote the J number of inequality and K number of equality constraints, respectively.

New developments, operating practices, safety, economics, and competitive global mar-

ket scenario motivates increasing use of the optimization tool. Computational powers with

improving technology supports the development in the area of optimization. Evolutionary

computation is becoming the most proven method for Global optimization of complex prob-

lems. Amongst them, Genetic Algorithm (GA) has become more popular being robust, flexible

and relatively efficient. Further, GA has become also more attractive because of the following

2 1. Introduction

reasons:- (1) The local information as such derivatives is not required, (2) GA is population

based and had multiple starting points rather than only one, (3) Parallel search at multiple

locations is carried out within the search space, (4) Stochastic search mechanism is utilised

and, (5) Discontinuous and noisy functions (Coello et al., 2006; Deb, 2001; Goldberg, 1989;

Holland, 1975) can be easily accommodated. Thus, more challenging and computationally

expensive optimization problems can be solved effectively by GA because of above mentioned

noteworthy features.

The optimization problems with multiple objectives can be sub-classified in two groups,

with conflicting objectives and non-conflicting objectives. When objectives are non-conflicting

with each other, they can be transformed to single objective and, the resulting problem is

conventional single objective optimization problem. When the multiple objectives are con-

flicting with each other, then they can be transformed to single objective without loosing

any information. The type of optimization problem with multiple conflicting objectives is

multi-objective optimization, which will have multiple solutions known as pareto optimal

solution. Most real-world problems have multiple conflicting objectives and complex search

space. Conflicting objectives give rise to a set of compromise solutions, instead of a single

optimal solution denoted as pareto-optimal set. None of the pareto-optimal solution can

be said to be better than the others. The search space for multi-objectiveness optimization

problems are generally too large and complex, hence, efficient optimization strategies are

required for solving these problems.

The population based GA can converge to the multiple solutions simultaneously in a

single run, which is a promising feature for solving multi-objective optimization problems.

Moreover, they are very simple and relatively efficient. Because of these features, the GA

have become popular for optimization solution of complex problems. However, the GA is

designed unconstrained by nature, and hence, require additional mechanism for the constraint

handling. Some times the simple GA may fail to even obtain a satisfactory solution for few

problems. This problem can be partially addressed by hybridization of the GA with other

optimization methods.


1.1 Motivation 3

1.1 Motivation

Evolutionary algorithms (EAs) have been widely accepted for solving several practical op-

timization applications in engineering. However, they are often criticized for the large

computational time, as well as, for their inefficiency to handle the constraints. To overcome

these problems, hybridization of EAs with the other optimization algorithms can be explored.

Since, there are several local search optimization algorithms which have good local conver-

gence and constraint handling capacity, the feature of hybridization adds significant value

to the EAs (Krasnogor and Smith, 2005). Several reported literature can be found for the

successful applications of such hybrid approach for solving complex optimization problems

(Nabil, 2016; Mohamed, 2015).

Sankararao and Gupta (2007) observed that for ZDT4 test function, binary coded NSGA-II

do not converge to true pareto front. They noted that "It may be mentioned here that though

the binary coded NSGA-II fails to converge to the global optimal solution, for this test problem,

the real coded NSGA-II does indeed, converge to the correct pareto solutions in 100,000

function evaluations". This observation initiated the thought of hybridizing real coded and

binary coded genetic algorithms to add robustness for solving MOO problems. It may be

noted that having two parallel populations evolving simultaneously, one binary coded and

another real coded can contribute to effiecient evolutionary algorithm. Zhou et al. (2011)

highlights that competitive/co-operative co-evolution is one approach to improve convergence

and robustness of MOO. This supported our hypothesis of using two parallel populations.

Patel and Padhiyar (2010) explored the concept of Alien in their work, which is propose to be

used here to exchange information between the parallel populations.

The second difficulty in using EAs for MOO is handling constraints. Bounds on decision

variables are mandatory requirement of all the EAs and they are part of their original design

nature. EAs find difficulties in handling decision variables without limits and expects the user

to specify it. Even the first step of EA, initiation of population can not be performed without

specifying the limits of decision variables. Engineering application generally have bounds on

decision variable naturally coming from problem domain, hence, constraints of bounds on


4 1. Introduction

decision variables are not crucial for EAs. EAs can also handle discrete and discontinuous

variables easily. The need is felt for additional mechanisms or modification in algorithm

for constraints other than bounds on decision variable. Equality or inequality constraints

should be considered. There are different constraint handling mechanisms available for MOO

using EAs (Mezura-Montes and Coello Coello, 2011; Deb, 2000; Sorkhabi et al., 2018; Jiang

et al., 2018; Patil et al., 2019; Strauch et al., 2019). Box-complex method is one of the oldest

and effective method for constraint optimization (Box, 1965). Patel and Padhiyar (2015)

modified GA using Box-complex method and applied it for optimal control problems. They

hybridized GA and Box-complex method to improve convergence. Zade et al. (2017) devel-

oped constrained handling by hybridizing cuckoo search algorithm with Box-complex method.

The use of Boundary Inspection approach to add constraint handling feature to the Par-

allel Universe Alien GA has been proposed. The approach developed is partly based on the

Box-complex method. The proposed algorithm uses one feasible point in place of centroid

of the feasible points in conventional method. The methods proposed is further enhanced

using automated selection of projection parameter. The two proposed modifications in GA for

MOO: (i) Parallel Universe Alien GA and (ii) Boundary Inspection approach for Constraint

handling will make it more robust and applicable to practical applications.


Accuracy - how close the generated non-dominated solutions are to the best known prediction.

(2) Coverage - how many different non-dominated solutions are generated and how well they

are distributed. (3) Variance for every objective - the maximum range of non-dominated

front, covered by the generated solutions. Performance metrics are important performance

assessment measure, which also allows to compare algorithms and to adjust their parameters

for better results. The classification into three categories is as under:- metrics evaluating

closeness to the pareto optimal front (convergence), metrics evaluating distribution (diversity)

amongst non-dominated solutions and, metrics evaluating convergence and diversity (Deb,

2001). Two critical issues normally taken into consideration while evaluating performance of

multi-objective optimization algorithms are: distance between obtained solutions and, spread


1.2 Key Issues in Multi-Objective Search 5

and uniformity among the obtained solutions.

All the proposed strategies in the present work are first tested using various benchmark test

problems. However, the proposed strategies are also validated using engineering problems.

Solving extruder screw design optimization problem is quite challenging, hence, the proposed

strategies are validated for solving MOO problems. There is a significant scope of utilizing

the multi-objective optimization tool for enhancing the rubber extrusion process efficiency.

1.2 Key Issues in Multi-Objective Search

Major problems that must be addressed when any evolutionary algorithm is applied to multi-

objective optimization are:

1. To accomplish fitness assignment and selection, respectively, in order to guide the search

towards the Pareto-optimal set.

2. To maintain a diverse population in order to prevent premature convergence and achieve

a well distributed and well spread non-dominated set.

In multi-objective evolutionary algorithms (MOEAs), generally objective function values

are directly used for fitness function. The selection processes are based on the classical aggre-

gation techniques, or direct use of the concept of pareto dominance. Instead of combining

the objectives into a single scalar fitness value, the MOEAs switches between the objectives

during the selection process. Each time an individual is chosen for reproduction, potentially

a different objective will decide which member of the population will be copied into the

mating pool. Schaffer (1985) proposed filling equal portions of the mating pool according to

the distinct objectives. Fourman (1985) implemented a selection scheme where individuals

are compared with a specific order of the objectives. Kursawe (1991) suggested assigning

a probability to each objective which determines whether the objective will be the sorting

criterion in the next selection step. The probabilities can be user defined or chosen randomly.

All of these approaches have a bias towards extreme solutions and are sensitive to non-convex

pareto-optimal fronts.


6 1. Introduction

Aggregation with parameter variation is a MOEA built on the traditional techniques for

generating trade-off surfaces. The objectives are aggregated into a single parameterized

objective function. The parameters of this function are not changed for different optimization

runs, but instead systematically varied during the same run. Goldberg (1989) suggested the

concept of calculating fitness on the basis of pareto dominance. The idea was taken up by

numerous researchers, resulting to several Pareto-based fitness assignment schemes. The

dominance based MOEAs are theoretically capable of finding any Pareto-optimal solution.

The dimensionality of the search space influences its performance.

Evolutionary optimization algorithms for multi-objective optimization searches multiple,

widely different solutions. Hence, maintaining a diverse population is crucial for the efficacy

of an MOEA. Elitism is the concept used to maintain diversity. A simple EA tends to converge

towards a single solution and often loses solutions due to selection pressure, selection noise,

and operator disruption. Elitism plays an important role in evolutionary multi-objective

optimization. The incorporation of elitism in MOEAs is more complex compared to single

objective EAs. Two basic elitist approaches are used in MOEA. One concept is to copy those

individuals from current population automatically to new population whose encoded decision

vectors are non-dominated. The second concept is to maintain an external set solutions whose

encoded decision vectors are non-dominated among all the solutions generated so far. Both of

these elitist policies may also be applied in some cases. Cooperative co-evolution, competitive

co-evolution and distributed evolution are the other strategies conventionally used to maintain

diversity in the population for MOEAs.

1.3 Major Contributions

Major contributions of the thesis work are summarized in the following three subsections.

1.3.1 Parallel Universe Alien Genetic Algorithm (PUALGA)

Hypothesis to modify GA using two sub-populations, one real coded and another, binary

coded was developed for this work. It was called Parallel Universe having different types


1.3 Major Contributions 7

of genetic encoding. Best members from binary coded population known as Alien members

go to real coded population and take part in evolution. Aliens will transfer the information

from one sub-population (Universe) to another; naming this concept of evolutions as Parallel

Universe Alien Genetic Algorithm (PUALGA). This approach increases robustness without

any additional computational burden by combining the capacity of both, binary and real coded

GAs. In fact, dividing the population in sub-population will reduce the calculations needed for

sorting and selection, and hence, may increase the overall efficiency of the algorithm. Though,

the proposed algorithm can be used with any population based evolutionary optimization,

GA was chosen to be used to demonstrate the clear benefits of the proposed concept of

hybridization.

The proposed algorithm has two specific tuning parameters in addition to conventional

parameters of GA. The size of binary and real coded population and number of alien transferred

at each generation are these two parameters. Sensitivity analysis needs to be carried out for

these two new algorithm parameters before investigating the performance of the algorithm.

The performance is evaluated using benchmark test functions and is compared with well

established algorithms to demonstrate the effectiveness of the proposed algorithm. The

algorithm is discussed in detail along with sensitivity analysis and performance evaluation in

chapter 4.

1.3.2 Boundary Inspection Approach for Constrained handling

A generalized constraint handling approach was also developed for population based EAs

using Boundary Inspection (BI) approach. The BI approach converts every infeasible member

to a feasible one during the evolution process. The algorithm attempts to move infeasible

point in a direction joining an infeasible point and a feasible point such that we reach within

feasible area. At every generation, using this approach all infeasible members are converted

to feasible members by moving towards randomly selected feasible point. The parameter

deciding the location of the new point is used from a predefined pool of values based on its

success history.


8 1. Introduction

A predefined ensemble of parameter λ was proposed to be used to locate the new point on

the line joining an infeasible point and the corresponding feasible point selected. Each value

in the ensemble is given equal opportunity during initial learning period. The success count by

each value in the learning period is converted to success probability, which is used in the next

learning period. During the learning period the success probability is kept constant. Value

of parameter λ to locate the new point is selected based on its success probability. Thus, the

value of parameter λ generating feasible point will be automatically preferred over the failing

value. This concept will automatically take care of tuning the value of parameter λ to serve

both purposes: (i) the generation wise parameter tuning during evolution and (ii) problem

specific tuning. The BI approach algorithm for constraint handling is discussed in detail along

with ensemble of parameter for automated selection of parameter and performance evaluation

in chapter 5.

1.3.3 Multi-Objective Optimization Problem Formulations for Rubber Extruder Screw

Design


considering temperature dependent viscosity modelled using Carreau-Yasuda model. The

model solution algorithm is also proposed and tested to converge velocity and temperature

profiles within the extruder channel. This validated model is used for optimization of screw

design parameters and temperature profile simultaneously to maximize throughput while

minimizing power consumption. The temperatures of the material under process within the

extruder and residence time distribution of product are also tracked for assured quality of

product. The screw helix angle, channel depth, and screw speed are used as manipulated design

parameters along with barrel temperature profile. Best screw geometry, screw speed and barrel

temperature profile are obtained using the proposed multi-objective optimization algorithm.

These multiple optimum solutions assist the decision maker in selecting an appropriate design

which is the best design solution considering the practical situation. All the relevant aspects

have been discussed in chapter 6.


1.4 Objectives of Research 9

1.4 Objectives of Research

The current research focuses on the following three objectives:

• Generating GA for multi-objective optimization problem, which has a high probability

of providing global optimum solution at the same time in less computational time.

• The developed algorithm will be tested using benchmark multi-objective optimization

problems.

• The algorithm will be tested on Extruder design optimization for maximization of

throughput, minimization of power consumption and, betterment of selected properties,

all conflicting objectives.

Taking into consideration above mentioned three objectives, development of GA program

with modifications in the existing algorithm has been proposed to make it more robust and

efficient. The developed program will be tested with benchmark test functions. The multi-

objective optimization application for rubber extruder throughput maximization and power

minimization will be developed and solved using proposed algorithm.

1.5 Structure of the Thesis

Literature review relevant to the current research work is presented in chapter 2. It includes

the multi-objective optimization basic concept, classical and evolutionary approaches. The

hybridization and constraint handling for evolutionary approaches is also reviewed. The rubber

extruder modelling and design is reviewed along with rubber rheology focusing on throughput

and power relationships. In chapter 3, Parallel Universe Alien Genetic Algorithm (PIALGA) is

presented as a new proposed hybridization approach for improving computational efficiency of

Genetic Algorithm. Its performance was demonstrated for multi-objective optimization(MOO)

using benchmark test functions. Sensitivity analysis and parameter tuning are also carried out

for the proposed algorithm. In chapter 4, proposed constrain handling mechanism is presented

for population based methods and implemented it in Genetic Algorithm. The proposed concept

is Boundary Inspection approach, which is also tested for MOO using benchmark constraint


10 1. Introduction

MOO test functions and design applications. The algorithm is further enhanced with the

concept of ensemble for automating tuning of parameter. Chapter 5 presents modelling of

rubber extruder developing throughput and power correlations using finite element approach

along with the solution strategies. The implementation of Rubber Extruder model and its

optimization using PUALGA and validation of results is presented in chapter 6. Conclusions

are drawn based on the work carried out and is discussed in detail in chapter 7 along with the

scope of future work.


11

Chapter 2

Literature Review

Optimization is a process of finding the best out of all possible solutions under the given

situations. The situations under which the best solution is found are constraints for optimiza-

tion. The criteria of optimization deciding the best is objective function. We use optimization

in almost all our decisions without realizing. Simple things like time management, finding

job, study and investment are examples of optimization applications. Optimization has many

applications in engineering, science, business and, economics, where quantitative models and

optimization methods are employed to find feasible solutions. Efficiency of manufacturing

and engineering activities can be improved by using optimization in design and operations.

As economy, energy and environmental landscapes are continuously changing, there is always

a scope for optimizing the current industrial operations. Formulation of optimization problem,

simplification/dressing, solution of optimization problem and validation of the solution are

the basic steps of optimization process. Different optimization methods are used for solving

problems based on the nature of formulated problem after simplification transformation.

Optimization methods are classified into two categories, namely direct and indirect (or

gradient based) methods. In direct methods, only objective function and constraints values

are used to guide the search process. On the other hand, first and/or second derivatives of

the objective function and/or constraints are used to guide the search operation in gradient

based methods. The direct search methods are relatively slow in convergence compared to the

indirect methods, since they do not use derivative information. However, direct methods can be

12 2. Literature Review

applied easily to the optimization problems without doing changes in the algorithm. Gradient-

based methods have faster convergence rate compared to the direct methods of optimization,

but can not be applied to non-differentiable problems. The difficulties of traditional direct and

indirect methods face are:

• They tend to get stuck to local solution.

• Convergence to an optimal solution is quite dependant on the initial guess values.

• They are not efficient for solving the problems having discrete variables.

• They are not suitable for parallel computing.

• An algorithm that is found efficient in solving one optimization problem may not be

efficient in solving another problem.

Evolutionary Algorithms (EAs) can overcome the above difficulties, hence they are be-

coming the most proven method for Global optimization of complex problems (Patel and

Padhiyar, 2015). EAs use the principle of survival of the fittest to generate better solutions

using operators emulated from natural evolution. Such processes lead to the evolution of the

population of individuals that are more suitable to their environment. In spite of the large

popularity of the EAs in the recent past, they are criticized for slow convergence rates. Hence

there is always a requirement for the improvement in their computational efficiency. Various

updates in such evolutionary algorithms (EAs) have been proposed in the open literature

to increase the convergence rate and probability of reaching to the global optimum. We

propose hybridization of two types of Genetic Algorithm (GA), binary coded and real coded

for enhancing the performance. Since the evolutionary algorithms are naturally designed for

unconstrained optimization problems, they require an additional mechanism for constraint

handling. Boundary Inspection approach is used along with GA for constrained optimization.

GAs can handle single and multi-objective optimization problems. Even with the devel-

opments in the computational powers of computers, solving the complex multi-objective

problems requires very long time. There is always a need for development of robust and

computationally efficient algorithms for large and complex problems. This research focus

on upgrading the GA to enhance the convergence and constraint handling capabilities for


2.1 Multi-Objective Optimization 13

multi-objective optimization. The proposed approaches are tested by benchmark test functions

and validated using rubber extruder screw design application.

2.1 Multi-Objective Optimization

Multi-objective optimization (MOO) is a method of optimization, which can deals with multi-

ple conflicting objectives. Few examples of conflicting objectives are: working and operating

cost, price and features, selectivity and yield, profit - environmental impact - safety cost. MOO

problems will have conflicting objectives and create a set of solutions (showing trade-offs

among the objectives), named as pareto optimal solutions. None of these pareto optimal

solution can be said to be better than the others with respect to all the objectives (Steuer,

1989). In the pareto optimal solutions for MOO problems, one objective can be improved only

by compromising the other objective. The pareto solutions for a bi-objective optimization

problem for maximizing througput and minimizing the power consumption is shown in Fig.

(2.1). The two end points of the pareto line corresponds to the minimum power consumption

and maximum throughput. Note that both the points can be obtained by solving two distinct

single objective optimization problems. All the points on the pareto line are the non-dominated

solutions. All the point above and left to this line are dominated by all the points on pareto

front. Note that all the points on the pareto front can be obtained simultaneously by solving a

population based evolutionary multi-objective optimization algorithm.

Optimization methods are claasified by Deb (2001) into two major categories: 1) Classical

methods and, 2)Evolutionary methods. A single random solution is used by classical methods,

which is updated at every iteration, to find the optimal solution by the deterministic procedures.

Classical methods are further sub-classified into two distinct groups: direct methods and gradi-

ent based methods. Direct methods use a objective function and a constraints value to find the

optimum. whereas, gradient based methods use the first and second derivative of the objective

function and/or constraints to find the search direction and optimal solution. MOO problems

can be solved by many methods. Most of them use the technique of converting the MOO

problem into one or multiple single objective optimization (SOO) problems. Each of these

SOO problem uses a scalar function, which is derived from the multiple objectives. Rangaiah



Figure 2.1. Pareto front for maximum throughput and minimum power for Extruder

(2017) says that there are different ways of defining a scalarizing function and, therefore there

exists multiple MOO methods. Though the scalarization mechanism is conceptually very

simple, the resulting SOO problems may be difficult to solve.

The MOO methods can be classified based on the involvement of the decision maker.

They are divided into two categories: preference-based methods and generating methods

(Diwekar, 2008). Another classification approach is based on generation of many Pareto-

optimal solutions and, the role of the decision maker (DM) in selecting the MOO solutions.

The classification, adopted by (Diwekar, 2008) is shown in Fig.(2.2).

Based on the experience and the information not considered in the MOO problem formu-

lation, DM can select the Pareto-optimal solutions. As shown in Fig. (2.2), MOO methods

are classified into two main groups: generating methods and, preference based methods. The

generating methods produce one or more Pareto-optimal solutions, but without taking any

inputs from the DM. The solutions generated are given to the DM for selection. Whereas the

preference-based methods take advantage of the preferences given by the DM at different

stage(s) for solving the MOO problem.

Generating methods produce one or more pareto-optimal solutions without any help of the



Figure 2.2. Classification of Multi-objective Optimization Methods (source: Rangaiah (2017)

decision maker(DM). The pareto obtained by these methods contain all the possible trade-off

information among all the objectives, which are provided to the DM for making a choice.

Generating techniques can further be divided into three sub-groups, namely, no-preference

methods, a posteriori methods using the scalarization approach and a posteriori methods using

the multi-objective approach. A posteriori methods are further classified in two groups, using

scalarization approach and using multi-objective approach. There are many ways of defining

a scalarization function, and hence many MOO methods exist. Although the scalarization

approach is conceptually simple, the resulting SOO problems with the augmented function

may not be easy to solve. Moreover numerous SOO problems are required to be solved for

generating the entire pareto front with this approach. Fu and Diwekar (2004) present an

approach for minimizing number of SOO problems to generate a pareto using the principles

of probabilistic uncertainty analysis.

A posteriori methods rank intermediate solutions using objective function values to find

multiple Pareto-optimal solutions for MOO. Population-based methods like non-dominated

sorting genetic algorithm, multi-objective differential evolution and multi-objective simulated



annealing belongs to posteriori methods. All these methods generate many Pareto-optimal so-

lutions, which are available to the DM, who review and select one of them for implementation.

The involvement of DM in all above methods is after getting the Pareto optimal solutions,

hence they are named as - a posteriori methods.

The preference-based methods takes into account the preferences defined by the DM at

intermediate stage(s) in solving the MOO problem. They are sub-classified into two categories:

a priori methods and interactive methods. In a priori methods, the preferences defined by

the DM are considered in the basic formulation of the SOO problem. Examples of a priori

methods are: value function methods, lexicographic ordering and, goal programming. The

value function methods formulate a SOO function, involving the original objective func-

tion values and preferences defined by the DM for optimization before solving the problem.

Weighting method is an classical example of value function methods. Lexicographic ordering

expects the DM to arrange the objectives according to their importance for solution by a SOO

method. In goal programming, the DM provides an parameter defining the aspiration level

for each of the objectives. The appropriate SOO problem is then formulated and solved. The

interactive methods provide one or multiple Pareto-optimal solutions at each stage of evolution.

Interactive surrogate worth trade-off method and the NIMBUS method are claassical examples

of interactive MOO methods. Vallerio et al. (2015) present an interactive multi-objective

framework to optimize dynamic processes.

Interactive methods require interaction with the DM during the search of the MOO problem

solutions. At the end of an iteration of an interactive method, DM reviews the obtained

Pareto-optimal solution(s) and suggests the changes required in each of the objectives. The

preferences suggested by the DM are incorporated in the problem formulation and the modified

optimization problem is solved for the next iteration. The interactive methods provide multiple

Pareto-optimal solutions as final solution set. Interactive surrogate worth trade-off method

and the NIMBUS method, belonging to this category are be applied to several chemical

engineering applications. Relative merits and limitations of the groups of these MOO methods

are summarized in table (2.1). Few MOO methods can be categorised in multiple groups



like, weighting method can be classified as a posteriori methods as well as a value function

methods in the a priori group. The adapted versions of ε-constraint method from a posteriori

group and goal programming from a priori group can be also classified as interactive methods.

Thus, the above classification of MOO methods is little subjective.

2.1.1 Classical Methods for MOO

Classical methods for MOO aggregate the multiple objective functions into a single objective

function. They use an analogy, which is similar to decision making before search. The

parameters of the aggregated function are not defined by the decision maker, but they are

systematically chosen by the optimization algorithm. Multiple runs with different parameters

are carried out to achieve a set of solutions to generate the Pareto-optimal set. The weighting

method (Deb, 2001), the constraint method (Cohon, 2004), goal programming (Steuer, 1986),

and the minmax approach (Koski, 1984) are examples of classical methods. The classical

optimization methods like weighting and constraint show following difficulties:

• Sensitivity of the method to the shape of the Pareto-optimal front.

• Problem knowledge is expected, which may not be available.

In weighting method the original multi-objective optimization problem is converted to an

single objective optimization problem by using linear combination of the objectives:

maximize y = f (x) = w1 · f1(x)+w2 · f2(x)+ ...+wk · fk(x)

sub jectto x ∈ X f

(2.1)

Where, wi are normalized weights (∑wi = 1).

Solving the optimization problem formulated in equation (2.1) for different weight combi-

nations yields different sets of solutions. This method generate the Pareto-optimal solutions

which can be easily shown, when an exact optimization algorithm is used and all the weights

are positive. Assume that a feasible decision vector a maximizes f for a given weight com-

bination and is not Pareto optimal, then there is always a solution b which dominates a, i.e.

f1(b)> f1(a) and fi(b)≥ fi(a) for i = 2, ...,k. Therefore, f (b)> f (a), which is a contradic-



Table 2.1. Main Features, Merits and Limitations of MOO Methods (Source: Rangaiah (2017))

Methods Features, Merits and LimitationsNo PreferenceMethods (e.g.,global criterion andneutral compromisesolution)

These methods do not require any inputs from the decisionmaker either before, during or after solving the problem.Global criterion method can find a Pareto-optimal solution,close to the ideal objective vector.

A Posteriori Meth-ods using Scalariza-tion Approach (e.g.,weighting and ε-constraint methods)

These classical methods require solution of SOO problemsmany times to find several Pareto-optimal solutions. ε-constraint method is simple and effective for problems witha few objectives. Weighting method fails to find Pareto op-timal solutions in the non-convex region although modifiedweighting methods can do so. It is difficult to select suitablevalues of weights and ε . Solution of the resulting SOOproblem may be difficult or non-existent.

A PosterioriMethods UsingMulti- ObjectiveApproach (manybased on evolu-tionary algorithms,simulated anneal-ing, ant colonytechniques etc.)

These relatively recent methods have found many applica-tions in chemical engineering. They provide many Pareto-optimal solutions and thus more information useful for de-cision making is available. Role of the DM is after findingoptimal solutions, to review and select one of them. Manyoptimal solutions found will not be used for implementation,and so some may consider it as a waste of computationaltime.

A Priori Methods(e.g., value func-tion, lexicographicand goal program-ming methods)

These have been studied and applied for a few decades.Their recent applications in chemical engineering are lim-ited. These methods require preferences in advance fromthe DM, who may find it difficult to specify preferenceswith no/limited knowledge on the optimal objective values.They will provide one Pareto-optimal solution consistentwith the given preferences, and so may be considered asefficient.

Interactive Methods(e.g., interactive sur-rogate worth trade-off and NIMBUSmethods)

Decision maker plays an active role during the solutionby interactive methods, which are promising for problemswith many objectives. Since they find one or a few optimalsolutions meeting the preferences of the DM and not manyother solutions, one may consider them as computationallyefficient. Time and effort from the DM are continuallyrequired, which may not always be practicable. The fullrange of Pareto optimal solutions may not be available.



Figure 2.3. Graphical interpretation of the weighting method (left) and the constraint method (right). (Source:Zitzler (1999))

tion to the assumption that f (a) is maximum. That is the reason non-convex solutions can

not be obtained using this method. This the main disadvantage of this technique: it cannot

generate all Pareto optimal solutions with non-convex trade-off surfaces. This is demonstrated

in Fig. (2.3) for the embedded system design example. As shown graphically, the optimization

process targets to move the line upwards until no feasible objective vector is better than

it (above it) and minimum one feasible objective vector is on it. The graphical procedure

demonstrates that the points B and C will never maximize f . Increasing the slope, D achieves

a greater value of f (upper dotted line); decreasing the slope, A will have a greater f value

than B and D (lower dotted line).

The ε-Constraint method is not biased towards the convex portions of the pareto front.

It transforms k−1 of the k objectives into constraints. The remaining one objective, which

can be chosen arbitrarily, is the objective function to be solved for k− 1 constraints. The

formulation of Constraint Methods can be represented as:

maximize y = f (x) = fh(x)

sub jectto ei(x) = fi(x)≥ εi, (l ≤ i≤ k; i 6= h)

x ∈ X f

(2.2)



The lower bounds and, εi are the parameters used by the optimizer to find multiple Pareto

solutions. The ε-Constraint method can obtain the solutions associated to non-convex parts of

the pareto curve, as presented in Fig. (2.3) on the right. By specifying h = 1 and ε2 = r (solid

line), the solution represented by A become infeasible regarding the extended constraint set.

At the same time the decision vector related to B, maximizes f within the remaining solutions.

Fig. (2.3) shows the problem with the technique. It represents that if the lower bounds are not

chosen appropriately (ε2 = r′), we may not get any feasible solution. To avoid this condition,

a range of values suitable for the εi need to be known.

The classical methods require multiple runs to obtain an approximation of the Pareto

front. As these runs are performed independently, it contribute to the high computation

overhead. Evolutionary algorithms (EAs) are gaining more importance compared to the

classical methods. EAs can handle large search spaces and, generate multiple solutions in a

single optimization run. EAs can be implemented such that, both of the previously discussed

difficulties of classical methods are avoided. The evolutionary MOO algorithms are discussed

in next subsection.

2.2 Evolutionary Multi-Objective Optimization Algorithms

Population based EAs have become significantly popular and find an edge over the classical

methods owing to their ability to converge the entire population to the optimal pareto front

in a single run. This property of the EAs has gained significant attention for multi-objective

optimization applications in the past two decades (Deb, 2001; Coello et al., 2006; Rangaiah

and Bonilla-Petriciolet, 2013). A good MOO EA is expected to provide (1) Accuracy - the

closeness of the generated solutions to the true pareto solutions. (2) Coverage - distinct

non-dominated solutions covering the true pareto front, and (3) Distribution- uniformity of the

obtained solutions over the true pareto front.

Schaffer (1985) proposed the first implementation of real multi-objective evolutionary algo-

rithm (vector-evaluated GA or VEGA). Schaffer changed the simple GA selection, crossover,

and mutation replacint it by independent selection according to each objective. The proposed


2.2 Evolutionary Multi-Objective Optimization Algorithms 21

selection procedure is repeated for each objective to contrbute a portion of the mating pool.

The entire population is randomly mixed before applying the crossover and mutation operators.

The shuffeling ballances the mating of individuals from different subpopulation groups. The

algorithm proposed by Schaffer worked efficiently for few cases, but in some cases it suffered

from the bias towards few individuals or a region. This results in to inferior coverage, which

is one of the goal of MOEO. Significant contribution was not notices for many years after the

pioneering work of Schaffer, till the non-dominated sorting procedure suggested by Goldberg

(1989). Since an EA needs to define fitness for reproduction, the skill is to define a single

metric from the number of objective function values. Goldberg suggested to apply the idea of

domination to allocate multiple copies of non-dominated individuals in a population. Since the

diversity is an important parameter of another concern, he proposed to use a niching strategy

between solutions of a non-dominated group. Taking insight from Goldbergs’ work, three

independent researcher groups developed different versions of multi-objective evolutionary

algorithms. These algorithms developed by different researchers, differ in the way a fitness is

assigned to each individual.

The original work on the evolutionary multi-objective optimization was carried out by

Schaffer (1985). However, the Schaffer’s EA was biased towards few points on the pareto

front, which was taken care by Goldberg (1989) and Srinivas and Deb (1994). Since then,

Non-dominated Sorting Genetic Algorithm (NSGA) proposed by Srinivas and Deb (1994) is

been widely used. Outline of the earlier works for multi-objective optimization using EAs

can be found in Fonseca and Fleming (1993), Coello (1999), and Coello et al. (2006). They

present a comprehensive survey and a critical review of multi-objective EAs. The more recent

review on MOO is presented by Arora (2017). Pareto archived evolutionary strategy (PAES)

Knowles and Corne (2000) and strength pareto evolutionary algorithm (SPEA-2) Zitzler et al.

(2001) are also prominent EAs for solving MOO along with non-dominated sorting genetic

algorithm (NSGA-II). All these algorithms use the concept of pareto-dominance. NSGA-II is

the most popular and widely accepted MOO Algorithm. It is applied in the various fields of

science and engineering in its original and modified forms. NSGA-II has evolved with many

new variants, which attempted to reduce complexity and enhance its convergence to the true



pareto front (Hossein et al., 2011; Fang et al., 2008; Tran, 2009; Jensen, 2003; Zhang et al.,

2015).

Fonseca and Fleming (1993), developed a multi-objective GA (MOGA), where they clas-

sify the whole population based on different non-dominated classes. They allocate rank one to

the individuals of the first (best) class. The rest individuals are ranked based on how many

solutions (say k) dominate a particular solution. That solution is allocated a rank one more

than k. Therefore, implementing this ranking procedure, it is possible that there exist many

solutions having the same rank. The selection procedure then utilises these ranks to select or

delete blocks of points to create the mating pool. Along with ranking procedure, MOGA uses

a niching method to distribute the population across the Pareto-optimal region. They have

used niching on objective function values in place of performing niching on the parameter

values.

Horn et al. (1994) applied Pareto domination tournaments in place of non-dominated

sorting and ranking selection method in their niched-Pareto GA (NPGA). A set comprising

of a specific number (tdom) of randomly selected individuals is created for comparison, from

the population at the beginning of each selection process. Two random individuals are cho-

sen from the population for selecting a winner according to the following procedure. Both

individuals are tested by comparing with the members of the comparison set for domination

with respect to all the objective functions. The non-dominated point is selected if it is the

only non-dominated and the other is dominated. A niche count is found to guide the selection

for each individual in the entire population, if both are either non-dominated or dominated.

The niche count represents the number of points in the population within a certain distance

(σshare) from an individual. The individual with least niche count is preferred. Since the

non-dominance is computed by using an individual with a randomly chosen population set

of size tdom, the performance of this algorithm greatly depends on the parameter tdom. True

non-dominated (Pareto-optimal) points can be obtained, if a proper value of the parameter

tdom is chosen.


2.2 Evolutionary Multi-Objective Optimization Algorithms 23

Similar to the MOGA, Srinivas and Deb (1994) developed a non-dominated sorting GA

(NSGA). NSGA differs from MOGA in two ways: fitness assignment procedure and the mech-

anism of niching. Once the population is classified for non-domination, best non-dominated

group is assigned a dummy fitness value equal to N (population size). A niche count for

each individual of the best class is obtained using parameter values instead of the objective

function values. The niche count represents a qualitative number of individuals in the vicinity

of the solution. For each individual, a shared fitness is obtained by dividing the assigned

fitness N by the niche count. The smallest shared fitness value F min1 is counted for further

use. Thereafter, the second class of non-dominated solutions are obtained and assigned a

dummy fitness value equal to F min1 − ε1 (where ε1 is a minor positive number) is assigned to

all individuals. Niche counts of all the individuals within this group are established and the

shared fitness values are obtained. This process is carried out till all solutions are assigned

a fitness value. This fitness assignment procedure conforms two aspects: (i) a dominated

solution is assigned a lower shared fitness value compared to any solution which dominates

it and (ii) In each non-dominated group, diversity is maintained. On a number of test prob-

lems and real-world optimization problems, NSGA has successfully obtained wide-spread

Pareto-optimal solutions or near Pareto-optimal solutions. The main difficulty inf NSGA

implementation is the selection of the niching parameter, which represents the maximum

distance between two neighbouring Pareto-optimal solutions. Although most researches used

a fixed value of the niching parameter, there exists case studies where an adaptive sizing

strategy has been suggested (Fonseca and Fleming, 1993).

Tan et al. (2002) have done performance assessments and comparisons for EAs for Multi-

objective Optimization. Evolutionary techniques for multi-objective optimization are recently

acquiring significant attention from researchers in different fields due to their effectiveness and

robustness in probing a set of trade-off solutions. Unlike conventional methods that aggregate

multiple objectives to form a resultant scalar objective function, evolutionary algorithms

with adapted breeding schemes for MO optimization are capable of andling each objective

component individually and guide the search in finding the global Pareto optimal set. Non-

dominated sorting (Deb et al., 2002), rank based sorting (Qu and Suganthan, 2010), and



evolution with decomposition (Jiao et al., 2013; Zhao et al., 2012) are the prominant evolving

approaches for solving the MOO problems. The dominance based ranking of populations

(Deb et al., 2002) requires repeted comparisons of members for sorting and hence it is

computationally costly. The reason for performance degradation with increasing dimensions

for well established Evolutionary Multi-objective Optimization Algorithms (EMOAs), NSGA-

II and SPEA2 was demonstarated by Coello et al. (2006). Lu and Yen (2003) developed a

rank-density-based genetic algorithm (RDGA) using the ranking procedure with automatic

accumulated ranking strategy and, a "forbidden region" concept. Qu and Suganthan (2010)

developed a sorting approach using the summation of normalized objective values along

with diversified selection. They observed this sorting mechanism to be faster and performing

better for both, the multi-objective evolutionary programming (MOEP) and multi-objective

differential evolution (MODE). Wang and Yao (2014) presents computationally less costly

corner sort algorithm for non-dominated sorting. All these algorithms use the concept of

pareto dominance for sorting and selection.

2.3 Evolutionary Optimization

Evolutionary algorithms are build on computational models of natural evolutionary processes:

selection, recombination, and mutation. Fig. (2.4) depicts an overview of a general evolution-

ary algorithm. Individuals, or the candidate solutions may be encoded as strings composed of

few alphabets, e.g. binary, integer, or real-valued, and an initial population is typically gener-

ated by randomly sampling these strings. The fitness value as a measure of its performance is

then computed for each candidate solution of the initial population. These fitness values are

then employed to bias the selection process during evolution. Fitter individuals are assigned a

higher probability of being selected for the reproduction compared to the individuals having

lower fitness values. Since the fitter individuals are selected for reproduction with higher

probability, the average performance of the population is expected to increase during the

evolution. It is possible that the individuals may be selected more than once at any generation

of the EA. New individuals are produced through the application of evolutionary operators

using the selected individuals. These new individuals also called the offspring are evaluated

for their fitness values, which subsequently have to compete the parent generation. This


2.3 Evolutionary Optimization 25

process of selection, reproduction, and evaluation are repeated until a specified termination

criteria is met. Typical termination criteria could be, a certain number of generations, the

variance of the fitness values of the individuals in the population, or any other user defined

criteria.

Figure 2.4. General Scheme of an Evolutionary Algorithm

Evolutionary methods mock the evolution principle of nature, resulting to a stochastic

search and optimization algorithm. It can surpass the classical method in many ways. Evo-

lutionary method (algorithm) uses a starting population of randomly created solutions at

each iteration, in place of using a single solution as in classical method. The population

is upgraded in each generation to finally converge to the optimal solution. Generating the

optimum solution set in a single simulation run is a special feature of the evolutionary methods

in solving multi-objective optimization problems.

EAs mimic the the natural and/or biological phenomena such as ants locate the shortest

route to a food source and birds find their destination during migration. The behaviour of such

biological species is followed by the key steps such as learning, adaptation, and evolution.

The pioneering work on such evolutionary computation reported in the literature was on the

genetic algorithms (GAs) (Holland, 1975; Goldberg, 1989). Despite their popularity, GAs

may require large computational efforts for converging to a near optimum solution. Moreover,

GAs may not even converge to a solution for numerous problems. There have been numerous



EAs proposed particularly in the recent past in an attempt to reduce the computational cost and

improve the quality of the solutions, especially being able to escape from converging to the

local optima. In addition to various improvised GAs, the recent developments in EAs include

differential evolution (Storn and Price, 1997), particle swarm optimization (PSO) (Shi and

Eberhart, 1998) , Ant colony systems (Dorigo et al., 1996), and shuffled frog leaping (Passino,

2002), firefly algorithm (Yang, 2010a), and Cuckoo search (Yang and Deb, 2009) method

to name few. One can refer these books (Gujarathi and Babu, 2016; Onwubolu and Babu,

2013; Yang, 2010b; Xinjie and Mitsuo, 2010) for more detail on various EAs. We present a

brief overview of hybrid EAs followed by constraint handling approaches for EAs in the next

subsection.

2.3.1 Hybrid Evolutionary Algorithms

Evolutionary algorithms have been widely accepted for solving several practical optimization

applications in engineering. However, they are often criticized for the large computational

time as well as for their inefficiency to handle the constraints. This is often attributed to the

inappropriate selection of the algorithm parameters. There is significant reported literature,

where the simple EA failed to attain the optimal solution Tseng and Liang (2006); Somasun-

daram et al. (2005); Lo and Chang (2000); Thakur (2014); Wang and Dang (2007); Mohamed

et al. (2012). This motivates for the hybridization of EAs with the other optimization algo-

rithms. Since there are several local search optimization algorithms which overcome the two

above mentioned problems with the EAs, their hybridization adds significant value to the EAs

Krasnogor and Smith (2005). The problem solving capability can greatly be enhanced when

two or more different methods are hybridized in a supportive manner (Ong and Keane, 2002).

Such hybridizations leverage the explorative advantage of the population based search and

exploitative nature of the local search algorithms (Ong et al., 2006).

In the past one decade, there has been significant increase of the research literature on

the hybrid EAs. This has been demonstrated by searching the number of publications in

the popular scientific databases, namely Scopus, ScienceDirect, and IEEE-Xplore using the

keywords "hybrid evolutionary algorithms". The search results are summarized in Table 2.2.



Note that the number of relevant papers could be smaller than those mentioned in the table

since no filtering was used in the search.

Table 2.2. Literature search count on various scientific databases with the key words "hybrid EvolutionaryAlgorithms"

Publication year ScienceDirect Scopus IEEE-Xplore2009-2019 20333 105014 182061998-2008 4660 16025 56641997 and earlier 1544 608 894All 26537 121647 24764

Hybridization of the EAs with the local search algorithms is also known as memetic

algorithms (MAs). MAs generally exhibit superior performance than the parent EAs or local

search algorithms. Several reported literature can be found for the successful applications

of such hybrid approach for solving complex optimization problems (Masegosa et al., 2013;

Nabil, 2016; Mohamed, 2015; Rocha et al., 2013; Kim and Liou, 2014; Cheshmehgaz et al.,

2013; He and Yen, 2014; Gujarathi and Babu, 2011). Two excellent review papers on MAs are

presented by (Chen et al., 2011; Neri and Cotta, 2012). These articles also present memetic

computing, methodologies, frameworks, and algorithms.

Hybridization can be incorporated at different stages, such as the initialization, evolution,

or selection in the population based EAs. While we do not focus on the exhaustive review on

these three level of hybridization a few of the representative implementations are mentioned

here. Rahnamayan et al. (2007) proposed hybridization at the initialization using opposition-

based learning and demonstrated that it helped accelerate the convergence compared to the

random initialization. Keedwell and Khu (2005) used cellular automata approach to provide

a good initial population to seed the GA and noticed that it enhanced the performance on

difficult problems.

For the hybridization at the evolution level, one of the popular techniques is hybridizing

an operator imported from a specific algorithm with another. One such effective operator that

enhances the performance of an EA is the cloning operator imported from the clonal selection



algorithm (De Castro and Timmis, 2002). Such a cloning operator enhanced the performance

in various EAs, such as the GA (Ludwig, 2012), Differential Evolution (Qin et al., 2010),

Particle Swarm Optimization (Hong, 2009) , Ant Colony Optimization (Gao et al., 2008),

Artificial Bee Colony Algorithm (Tien and Li, 2012), Harmony Search (Wang et al., 2009a),

Tabu Search (Layeb, 2012) , and Flower Pollination Algorithm (Nabil, 2016), Gravitational

Search (Gao et al., 2013).

Managing the population diversity through selection operator is one of the important goals

in the hybridization. Molina et al. (2005) divides the EA population in three sections and

apply the local individual search operator for the selection based on the fitness values. Similar

approach to maintain the diversity with the local search operators was attempted by Nguyen

et al. (2007) by dividing and sorting the population into an arbitrary number of levels. Both

these works show to provide better results than the conventional, random selection. The

detailed analysis of the tradeoffs between the computational time and fitness values when

applying such hybridization can be found elsewhere Bambha et al. (2004). The hybridization

is also applied in the form of multi-populations or ensembles (Park and Ryu, 2010; Yang

and Yao, 2008; Tang et al., 2007; Mezmaz et al., 2007). Such hybrid models exchange the

information among the different populations during the evolution. The improved explorative

and exploitive properties exhibited by such hybridization are presented by (Park and Ryu,

2010; Yang and Yao, 2008).

Another popular strategy to construct hybrid EAs is by combining the best features of

two or more EAs to form a hybrid algorithm. Trivedi et al. (2016) presents hybridization of

the two popular EAs, namely GA and DE for solving a nonlinear, high-dimensional, con-

strained, and mixed-integer optimization problems. Kim (2005) presented a hybrid GA with

the bacterial foraging for PID controller tuning purpose. The hybridization of GA and PSO

was employed and analysed by Grimaldi et al. (2004) and Deb and Padhye (2014) using

the unimodal functions and an electromagnetic optimization problem, respectively. Li et al.

(2015b) hybrided PSO and chemical reaction optimization for multi-objective optimization to

enhance the diversity with crowding distance mechanism.



We developed hybridization hypothesis to modify GA using two sub-populations, one real

coded and another, binary coded. We call the concept of Parallel Universe having different

encoding. Best members from binary coded population known as Alien members will go to

real coded population and take part in evolution. Alien will transfer the information from

one sub-population (universe) to another; we call this concept as Parallel Universe Alien GA

(PUALGA). This approach can increase robustness without any additional computational

burden by combining the capacity of both, binary and real coded GAs. In fact, dividing the

population in sub-population will reduce the calculations needed for sorting and selection and

hence will increase the overall efficiency of the algorithm. Though, the proposed hypothesis

can be used with any population based evolutionary optimization, we choose to use GA to

demonstrate the benefits of the proposed concept of hybridization. The review of constraint

handling for EAs is presented in next subsection.

2.3.2 Constraint Handling in Evolutionary Algorithms

Most real-world problems are constrained in nature and a possible criticism of the EAs has

been the lack of efficient and generic constraint handling feature. Evolutionary optimization

algorithms are unconstrained by nature and hence need additional mechanisms to handle

equality ad inequality constraints (Kramer, 2010). However, the EAs can handle the bound

constraints on decision variable more effectively since it is one of their design features. There

exist two excellent review articles on the constraint handling methods for EAs in the literature

Kramer (2010); Mezura-Montes and Coello Coello (2011). Few of the popular constraint

handling approaches used with the EAs are mentioned below. Note that the method of ignoring

infeasible solutions (Koziel and Michalewicz, 1999) and the method of decoders (Koziel and

Michalewicz, 1998) are computationally quite expensive and have become obsolete. Hence,

these two methods have not been covered in the following list of popular constraint handling

methods,

1. Feasibility rules

2. Penalty functions



3. Stochastic ranking

4. ε-constrained method

5. Multi-objective concepts

Method of feasibility rules is one of the most popular constraint-handling techniques for

long time, which was originally proposed by Deb (2000). In this mechanism, a set of three

feasibility rules are added to a binary tournament selection as follows:

• Comparing two feasible solutions, the one with the best objective function is selected.

• Comparing a feasible and an infeasible solution, the feasible one is selected.

• Comparing two infeasible solutions, the one with the lowest sum of constraint violation

is selected.

The popularity of this method for constraint-handling lies in its easy interfacing with

a variety of algorithms, without introducing new parameters. Although it was originally

proposed for GA (Deb, 2000), it is widely implemented in DE (Huang et al., 2006; Brest et al.,

2006; Elsayed et al., 2011), PSO (Cagnina et al., 2006, 2007) , Bacterial Foraging algorithm

(Mezura-Montes and Hernández-Ocaña, 2009) architecture to mention a few. Among various

variants of this method (Zielinski and Laur, 2008; Zielinski et al., 2008; Zielinski and Laur,

2006), Mezura-Montes and Coello Coello (2005) combined the method of feasibility based

rules with other mechanisms such as retaining infeasible solutions closer to the feasible regions

for active constraints.

Penalty function method for constraint handling converts a constrained optimization

problem into an unconstrained one using penalty functions. The transformation can be

expressed as follows,

φ(x) = f (x)+P(x) (2.3)

where, φ(x) is the augmented function, f (x) is the objective function to be minimized, and

P(x) is the penalty function, which is defined as:



P(x) =J

∑j=1

r j max(0,g j(x))+K

∑k=1

ck | hk(x) | (2.4)

where, g(x) are inequality constraints, h(x) are equality constraints, and r j and ck are

penalty factors.

In the penalty method, penalty parameter is multiplied with the extent of constraint viola-

tion and is augmented with the objective function. While it is the simplest method of handling

constraints, finding the appropriate penalty values is a challenging task. This problem is

partially addressed in the literature by updating the parameters adaptively (Coello and Efre’n,

2002). In one of the adaptive mechanisms, the penalty parameters can be updated using

the generation counter in the EAs (Joines and Houck, 1994; Kazarlis and Petridis, 1998).

Michalewicz and Attia (1994) updated the penalty parameters with the concept of cooling

factor that is used in simulated annealing method. The current best fitness solution in an

EA was used to update the penalty values (Rasheed, 1998). Penalty parameters can also be

updated based on the number of feasible and infeasible solutions in the population (Hamda and

Schoenauer, 2000; Hamida and Schoenauer, 2002). Augmented lagrangian method (Nocedal

and Wright, 1999) is a penalty method hybridized with the lagrangian multiplier method.

The Stochastic Ranking (SR) method Runarsson (2004); Wu and Yu (2001); Runarsson and

Yao (2000) is a technique that was originally proposed to overcome the inherent shortcomings

of penaly method of tuning the penalty parameters. In the SR method, the fitness value of

each individual is computed through a stochastic ranking procedure quite similar to a bubble

sort. Thus, the individuals are compared based on the constraint violation only to the adjacent

neighbourhoods in this method. After its first appearance in the year 2000, the SR has been

used with DE Fan et al. (2009); Mezura-Montes et al. (2005); Zhang et al. (2008); Liu et al.

(2009b,a), PSO Ali et al. (2012); Jian et al. (2008); Pulido and Coello (2004) and Ant colony

optimization Leguizamon and Coello (2007); Fonseca et al. (2007) for constraint handling.

ε-constraint method (Takahama and Sakai, 2006) employs fitness assignment process

similar to the superiority of feasible solutions method, but with an adaptive relaxation in

constraint violation for initial few generations using the ε parameter. The ε parameter in this



method creates a space to accommodate more infeasible solutions in the population during the

early stages of evolution. The ε value is updated according to the following equations:

ε(0) = υ(xθ )

ε(k) =

ε(0)(

1− kTc

)cp

0 < k < Tc

0 k ≥ Tc

(2.5)

where υ(xθ ) is the overall constraint violation using equation top θ th individual at initial-

ization; and cp is a parameter is to be [2,10].

The ε-constrained method and its predecessor, namely the α-constrained method (Taka-

hama and Sakai, 2005a), have been widely employed to different EAs such as GAs (Takahama

and Sakai, 2004), PSO (Takahama and Sakai, 2005b), hybrid PSO-GA (Takahama et al., 2005)

and DE (Wang and Li, 2010; Takahama and Sakai, 2006).

Constraint handling method using multi-objective optimization techniques treats each

constraint as an objective. Thus, any multi-objective optimization method can be employed

to the resulting MOO problem. Liang et al. (2010); Li et al. (2010); Gong and Cai (2008);

Coello (2000); Mezura-Montes and Coello (2008) employ such MOO methods. In a variation

of this method, Wang et al. (2007a) augmented all the constraints violation in one objective,

while keeping the original objective intact. This concept is more suitable when the number

of constraints is large to avoid significantly more challenging problem with many-objective

optimization problem (Liu et al., 2007; Wang et al., 2007b, 2008, 2009c,b; Wang and Cai,

2010; Li et al., 2008; Venter and Haftka, 2010; Wang et al., 2010).

Constraint handling becomes even more crucial and complex in multi-objective EAs.

Singh et al. (2010) extended simulated annealing for multi-objective constrained optimization

problems. Yang and Deb (2014) used constrained method and adaptive operator selection in

multi-objective evolutionary algorithm based on decomposition (MOEAD). Yang and Deb

(2013) proposed a cuckoo search algorithm for multi-objective optimization under the complex

non-linear constraints. The constrained multi-objective optimization techniques are studied

in detail by Qu and Suganthan (2011a). They compared three constraint handling methods


2.4 Rubber Extruder Design Optimization 33

with the ensemble of those three constraint handling methods. They ensemble self-adaptive

penalty, superiority of feasible solution, and ε-constraint methods.

We developed a generalized constraint handling approach for population based EAs using

Boundary Inspection (BI) approach. The BI approach converts every infeasible member to a

feasible one during the evolution process. The algorithm attempts to move infeasible point in

a direction joining an infeasible point and a feasible point such that we reach within feasible

area. At every generation using this approach all infeasible members are converted to feasible

members by moving towards randomly selected feasible point. The parameter deciding the

location of the new point is used from a predefined pool of values based on its success history.

We use rubber extruder design optimization application to test the performance of proposed

algorithm along with benchmark test problems.

2.4 Rubber Extruder Design Optimization

Extruder is the machine that force rubber compound through a die under controlled conditions

of temperature and pressure, rate and homogeneity to give a continuous length of material

having the shape of the fitted die. Extrusion is used when the rubber compound is to be

shaped in continuous length of constant cross-section. The cross-section may be solid, hollow,

symmetrical or complex. It is in disputably the most important piece of equipment in polymer

processing industry including rubber plastic and food processing. Material to be extruded

can be in solid state or molten state. The extruder can be considered as a modular machine,

where all the components are interchangeable assembled in to a complete extruder to meet

the customer’s requirement. Extruders are widely used in the rubber industry in a verity of

applications. Extruders can be classified in different ways.

There are two basic types of extruders: continuous (screw extruder) and discontinuous

(batch type extruder/ram extruder). Continuous extruder utilizes a rotating member for

transport of material. Batch extruder generally has a reciprocating member to cause the

transport of the material. Extruders can be classified according to the feedstock temperature

necessary for successful operation. Second, it identified with type of application. First

category can be further classified as hot-feed and cold-feed extruder. To obtain high output



rate and good dimensional control by using an extruder for long runs for rubber compound

having a narrow range of flow properties, the screw, the head and the die design is crucial.

Selection of feed, haul off equipment and, control system is important for maintaining a good

dimensional control to accommodate minor variations in the feed materials. The cold-feed

extruder is advantageous compared to hot-feed extruder considering following points:

• Lower capital cost of equipment

• Reduced labor cost

• Good temperature control

• Efficient dimension control of Extrudate

• Capability for handling a wider range of rubber compound

Extrusion process is the technique of preforming unvulcanized rubber compounds by forc-

ing material through extruder dies, to gain desired shapes and sizes. For manufacturing of long

length of rubber products, extrusion process is widely used in rubber and polymer processing

industries. It is usually used to produce profiles such as window and door seals, tubes and tire

treads. Extrusion involves multiple complex phenomena, such as typical rheological behaviour

and fluid flow with free surfaces. The task of design engineer for extrusion process is to find

the screw and die geometry and the process conditions (flow rate and temperature) which

enable a stable flow of high precision and high quality extrudate profile. Five to six design

iterations are needed to set up a new extrusion line if similarity to an existing product can be

exploited, whereas ten to fifteen design iterations are needed for a totally new product.

Successful installations of an extrusion line is a very long, cumbersome and costly itera-

tive procedure. Additional aspect to be consider while designing the die is that, the wasted

rubber cannot be easily recycled. Therefore a prediction by means of numerical simulation

could dramatically improve the extrusion die design process. The associated problems are

complex and require an application of state of the art technologies to bring out the solution in

a reasonable time. The screw is a heart of extruder design, and it rotates inside a heated barrel.

The polymer compound flows through gravity in the hopper and progresses along the axis of



the helical screw due to wall friction forces. Conduction and dissipated of heat influences

the melting of compound near the inner barrel wall. The softened material creates a helical

recirculating path and assemble in a pool, segregated from the surviving solids. This fluid

uniformly mixed, pressurized and forced to get through the die, where shape is given before

being quenched. Modelling of the extrusion process is attained by sequentially connecting the

individual stages with appropriate boundary conditions. Each zone in extruder is described by

the mass conservation, momentum and energy balance equations together with the equation

representing the melt rheological behaviour.

The extrusion process consists of forcing a rubber compound by using screw extruder

through feeding channels and extrusion dies, which may have complex geometry. The chan-

nels are responsible to condition the flow of rubber compound parameters like velocity and

temperature. It also distribute the flow rate of different blends in the case of co-extrusion. The

role of extrusion die orifice is to produce a profile with the required geometry. The critical

part of extruder is designing a screw. There are two major approaches to the design and

optimization of extruder screws: analytical and experimental. The analytical approach uses

mathematical models and computer analysis. The experimental approach use either production

equipment or specialist small scale laboratory machines.

The extruder screw is one of the most crucial parts of a single screw extruder. The screw is

a threaded shaft, which lies co-axially and horizontally inside the barrel. It has, as a rule, right

hand threads with double helix starting point. The double thread distributes the compound

evenly about the axis and it is retained in the extruder for a shorter time than would be the case

with a single thread of the same pitch. It is connected with a motor and rotates anti-clockwise

and the compound moves forward along the flights. There are two types of the screws: torpedo

type and full flighted, with flat or pointed end. Torpedo type screw is so named because of

the torpedo like extension at the extrusion end. This extension consists of a section having

an outside diameter which is larger than the root diameter and length several times than

screw outside diameter. This ensures mechanical mixing, conducted heat distribution, and

controllable frictional heat. The torpedo type is mainly used in plastics. In full flighted screw,



the flight runs clear to the end. Flight pitches may be constant or variable. Design of a screw

depends on the extrusion rate, nature of die, material stock, etc. As the pressure of compound

at the discharge end is to maintain output, screw should have lower volume in flights at the

discharge end. There are four ways to achieve this :

1. by reducing the pitch of the screws.

2. by reducing the depth of the base of the screw.

3. by reducing the overall diameter of screw and barrel.

4. by increasing the number of starts in the screw.

Computer software are available which simulates the passage of the polymer material

through a single screw extruder. The simulation program calculate the melting rate, the melt

temperature, the mass flow rate and the power consumption using the mathematical model.

Developing a model which is accurate enough to represent the real process and same time easy

to solve is an art. Different modelling approaches for Rubber Extruder modelling are reviewed

in the next subsection. Relations of throughput power for rubber extruder are reviewed in

subsequent subsection.

2.4.1 Modelling of Rubber Extruder

Vera-Sorroche et al. (2014a) developed model for single screw extruder to study the effect

of polymer rheology on the thermal efficiency of the extrusion process. Authors studied the

effect of HDPE rheology and processing parameters on the thermal efficiency of the single

screw extrusion process. Variation in radial melt temperatures across the die flow path were

noticed to be dependent on the screw geometry, screw revolution speed, set temperature and

viscosity of polymer. Poorer temperature homogeneity and larger fluctuations were observed

for single flighted extruder screws compared to a barrier flighted screw with a spiral mixer.

Bulk temperature and the quantum of temperature variations increased with increasing melt

viscosity. Specific energy consumption was much dependent upon polymer melt viscosity.

Shin and White (2000) developed mathematical model for non-isothermal non-Newtonian

flow of rubber compounds in a pin barrel extruder. They observed that both shear thinning



behaviour and viscous dissipation induced non-isothermal behaviour reduce the pumping abil-

ity compared to an isothermal Newtonian fluid. Pin barrel extruder showed good agreement

with experimental results for the non-isothermal non-Newtonian behaviour for three rubber

compounds in the laboratory extruder; a passenger tire tread compound based upon SBR and

BR (PTT), a NR based truck tire tread compound (TTT), and an NBR based mechanical goods

compound (hose).

Wilczynski et al. (2018) conducted experimental and theoretical studies on the single-

screw extrusion of wood-plastics composites and developed computer model of single-screw

extrusion that considered solid conveying, melting based on the wood flour content, melt

flow in the screw, and melt flow in the die. They conducted experimental research on the

flow and melting of polypropylene based composites with selected wood flour content in the

single-screw extruder and using the experimental results developed elementary models of the

process. Integrating these elementary models developed a global model of the process. They

applied 3D non-Newtonian finite element method on screw pumping properties to model the

melt flow in the screw metering section. The model can be used to predict the extrusion output,

pressure and temperature profiles, melting profile, and power consumption. The proposed

model was successfully validated by experimental results. The pressure predicted by model

was observed to be little higher than the experimental values. They observed that the slip at the

screw and the die plays an crucial role in extruder operation. They noticed that when the slip at

screw/barrel surface increases, the extrusion output and pressure decrease. They also noticed

that when the slip at the die increases, the extrusion output increases and the pressure decreases.

Product quality and output rate are remarably impacted by screw speed and heating of the

barrel and screw in rubber extruder. Applying appropriate heating can increase throughput

with low material temperatures and sufficient thermal and material homogeneity. Overheating

of material may cause thermal degradation or vulcanisation of the rubber compound during

extrusion. Brockhaus and Schöppner (2015) studied flow behaviour and temperature patterns

within the screw channel using numerical flow simulations of non-isothermal shear-thinning

melt flows. They considered dissipative heating in the screw channel. They observed that the



specific throughput increased with rising screw temperature as well as rising barrel tempera-

ture. Though the screw temperature had a significantly greater influence on throughput. They

noticed that a high barrel temperature is not helpful for increasing throughput. A high barrel

temperature has a noticable influence on melt temperature than a screw temperature at the

same level. They concluded that a high throughput with low melt temperature can be attained

by a high screw temperature and low barrel temperature. Vignol et al. (2005) developed

simplified model for the estimation of mass flow rate and pressure at the exit of single-screw

extruder depending on the material properties and extruder operating conditions. The model

was developed using experimental data and predictions using FLOW 2000, a commercial

extrusion simulator. The one of the objective of the model is to get fast decision making

related to the extruder operating conditions during raw material changes. They noticed that

these comprehensive models are more lucrative than computational packages commercially

available and are enough accurate compared to conventional analytical equations, which do

not take into account solids conveying and non-Newtonian behaviour of the polymers.

Computer soft wares are available to simulate the passage of polymer material through a

single screw extruder. The simulation program computes the melting rate, the melt tempera-

ture, the mass flow rate and the power consumption. The mathematical model is developed

based on the assumption that there is no slip of the polymer melts at the walls of the screw

and barrel. It is also possible to experimentally evaluate the degree of slip occurring by the

help of the instrumented extruder. The ratio of the actual output of the screw divided by the

theoretical output, based on the screw geometry, is defined as a dimensionless parameter

which can be used to measure the wear of the screw. The larger the wear, the larger the

back-flow over the flight, reducing the screw efficiency. The maximum efficiency of the

screw, with no slip and no screw wear (Pure Drag Flow) gives the value of this dimensionless

parameter to be 0.5. Abeykoon et al. (2014) used FLOW 2000 to estimate the extruder total

power for the Polystyrene. They used the same processing conditions for the experiments and

simulations. They selected the frictional coefficients of material-barrel and material-screw

which resulted in good fitting between experimental and estimated mass throughput values

by trial and error procedure. Frictional coefficient of 0.43 for material-barrel and 0.2 for



material-screw gave best results. Screw can not be be designed without having all the thermal

and rheological properties of the rubber material to be processed. For the screw design to be

precise, the physical properties of the polymer must be known accurately. Lee’s Disk method

is the simplest method to estimate the thermal conductivity of the polymer. Generally 1 kg

of material is required to predict all the essential thermal and rheological properties of the

polymer, which are the input data for the simulation program.

Ha et al. (2008) analysed the liquid-state melted rubber flow near the die region during

the extrusion forming process of automobile weather strips using finite element thermal flow

analysis. They investigated flow velocity, temperature and pressure fields for flow of the

melted rubber material with respect to the inlet flow rate and the wall slip condition. They

used the power-law and Arrhenious-law models to represent the shear viscosity of the melted

rubber flow. They used the least-square fitting of experimental values for associated param-

eters. Lipár et al. (2013) developed a numerical model of an extruder using finite element

approximation, with the inputs being powers of heaters and output the extruder temperature

field. They developed the model using ANSYS Polyflow and validated experimentally. They

used this model as a plant for controller tuning and testing set points.

Ferretti and Montanari (2007) proposed a finite-difference method for solving the down

channel velocity in a single screw extruder for Newtonian fluids. They developed an effective

and easy procedure to obtain the velocity field. The model is user-friendly straightforward

and easy to apply for industrial and research applications. The model is implemented in MS

Excel which makes is more interesting and useful for analysis purposes. The tool is useful for

constructing the screw characteristics and the analysing the extruder performance. The authors

validated model comparing the predicted down channel velocity data with the analytical data

published in the open literature. Mesh size is the important parameter influencing the accu-

racy of the results. The tool is useful for the purposes of screw design and extrusion simulation.

Marschik et al. (2017) developed an mathematical relationship for computing the pumping

capability of power law fluids in three-dimensional screw channels under isothermal con-



ditions. They represented the three-dimensional extruder flow by four dimensionless input

variables: the height to width ratio, the pitch to diameter ratio of the screw channel, the

power law index and, the pressure gradient in the down channel direction. The approximated

mathematical results by a hands on optimization algorithm using symbolic regression based

on genetic programming. They observed high precision for the output pressure gradient

relationship. The analytical approximation allows fast and stable estimation of the three

dimensional flow characteristics for power law fluids in single screw extruder without using

mathematical methods. Due to the algebraic nature of the model equations developed, it can

be easily used with any computational calculation software. The heuristic model can also

be used for developing optimal design of extruder screws. The authors are extending the

approach for predicting the temperature development along the screw channel considering the

thermal effects.

Dong et al. (2012) developed a program using the incompressible smoothed particle hy-

drodynamics method to simulate the non-Newtonian fluid flow in the mixing section of a

single screw extruder. They modelled the transverse flow in the extruder by one 2D lid-driven

cavity flow and used the power law model for the viscosity of the fluid. They simulated shear-

thinning, shear-thickening and Newtonian fluid in the single screw extruder and analysed the

velocity profile along the centre of screw extruder comparing with the theoretical solution.

They noticed the method to be accurate and effective. The method can be extended for the

simulation of the complex systems using 3D modelling.

Abeykoon et al. (2011) used a thermocouple mesh procedure to evaluate the die melt

temperature profile of a single screw extruder. They presented a static non-linear polynomial

model to forecast the die melt temperature profile using the measured process parameters. The

model predictions using the proposed model were noticed to be in agreement with formerly

noted experimental discoveries. The authors used the model to evalaute the optimum process

settings to accomplish the chosen average die melt temperature, reducing melt temperature

variance across the melt flow. Abeykoon et al. (2014) proposed a method to model the die

melt temperature profile in polymer extrusion as a function of computable process variables



(screw speed and barrel set temperatures) under dynamic processing conditions. The authors

noticed the significant influence of screw speed and barrel set temperature on the extent of

melt temperature and temperature homogeneity across the melt flow. The metering zone

temperature are the most crucial for melt temperature and the temperature homogeneity of the

extruder melt output. Authors obtained very good accuracy in predictions over wide operating

range using the proposed dynamic model. The proposed model is simple in structure and

suitable for real-time applications to build-up a control strategy to achieve the appropriate

melt flow homogeneity in polymer extrusion by adjusting the process settings.

Ghoreishy et al. (2005) combined the continuous penalty finite element scheme as well as

generalized Newtonian rheological model to resolve the governing equations of continuity

and momentum in three-dimensional cartesian coordinate system. The proposed combination

resulted to a robust and reliable model for the simulation of the flow of the polymer melts in

polymer processing operations. The proposed approach has limitations for low temperatures

applications like in rubber processing machines because the viscoelastic effect becomes more

prominent in this case and the proposed approach cannot completely cope with flow conditions

encountered in such temperature ranges. Ghoreishy et al. (2000) used finite element method

to develop a mathematical model for the simulation of the flow of thermoplastic elastomer

through extrusion dies. Ignoring the slip of the polymer melt on the solid surface resulted in

a drastic error in the predicted flow rate. The authors used the Navier’s slip condition in a

cylindrical coordinate system to develop a model. The good agreement between the experi-

mentally obtained flow rates and calculated values was obtained confirming the applicability

of the proposed model. The proposed model was developed and validated for the generalised

Newtonian constitutive equations, which needs modifications for the viscoelastic behaviour.

Rauwendaal (2004) developed the FEA model based program for analysis of flow and

heat transfer inside extruders. The program can predict three dimensional velocity profiles in

screw extruders along with the pressure and temperature fields. They observed that high melt

temperature regions form in the middle of the channel when the viscous heat generation is

predominant. It can be attributed to the thermal convection created by the recirculating flow



pattern in the screw channel. The melt temperature non-uniformities become more crucial

with bigger extruders, high viscosities, and large screw speeds. The melt temperatures inside

the screw channel rises significantly as the flight clearance increases and the high temperature

region enlarges towards the root of the screw. As the residence times at the screw root are quite

long, rise in temperature in that region may cause serious consequences leading to degradation

of the product. The analysis of results illustrate that it is vital to make sure that the flight

clearance between screw and barrel is within acceptable limits. The radial flight clearance less

than 0.003D, where D is the screw diameter used in practice.

Gopalakrishna et al. (1992) carried our mathematical simulation using the transport

phenomena resulting in the flow of a non-Newtonian fluid through a single screw extruder.

They observed that significant viscous dissipation within the material can lead to 100% rise

of material temperature above the imposed barrel temperature. The bulk temperature at the

die exit can increases upto 200% for viscous dissipation. For higher reaction rates, moisture

elimination and bonding because of the reaction takes place first at the screw root. Sharp

changes in viscosity results due to bonding taking place in the vicinity from the inlet along the

extruder channel. Pressure development and throughput are strongly interconnected. Rheology

of material also plays significant role in pressure development; Non-Newtonian behaviour

decreases the pressure developed at the die. The bulk temperature rises continuously from

the inlet to the outlet. Cooling may be needed to control the rise in bulk temperature to

maintain quality of product. Authors also developed the process to estimate the residence time

distribution (RTD) using particle traces numerically. The results of RTD profile shows very

good match with the experimentally observed RTD data. The temperature profiles in the screw

channel for the flow of highly viscous fluid also shows good compliance with experimental

data.

2.4.2 Throughput Power relations for Rubber Extruder

Extrusion is an energy demanding polymer production process, hence the process energy

efficacy become a key matter. Abeykoon et al. (2016) investigated the pattern of energy

usage and losses of each component in the extrusion for process energy optimization. The



focus of the study was to improving the energy efficiency in polymer processing maintaining

quality. They investigated the total energy utilization, drive motor energy utilization, power

factor and the melt temperature profile across the die melt flow of an large scale extruder with

three diverse screw geometries, three different polymer types and a wide range of processing

conditions. The results were in the accordance with the earlier findings such as: decrease

of the extruder specific energy consumption with screw speed and increase of melt thermal

fluctuations with the screw speed. They observed that the level and fluctuations of the extruder

power factor is based upon the material being processed. The level and magnitude of the

fluctuations of the extruder power factor depletes with the screw speed. These parameters are

dependent upon the polymer type, screw geometry and processing conditions. They noted that

the extruder specific energy consumption reduces with increasing screw speed, while specific

energy consumption of the drive motor may have either increasing or decreasing behaviour.

The energy demand by the heaters vary with the processing conditions. They observed the

link between the extent of energy demand from the heaters and the melt thermal fluctuations,

higher the energy demand of the heaters the higher the melt thermal fluctuations.

Abeykoon et al. (2014) investigated the total energy demand of an extrusion plant under

distinct processing conditions describing ways to optimize the energy efficacy. They carried

out detailed analysis for modelling of the energy utilization in polymer extrusion. Authors

experimentally observed the mass throughput, total energy utilization and power factor of an

extruder over varied processing conditions and developed empirical model for total extruder

energy demand using commercially available extrusion simulation software along with experi-

mental results. They observed that the extruder energy demand is linked with the machine,

material and process variables. The total power predicted by the simulation software was

lagging with an offset compared to the experimental results. Empirical models were observed

to be well agreement with the experimental measurements, which authors used in studying

process energy behaviour in detail and to identify ways to optimise the process energy efficacy.

They noticed that the screw geometry plays significant role in identifying energy requirement

depending upon the material being processed. It was also noticed that running the processes

at high speeds with a high power factor can obtain an improved process energy efficacy. The



optimum process operating point can be chosen using energy and thermal efficacies.

Vera-Sorroche et al. (2013) studied energy consumption per unit mass high density

polyethylene and noticed that the energy consumption is highly reliant upon throughput

but relatively independent of set temperature or screw geometry. Polymer extrusion is an

energy intensive process, hence it has high potential of optimization. Generally extrusion

process are often operated at suboptimal conditions. Extruder screw geometry and extrusion

variables should optimised to match the properties of individual polymers, but in practice

this is rarely achieved due to the lack of knowledge about the process. Extruder screw

design, screw speed and set temperature were observed to have a noticable effect on the

thermal homogeneity of the melt and process energy consumed. Lowest energy consumption

were required at the high extruder throughputs for investigated screw geometries and melt

temperature homogeneity was linked with extruder throughput and extruder screw geom-

etry. Barrier flighted screw with a spiral mixer exhibited better temperature homogeneity

and smaller fluctuations than single flighted extruder screws, but the barrier flighted screw

exhibited higher melt pressure fluctuation. The analysis of the results show that the single

screw extruders should be operated at the highest possible throughput to maximise efficiency.

The screw geometry may be chosen to optimise melt homogeneity. Vera-Sorroche et al.

(2014b) studied the the influence of HDPE rheology and processing parameters on thermal ef-

ficiency of the single screw extrusion process using an extruder. These result analysis reflected

that the rheological properties of the polymer significantly impact the thermal efficacy of the

extrusion process along with extruder screw design, set extrusion temperature and screw speed.

Lawal and Kalyon (1994) developed mathematical model for flows behaviour of viscoplas-

tic fluids exposed to different slip coefficients at the barrel and screw surfaces. Viscoplastic

fluids like plastic, composites, rubber and elastomer, and energetics industries, exhibit wall

slip, which can be manipulated by appropriate choice of construction materials, surface rough-

ness, and grooves. Barrel and screw surfaces can be conditioned to generate different slip

coefficients. The analysis reflected that the pressurization ability of the extruder reduces

with the increasing ratio of slip coefficient at barrel surface over screw surface. The study


2.5 Rheology of Rubber 45

indicated that over certain operating condition range the presence of wall slip at the screw

surface with no slip at the barrel surface will result into increased extrusion production rates.

The mixing ability of the extruder degrade with increasing slip ratio. The mathematical

model developed can be used to apply design expressions for viscoplastic fluids with known

properties, including wall slip coefficients. It can also be used to find out experimentally the

wall slip coefficient ratios using the extrusion hardware. The model can be used to introduce

improvements in engineering, design and optimization of extrusion lines.

Rheology of polymers plays very important role in design and simulation of extruder.

Screw design, barrel heating/cooling, power consumption and throughput are highly inter-

connected and strongly influenced by rhelogy, hence rubber rheology is reviewed in next

section.

2.5 Rheology of Rubber

Elastomers have the elastic properties of both an elastic solid and a viscous fluid. The

behaviour of viscoelastic materials can be expressed using Hooke’s law of elasticity, which

is appropriate for the linear behaviour of elastic solids, and Newton’s law of viscosity is

appropriate to the linear behaviour of viscous liquids. Robert Hooke was the main person to

observe a connection between force and deflection in linear elastic solids in 1678. He simply

stated that the force, F , is linearly proportional to the deflection, Dx which is written as

F = kDx (2.6)

where k is proportionality constant(spring constant), also named as the stiffness. Leonhard

Euler modified Hooke’s conception in 1727, who defined the force in terms of stress, F/A,

and the displacement in terms of strain, Dx/h, where h stands for the original length. The

relationship is expressed as:

F/A = G(Dx/h) (2.7)

In terms of stress and strain it can written as:



τxy = G(γxy) (2.8)

where τxy is the shear stress and γxy the corresponding shear strain.

Newton’s law of viscosity can be represented in the form:

F = c(dxdt

) (2.9)

where c is a viscous damping coefficient. It can be represented in terms of stress and strain

as follows:

τ = ηdedt

(2.10)

Where η is viscosity.

Newton and Hooke suggested their fluid and solid models. After a gap of almost two

centuries in 1867 James Clerk Maxwell attempted to model the behaviour of a body that

has combination of a viscous and an elastic force element during deformation. James Clerk

Maxwell published his article titled "On the Dynamical Theory of Gases" Maxwell (1867),

where, he introduced a model for a system that has both, elastic and viscous effects. The

model is based on the reality that when a stress τxy is imposed on the system, this stress is

identical for the fluid as well as solid elements, and the total strain is the sum of the elastic

strain, γηxy , and the viscous strain, γG

xy such that τxy = τηxy = τG

xy.

Maxwell’s linear differential equation given by:

dτxy

dt= G

dγxy

dt−

τxy

λ(2.11)

where λ = G/η , is relaxation time.

Maxwell’s linear differential equation can be solved for strain for constant stress as follows:

γxy =τxy

ηt +

τxy

G(2.12)

In the case of constant stress the material component experiences an instant deflection,

due to its elastic component, and continues to deform at a constant rate, due to its viscous



element. The continuous flow occurrence by the material under constant load is known as

creep. When the load is released, the accumulated elastic deformation is recovered but the

viscous deformation remains. Maxwell’s linear differential equation can be solved for stress

as follows:

τxy = Gγxy e−

tτ (2.13)

The fluids that does not behave in a Newtonian fashion between shear stress and shear

rate when it undergoes deformation is known as non-Newtonian. The relation between shear

stress and shear is non-linear. Polymer melts and polymers solutions and liquids in which fine

particles are suspended, are usually non-Newtonian. When viscosity decreases with increasing

shear rate, the fluid is called shear-thinning, whereas when the viscosity increases as the fluid is

subjected to a higher shear rate, the fluid is called shear-thickening. Shear-thinning fluids also

are called pseudoplastic fluids and shear-thickening fluids are called dilatants. Viscoplastic

is an another type of non-Newtonian fluid that will not flow when only a small shear stress

is applied. The shear stress must exceed a critical value (yield stress) for the fluid to start

flowing. They behave like solids when the applied shear stress is less than the yield stress.

Once it exceeds the yield stress, they flow just like an ordinary fluid. Another type of fluids

exhibit time-dependent behaviour, the viscosity vary with time at constant shear rate. This

includes both thixotropic and rheopectic fluids. The viscosity of a thixotropic liquid decreases

with time under a constant applied shear stress. However, when the stress is removed, the

viscosity will gradually recover with time. The rheopectic fluid viscosity increases with time

when a constant shear stress is applied.

Viscoelasticity is the property of a material to demonstrate both viscous and elastic prop-

erties under the same conditions when it undergoes deformation. Viscous materials present

resistance to shear flow and strain linearly with time when a stress is applied. The shear stress

of these materials depends on strain: when strain is applied and then released, they return

to their initial configuration. Some common and well-known viscoelastic materials include

paint, blood, ketchup, honey, mayonnaise, polymer melt, polymer solution and suspension,

shampoo, and corn starch.



Because of the molecular structure, polymers are the most complex fluids. Complex exper-

imental and mathematical exercise is required to model rheological behaviour of polymers.

Polymers are composed of macromolecules, and these large molecules have the ability to

slide past each other, hindered only by intramolecular forces and molecular entanglements.

Polymers tend to relax stresses that arise when subjected to a deformation. When a mass of

polymer is subjected to a stress, the molecules tend to move in an effort to relax those stresses.

If these stresses are caused by a constant strain, the initial stress by this deformation relaxes in

a given time interval, referred as relaxation time. Rheologists use the stress relaxation test to

interpret the viscoelastic behaviour of polymers.

Most polymers exhibit shear thinning, temperature and pressure dependent viscosities

(Osswald and Rudolph, 2014). The shear thinning effect is defined as the reduction in viscosity

at high rates of deformation. This phenomenon is explained by the fact that the molecular

chains are disentangled and stretched out at high rates of deformation and can therefore slide

past each other with more ease, which in turn lowers the bulk viscosity. To take these non-

Newtonian effects into consideration neglecting the viscoelastic effects, viscosity is defined as

a function of the strain rate and temperature as follows,

τ = η (γ, T ) γ (2.14)

Where, γ is the strain rate. This relationship is also known as generalized Newtonian fluid

model. Several models that comply with the generalized Newtonian fluid assumptions have

been proposed in the literature. They differ in their form and in the number of parameters

required to fit them to experimental results. These models are developed to obtain analytical

solutions for different flow scenarios encountered in polymer processing, and to allow storage

of the measured data with a minimum number of parameters (Tadmor and Gogos, 2006). The

flow behaviour of different fluids requires usage of different models; some fluids may be shear

thinning, others may be fluids that experience a yield stress and exhibit both behaviours. The

model is selected such that it best fits the measured viscosity data and at the same time is

appropriate for the specific application (process) and type of flow. Complex models that better



represent the rheological behaviour of the polymer can add significant difficulty to the analysis

of a flow field. Hence, it is very important to balance between the complexity of model and its

capacity to closely represent the experimental data. Some of the models used to represent the

viscosity of industrial polymers are presented here.

The Power Law model proposed by Ostwald and Auerbach (1926) is a very simple model

that accurately represents the shear thinning region in the viscosity versus shear rate curve,

but neglects the Newtonian plateau observed at small strain rates. The Power Law model

can describe the data of shear-thinning and shear thickening fluids. The Power Law model is

represented as:

η = m(T ) γn−1 (2.15)

where m is consistency index and n is the Power Law index. The Power Law index

represents the shear thinning behaviour of the polymer melt for n < 1. The Power Law index

n = 1 represents Newtonian behaviour and n = 0 represents a plug flow. The consistency

index may include the temperature dependence of the viscosity. The temperature dependence

of the consistency index can be considered using the following relation:

m(T ) = m0 e−a(T−T0) (2.16)

where, a is the sensitivity parameter representing the temperature dependence. Due to

its simplicity and capacity to represent wide range of polymers and foods, it is very popular

and commonly used model for computational applications. Although the power law model is

popular and useful, its empirical nature should be accounted while using it. One of the reasons

for its popularity is its applicability over the wide shear rate range (101−104 s−1) obtained

with many commercial viscometers. One limitation of the power law model is that it does not

describe the low-shear and high-shear rate constant-viscosity data of shear-thinning materials.

The Bingham Model is a two-parameter empirical model that is applicable to the materials

that exhibit yield stresses τ0, below which the material does not flow. Polymer emulsions and



slurries are common examples of Bingham fluids. Bingham fluid behaves like a Newtonian

liquid above the yield stress and can therefore be represented as follows:

η =∝ (or γ = 0) for τ ≤ τ0

η = µ0 +τ0

γfor τ > τ0

(2.17)

where, µ0 is the Newtonian viscosity for overcoming yield stress. The model indicated

that a critical level of stress must be attained for flow to initiate.

The Herschel-Bulkley Model is used to represent the behaviour of fluids that have a yield

stress like a Bingham fluid, but otherwise exhibit shear thinning behaviour. The model is

represented as:

τ = τ0 +m γn for τ ≤ τ0

η =τ0

γ+m γn−1 for τ > τ0

(2.18)

where, µ0 is the Newtonian viscosity for overcoming yield stress, m is the consistency

index, and n is the Power Law index. Like Bingham model, Herschel-Bulkley model also

demand that a critical level of stress must be achieved to initiate flow, below which the material

behaves like a solid. When the stress applied is more than the critical stress value, the material

behaves like a Power Law fluid. Like the Power Law model, n < 1 shows shear thinning, n > 1

shear thickening, and n = 1 brings the model to the Bingham and Newtonian flow above the

critical yield stress.

The Bird-Carreau-Yasuda Model developed by Bird and Carreau (1968), Carreau (1972)

and Yasuda et al. (1981) that accounts for the observed Newtonian plateaus and fits a wide

range of strain rates. It contains five parameters:

ηγ −η∝

η0−η∝= (1+ |λ γ|a)

(n−1)a (2.19)

where, η0 is the zero shear viscosity, η∝ is an infinite shear rate viscosity, λ is a time

constant, and n is the Power Law index. The parameter a accounts for the width of the



transition region between the zero shear viscosity and the Power Law region. Neglecting the

infinite shear rate viscosity the model reduces to a three parameter model as follows:

η(γ) =η0

(1+ |λ γ|a)(n−1)

a

(2.20)

The Cross-WLF model is a six parameter model which considers the effects of shear

rate and temperature on the viscosity. Like the Bird-Carreau-Yasuda model, this model also

describes both Newtonian and shear thinning behaviour. The shear thinning part is modelled

by the general Cross equation (Cross, 1965). The Cross-WLF model is the most used model

by injection moulding simulation software, because it offers the best fit to most viscosity

data (Hieber and Chiang, 1992). The cross model was popular and earlier alternative to the

Bird-Carreau-Yasuda model. The Cross-WLF model is represented as:

ηγ −η∝

η0−η∝=

1

1+(K γ)1−n (2.21)

where, η0 is the zero shear viscosity, η∝ is infinite shear viscosity, K is time constant, and

n is the Power Law index. The Cross model reduces to Power Law model for ηγ << η0 and

ηγ >> η∝.

Neglecting the infinite shear viscosity, the Cross model can be written as:

η(γ) =1

1+(K γ)1−n (2.22)

The zero shear viscosity is modelled with the WLF equation:

η0(T ) = D1 exp[

A1(T −D2)

A2 +T −D2

](2.23)

where, D1 is the viscosity at a reference temperature; D2 , A1 and A2 are the temperature

dependency parameters.

There are several viscosity models available which can represent the flow behaviour of

polymers. Among all the available models, Modified Cross and Carreau-Yasuda models are

the most popular and widely used models. Bansal et al. (2013) compared these two models on



the basis of percentage die swell in the extruded profile and concluded that the Carreau-Yasuda

viscosity model predicted the extrudate swell better than Modified Cross model.

2.5.1 Effect of Temperature on Rheology of Rubber

The temperature dependence of the viscosity can be expressed as a function of shear rate

written as:

η (T, γ) = f (T )η(γ) (2.24)

The function f (T ) for small variations in temperature can be approximated using an

exponential function as follows:

f (T ) = exp [−a(T −T0)] (2.25)

where, a is the temperature sensitivity of the viscosity, T is the temperature at which the

viscosity is to be calculated, and T0 is the reference temperature at which the viscosity is

known.

There are Arrhenius shift and the WLF shift models that can also be used to account for

temperature dependence of viscosity. The Arrhenius shift model for semi-crystalline polymers

can be written as follows:

aT (T ) =η0(T )η0(T0)

= exp[

E0

R

(1T− 1

T0

)](2.26)

where, E0 is the activation energy, T0 is the reference temperature, and R is the gas

constant. Using this model, the measured viscosity at different temperatures can be used

to generate a master curve at a required temperature. This model is valid for temperatures

T > Tg + 100 K, below which the free volume effects dominate the behaviour. For lower

temperatures the dependence of the viscosity of amorphous thermoplastics is best described

by the Williams-Landel-Ferry (WLF) model (Williams et al., 1955; Ferry, 1980) as follows:

logaT (T ) = logη0(T )η0(T0)

=−C1(T −Ts)

C2 +T −Ts(2.27)



where, η is the viscosity of the polymer at any given temperature T in relation to a

reference viscosity at a reference temperature, Ts. This equation holds true only in the

zero shear viscosity region for polymers. Tg is much lower than the polymer processing

temperatures, hence Van Krevelen and Te Nijenhuis (2009) proposed a better alternative for

Ts using, Ts = Tg +43 K. Generally the viscosity is not known at the reference temperature

Ts, but known at a temperature in the processing temperature range T ∗. Hence, a second

shift between measurement or processing temperature T ∗ and the reference temperature Ts is

required when we use this model.

2.5.2 Effect of Pressure on Rheology of Rubber

The effect of pressure on viscosity is well known and can be incorporated into existing models.

The Power Law model can be used with pressure sensitivity factor b, proposed by Barus

(1893). The pressure shift factor is defined as follows:

ap(p) =η0(p)η0(p0)

= [b(p− p0)] (2.28)

The power law model accounting for temperature and pressure variation will be as follows:

η(T, γ, p) = m0 · exp[−a(T −T0)] · exp[b(p− p0)] · γn−1 (2.29)

The opposing signs of the temperature sensitivity factor a, and pressure sensitivity factors

b reflect that the viscosity increases with decreasing temperature and increasing pressure.

Cogswell (1981) related a viscosity change to a change in density. He noted that a temperature

reduction and an increase in pressure will increase both density and viscosity. Based on that

assumption, WLF model is developed using WLF-temperature shift in combination with the

Bird-Carreau-Yasuda model. The zero shear viscosity is modelled with the WLF equation :

η0(T, p) = D1 exp[

A1(T −Tc)

A2 +T −Tc

](2.30)

where, T c = D2 +D3 · p and A2 = A2 +D3 · p.



2.6 Summery

Evolutionary multi-objective optimization (EMOO) is an important and useful field of research.

Developing algorithms to solve real-life multi-objective problems is one of the core area of

research in engineering optimization field. During the World Congress of Computational

Intelligence (WCCI) in Vancouver 2006, EMOO has been evaluated as one of the three fastest

growing fields of research and application amongst all computational intelligence fields. The

Evolutionary Optimization (EO) algorithms use a population-based approach in which, the

iterations are performed on a set of solutions (called population) and, more than one solution

is generated at each iteration. The main reasons for the popularity of EO algorithms are as

follows: (i) They do not require any derivative information; (ii) EO algorithms are relatively

simple to implement; (iii) EO algorithms are flexible and robust, i.e. they perform very well

on a wide spectrum of problems. The use of a population in EO algorithms has a number of

advantages: (i) it provides an EO procedure with a parallel processing power, (ii) it allows EO

procedures to find multiple optimal solutions, thereby, facilitating the solution of multi-modal

and multi-objective optimization problems and, (iii) it provides an EO algorithm with the

ability to normalize decision variables (as well as objective and constraint functions) within

an evolving population using the best minimum and maximum values in the population.

The shortcoming of working with a population of solutions is the computational cost and

the memory necessary for the execution of the iterations. EO are computationally expensive

hence, there is always a need for improvement of the computational efficiency of the algorithm.

Most engineering applications are multi-objective and constrained in nature hence, an efficient

constraint handling mechanism is also an important part of the algorithm along with compu-

tational efficiency. Due to the limitations of algorithms, engineering optimization problems

are sometimes solved as single objective optimization problems fixing the preferences before

optimization. Multi-objective optimization solutions gives a full spectrum of solutions to the

decision maker, which can be used to select one best solution knowing the compromise made

in selection. Literature review clearly reflects that very few applications in Rubber technology

have been optimized as multi-objective optimization problem. Rubber extruder screw design

is an important and complex optimization problem. Optimum design with minimum energy


2.6 Summery 55

consumption and maximum throughput can be obtained by formulation and solving rubber

extruder screw design problem as multi-objective optimization formulation.


57

Chapter 3

Parallel Universe Alien Genetic

Algorithm (PUALGA) for

Multi-Objective Optimization

Evolutionary Algorithms (EAs) are becoming the most proven method for Global optimization

of complex problems. Design of EAs allows implementation of bound constraints naturally,

but they are often criticised for their exhaustive computational requirements and limitations

for handling constraint functions. Even with the developments in the computational powers

of computers, solving the complex multi-objective problem requires very long time. There

is always a need for development of robust and computationally efficient EAs for large

and complex problems. Hybridization is one approach used to improve performance of

EAs. Two EAs are hybridized using binary and real coded sub-populations. Search space

exploration capability of binary coded GA is explored, combining it with real coded GA.

Though different EAs can be used for hybridization; GA is used explicitly for both sub-

population evolution. Sub-populations exchange information through Aliens from binary

population to real population. This concept implemented in GA framework is presented as

Parallel Universe Alien GA (PUALGA). The proposed algorithm is tested using benchmark

multi-objective test problems and statical analysis of result are presented. The results obtained

are compared with jumping gene adaptation in GA and native GAs used in hybrid, which

show consistent improvement in performance of proposed algorithm.

58 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization

3.1 Introduction

Most engineering optimization problems are complex in nature and multi-objective. Multi-

objective optimization (MOO) is a group which handles multiple and conflicting objectives

simultaneously. When objectives are conflicting, achieving the optimum for one objective

requires some compromise on one or more other objectives. The relevance and importance of

MOO is increasing due to increasing complexities in the design and operation of processes.

MOO problems with conflicting objectives will have a set of solutions (representing trade-offs

among the objectives), which are called Pareto optimal solutions (non-inferior solutions), of

which none can be considered better than the others with respect to all objectives (Steuer,

1989). There are two main goals in MOO: (a) to find a set of solutions as close as possible to

the true optimal Pareto front and, (b) to find a set of solutions as diverse as possible. First goal

is common for any optimization problem whereas second goal is specific to multi-objective

optimization problem.

Schaffer (1985) was the first to implement a real multi-objective evolutionary algorithm

(vector-evaluated GA - VEGA). Schaffer reconstructed the simple genetic algorithm (with

selection, crossover, and mutation) by designing independent selection cycles according to

each objective. No remarkable study was observed for almost a decade after the initial work of

Schaffer, till a non-dominated sorting procedure was developed by Goldberg (1989). Goldberg

proposed to apply the concept of domination to keep more copies to non-dominated individu-

als in a population. Since diversity is of further concern, he proposed the use of a nitching

policy among solutions of a non-dominated group. Getting this clue, researchers developed

different versions of multi-objective evolutionary algorithms. Basically, these algorithms

differ in the way fitness is assigned to each individual. Srinivas and Deb (1994) developed a

non-dominated sorting GA (NSGA) which became very popular. The Non-dominated Sorting

Genetic Algorithm-II (NSGA-II) was presented as an upgrade of NSGA Deb et al. (2002).

Kasat and Gupta (2003) developed Jumping Genes (JG) to the binary coded NSGA-II for

multi-objective optimization. Guria et al. (2005) refined binary coded jumping gene (JG)

using fixed length of JG, named as adapted jumping gene(aJG), which is one of the algorithm

used for comparison with the proposed algorithm. NSGA-II is computationally productived


3.1 Introduction 59

than the former algorithms, and its performance is much better, so it acquired popularity

and became a benchmark multi objective evolutionary algorithms (MOEAs). A survey and

concise information for MOEAs is presented by Li et al. (2015a), Zhou et al. (2011) and

Van Veldhuizen and Lamont (2000).

Evolutionary algorithms (EAs) have been widely accepted for solving several practical

optimization applications in engineering. However, they are often criticized for the large

computational time as well as for their inefficiency to handle the constraints. This motivates

for the hybridization of EAs with the other optimization algorithms. Since there are several

local search optimization algorithms which overcome the two above mentioned problems

with the EAs, their hybridization adds significant value to the EAs (Krasnogor and Smith,

2005). Several reported literature can be found for the successful applications of such hybrid

approach for solving complex optimization problems Nabil (2016) and Mohamed (2015).

Sankararao and Gupta (2007) observed that for ZDT4 test function, binary coded NSGA-II do

not converge to true pareto front. They noted that "It may be mentioned here that though the

binary coded NSGA-II fails to converge to the global optimal solution, for this test problem,

the real coded NSGA-II does indeed, converge to the correct pareto solutions in 100,000

function evaluations". This observation initiated the thought of having two parallel populations

evolving simultaneously, one binary coded and another, real coded; which can contribute

to have robust evolutionary algorithm. Zhou et al. (2011) highlights that competitive/co-

operative co-evolution can be an approach to improve convergence and robustness for MOO.

This supported our hypothesis of using two parallel populations. Patel and Padhiyar (2010)

explored the concept of Alien in their work, which is proposed to be used here to exchange

information between the parallel populations.

Hybridization of binary coded and real coded GA is proposed to be used in this work.

Binary coded GA can explore search space reducing the accuracy of encoding. Two par-

allel populations are created and evolved exchanging information. Members from binary

coded population go to real coded populations as Aliens and take part in evolution. This

approach combines the strengths of binary coded and real coded GA along with benefits



of parallel populations. Non-dominated sorting and crowding distance is used for selection.

The proposed algorithm followed by brief discussion about native binary and real coded GA

is discussed in next section. Following that, test functions and performance measures used

for performance evaluation are also discussed. The results are discussed in the next section

followed by conclusion of the work.

3.2 Binary and Real coded GA

Genetic Algorithm (GA) is a popular population based stochastic optimization technique for

single and multi-objective optimization problems. Though, it is a computationally costly

algorithm in comparision with the gradient based algorithms, it is a preferred tool for complex

problems and off-line investigation because of its capability of delivering possible global

optimum solution. There are two types of GA, binary coded, where the values of population

members are encoded as binaries (0, 1), or real coded, where the population members real

values are directly or indirectly used. The GA has four basic steps,for both real coded or

binary coded: initiation, selection, mutation and, crossover. GA starts with a initial popula-

tion generated randomly using the defined range of all the decision variables and uniformly

distributed over the entire solution space. This intial population of multiple members is then

passes through recombination and/or mutation to contribute diversity in the population to

get a better offspring(child). At each stage of GA, mutation and crossover operations are

executed to add diversity in the population members Deb (2001). This new child population

has the potential to generate better fitness value compared to the parent generation. The best

members from these two generations; parent and offspring survive to the next generations.

This completed one generation of evolution. The new generation then go through the same

crossover, mutation and selection processes. GAs are criticised for their slow convergence

rate but are appreciated for their capacity to handle complex problems.

There are modifications proposed in GAs for solving optimization problems to handle

the issue of slow convergence rate and to improve the probability of converging to the global

optimal solution. The conventionally used methods for the selection of population members


3.2 Binary and Real coded GA 61

are: roulette wheel, the tournament selection, stochastic remainder roulette wheel, the stochas-

tic universal sampling and, rank selection. A concise study of different selection methods

has been presented by Goldberg and Deb (1991). There are many approches for crossover

operation in binary coded GA: single-point, two-point, multi-point, and uniform crossover.

These binary crossover strategies are investigated in detail by Wu and Chow (1995). There

exists crossover operators like: linear, naive, blend, simulated binary, fuzzy recombination,

unimodal normally distributed, simplex and unfair average crossover for real coded GA (Deb,

2001). Mixed crossover operators are investigated in detail by Hasancebi and Erbatur (2000)

and Zaharie (2009). Overview for real parameter crossover and mutation operators is presented

by Herrera et al. (1998).

Along with crossover, mutation is also a crucial step in GA to add diversity in population.

Mutation supports GA to overcome the issue of getting trapped into a local minimum. Con-

ventinally used mutation techniques are: non-uniform, normally distributed and polynomial

mutation. There are mutation techniques presented and investigated in the literature for their

effect on diversity of population for convergence to the global optimum solution Grefenstette

(1986); Deep and Thakur (2007). The influence of starting population on convergence rate

and global solution is investigated by Goldberg et al. (1992) and Harik et al. (1999). The

crossover and mutation operators contribute diversity in the population resulting to the high

probability of convergence to global optimum solution. The selection method leads the GA to

achieve better convergence.

There have been noticable work to improve the Basic GAs in the literature. Kasat and

Gupta (2003) developed the concept of Jumping Genes (JG) to the binary coded NSGA-II

for multi-objective optimization. The basic feature of the proposed mechanism is that it

consists of a simple operation, where a transposition of gene(s) is actuated within the same

or a different chromosome of the GA population. Guria et al. (2005) enhanced binary coded

Jumping Gene (JG) using the fixed length of JG, denoted as adapted Jumping Gene(aJG). Jung

(2009) suggested a GA with a selective mutation policy based on the ranking of population

members. Specific groups with high and low ranked members are focused for selective



mutation for obtaining improved convergence. Modified DNA genetic algorithm presented by

Zhang and Wang (2013) uses DNA encoding, choose crossover and, frame-shift mutation for

parameter prediction for an oxidation process. Musharavati and Hamouda (2011) developed a

modified GA with a combination of a neighborhood search based mutation operator and an

additional threshold operator for neglecting untimely convergence. Patel and Padhiyar (2015)

modified genetic algorithm using Box-Complex method to improve convergence. There have

been numerous attempts to improve the Basic GAs in the literature. Pandey et al. (2014) has

presented an exhaustive review on different approaches implemented to prevent premature

convergence with their strengths and weaknesses.

Though there are many tuning parameters in GAs, determining proper values of parameters

is crucial. Selecting a very small population size increases the risk of converging to a local

minimum, where as, a larger population has more chance of finding the global optimum at the

expense of more CPU time. Same way accuracy of encoding decision variables plays crucial

role in binary coded GA. Choosing shorter chromosomes has more probability of exploring

search space during initialization and evolution (crossover and mutation) stages. Two parallel

populations, one binary coded and one real coded are proposed in this work. Binary coded

population is exploited to scan search space using shorter binary chromosome lengths and real

population is explored for convergence with desired accuracy. The binary population takes

care of global searching and supports the real population to escape any local optima. The

proposed binary real coded hybrid algorithm to explore search space is presented as Parallel

Universe Alien GA in the next section. It uses the concept of Alien members for information

exchange between these binary and real coded populations.

3.3 Non-dominated Sorting GA

A constrained multi-objective optimization problem consists of more than one conflicting

objective functions, as well as, a finite number of equality and/or inequality constraints. The

MOO problem can mathematically be defined for x ∈ Rn as follows,


3.3 Non-dominated Sorting GA 63

Min/Max fm(x) m = 1,2,3...,M;

Subject to g j(x)≥ 0 j = 1,2,3...,J;

hk(x) = 0 k = 1,2,3...,K;

x(L)i ≤ xi ≤ x(U)i i = 1,2,3...,n.

(3.1)

The problem has M objective functions with J inequality constraints, K equality constraints

and bounds on decision variables.

Goldberg (1989) suggested moving the population toward true pareto front by using a

selection mechanism that favours solutions which are non-dominated. He used fitness sharing

and nitching as a diversity maintenance mechanism. Srinivas and Deb (1994) proposed an

algorithm using non-dominated sorting proposed by Goldberg and called it the Non-dominated

Sorting Genetic Algorithm (NSGA). The NSGA algorithm is based on several layers of classi-

fications of the individuals as suggested by Goldberg (1989). Before selection is performed,

the population is ranked on the basis of non-domination, where, all non-dominated individuals

are classified into one category with a dummy fitness value, called the rank of pareto front.

Then, this group of classified individuals is ignored and, another layer of non-dominated

individuals is considered. The process continues until all individuals in the population are

classified. The process of ranking based on non-dominated sorting is illustrated in Fig. (3.1)

The disadvantage of the rank based non-dominated sorting selection mechanism is that, all

members of the first p pareto fronts have the same fitness values as far as survival selection

is concerned. Thus, the population members closely located to one another on those pareto

fronts will be selected in survival selection step. This phenomenon can hamper the diversity

of the population and hence hamper the convergence to the global optima. The NSGA is

an inefficient algorithm because of the way in which it classifies individuals. Further, the

single parameter fitness selection techniques such as roulette wheel and stochastic universal

sampling cannot be applied in NSGA framework, as only single fitness parameter is required in

these selection operators. Authors used adopted Stochastic remainder proportionate selection

for this algorithm. The pseudo code for NSGA and the algorithm is presented in Appendix A.2.



Figure 3.1. Ranking of population using non-dominated sorting

Deb et al. (2002) modified the NSGA algorithm to enhance the computational performance

of algorithm. The proposed NSGA-II algorithm outperformed and became the benchmark

MOEA. They use elitism and a crowded comparison operator that ranks the population based

on both pareto dominance and region density. This crowded comparison operator makes

the NSGA-II considerably faster producing very good results. In NSGA-II algorithm all

the population members of the previous and current generations are grouped into different

ranks of pareto fronts, with 1st rank pareto front nearest to the true pareto front. All the

members of the first p ranks of the pareto fronts are chosen for the next generation, if the sum

total number of members of the p pareto fronts are less than or equal to the population size.

Thus, these all members have the identical fitness values for survival selection. The rest of

the population members are chosen from the next higher rank pareto front based upon the

crowding distance. NSGA-II uses the crowding distance in the selection operator to keep a

diverse front by making sure that each member stays a crowding distance apart. The crowding

distance calculation for members in a non dominated pareto front is illustrated in Fig. (3.2).

The process of crowding distance calculation has following steps:

• Rearrange all members in the front in ascending order of the values of any one of the


3.4 Jumping Gene GA 65

Figure 3.2. Crowding distance calculation for population member in a pareto front (Source: Deb (2001))

objective function (fitness function).

• Find the largest cuboid (rectangle for two fitness functions) enclosing member i that just

touches its nearest neighbours in the objective function space.

• Crowding distance is half the sum of all sides of this cuboid.

• Large values are assigned to solutions at the boundaries (choice of this large value may

influence the convergence characteristics).

The selection based on non-dominated ranking and crowding distance keeps the population

diverse and helps the algorithm to explore the whole fitness landscape. The pseudo code of

the NSGA-II algorithm is shown in Appendix A.3. NSGA-II algorithm is used to compare

the performance of the proposed PUALGA. The NSGA-II algorithm can be implemented for

both, binary or real encoding, where in, binary encoded NSGA-II is used for performance

comparison.

3.4 Jumping Gene GA

NSGA-II uses the concept of elitism borrowed from nature, the better parents gets the chance

to take part in producing the next generation. In this algorithm, the diversity decreases because

of elitism, which needs to be maintained for better performance of MOEAs. Kasat and Gupta



(2003) developed an algorithm using the concept of Jumping Genes (JG) or transposons in

biology. It increases the genetic diversity in the population. The new algorithm developed

by them is being referred to as NSGA-II-JG. This adaptation exploits the benefits of elitism,

while still maintaining genetic diversity. Though the authors demonstrated the concept of JG

adaptation in GA for the multi-objective optimization algorithm (NSGA-II), it can also be

used for any other EA. Moreover, this adaptation can also be used to solve single objective

optimization problems.

Kasat and Gupta (2003) exploited two kinds of JG to adapt the binary coded elitist

non-dominated sorting genetic algorithm, NSGA-II. The two adaptations used by them, re-

placement and reversion are shown schematically in Fig. (3.3). These adaptations mimic

natural genetics. They proposed to use a probabilistic approach in implementing JG adapta-

tions. Randomly selected Pjump fraction of strings in the population are modified by the JG

operator. In replacement JG operator, a part of the binary string in the offspring population is

replaced with a randomly generated new binary string having the same length. The adaptation

string is generated using the same procedure as used for generating members of the initial

population. The two sites p and q in the original chromosome are selected using random

numbers between which replacement occurs. In case of reversion JG operator the binaries

between two sites p and q are reversed. Authors assume only a single transposon in any

selected chromosome for simplicity of the algorithm. The JG operators are introduced after the

mutation stage in NSGA-II. The detailed algorithm procedure and flow-chart of NSGA-II-JG

is provided in Appendix-B.

3.5 Proposed Parallel Universe Alien GA

Two sub-populations: one real coded and another binary coded, are proposed to be used in

this algorithm. This is called the concept of Parallel Universe having different encoding. Best

members from binary coded population known as Alien members will go to real coded popu-

lation and take part in evolution. Aliens will transfer the information from one sub-population

(universe) to another, terming this concept as Parallel Universe Alien GA (PUALGA). This

approach increases robustness without any additional computational burden by combining the


3.5 Proposed Parallel Universe Alien GA 67

Figure 3.3. Schematics of replacement and reversion JG adaptations for GA (Source: Kasat and Gupta (2003))



capacity of both, binary and real coded GAs. In fact, dividing the population in sub-population

will reduce the calculations needed for sorting and selection and hence will increase the overall

efficiency of the algorithm.

3.5.1 Proposed PUALGA algorithm

Though, the proposed algorithm can be used with any population based evolutionary optimiza-

tion, GA is chosen to demonstrate the clear benefits of the proposed concept of hybridization.

The implementation of propose algorithm is as follows:

1. Initialization of GA parameters: Population size (nPopul), binary population fraction

(bFr), number of generations (nGen), number of decision variables (nVar), maximum

and minimum bounds on decision variables (xMin, xMax), accuracy for binary encoding

(Acc), number of Aliens transferred every generation from binary population to real

population (nAl)

2. Generation of binary and real coded Population and fitness calculation.

3. Selection for nPopul members for binary and (nPopul−nAl) members for real popula-

tion. Add nAl members from binary to real population.

4. Carry out Crossover and Mutation for each Population.

5. Do Fitness calculation for each population.

6. Do Elitism selection for each binary and real population.

7. Alien member addition from binary to real coded population replacing the worst member

in real coded population.

8. Continuation of loop if maximum number of generations are not reached otherwise

continue the loop; go to step 3.

The algorithm flowchart for above discussed Parallel Universe Alien GA (PUALGA) is

presented in Fig. (3.4).


3.6 MOO test problems and Performance measures 69

Figure 3.4. Parallel Universe Alien GA Evolution Scheme

3.6 MOO test problems and Performance measures

For evaluation of the proposed algorithm, there are nine benchmark test functions in this

work. The selected functions here are: SCH (Schaffer, 1985), FON (Fonseca and Fleming,

1993), POL Poloni (1995), KUR Kursawe (1991) and ZDT1, ZDT2, ZDT3, ZDT4, ZDT6

test problems Zitzler et al. (2000) from past studies in this area. The details of the MOO

test functions are given elsewhere Deb (2001). For the sake of readers’ convenience, brief

details of the test problems are presented in table 3.1. All the selected problems have two

objective functions, which needs to be minimized. Every test function has certain difficulties



for multi-objective optimization.

Table 3.1. Details of MOO test functions (Source: Deb (2001))

ProblemNumber ofvariables, n,and bounds

Objective functions to be minimizedPareto frontnature andlocation

SCH1n = 1−1e5≤ x≤ 1e5

f1 = xf2 = (x−2)2

convexx ∈ [0,2]

FONn = 3−4≤ xi ≤ 4

f1 = 1− exp[−∑

3i=1(xi−1/

√3)2]

f2 = 1− exp[−∑

3i=1(xi +1/

√3)2] non-convex

x1 = x2 = x3∈ [−1/

√3,1/√

3]

POLn = 2−π ≤ xi ≤ π

f1 = 1+(A1−B1)2 +(A2−B2)

2

f2 = (x1 +3)2 +(X2 +1)2

A1 = 0.5sin1−2cos1+ sin2−1.5cos2A2 = 1.5sin1− cos1+2sin2−0.5cos2B1 = 0.5sinx1−2cosx1 + sinx2−1.5cosx2B2 = 1.5sinx1− cosx1 +2sinx2−0.5cosx2

non-convexdisconnected

KURn = 3−5≤ xi ≤ 5

f1 = ∑n−1i=1

[−10exp

(−0.2

√x2

i + x2i+1

)]f2 = ∑

ni=1(|xi|0.8 +5sinx3

i) non-convex

ZDT1n = 300≤ xi ≤ 1

f1 = x1

f2 = g[1−√

x1/g]

g = 1+9[∑

ni=2 xi

]/(n−1)

convexx1 ∈ [0,1]x2...n = 0

ZDT2n = 300≤ xi ≤ 1

f1 = x1f2 = g

[1− (x1/g)2]

g = 1+9[∑

ni=2 xi

]/(n−1)

non-convexx1 ∈ [0,1]x2...n = 0

ZDT3n = 300≤ xi ≤ 1

f1 = x1

f2 = g[1−√

x1/g− (x1/g)sin(10πx1)]

g = 1+9[∑

ni=2 xi

]/(n−1)

non-convexdisconnectedx1 ∈ [0,1]x2...n = 0

ZDT4n = 100≤ x1 ≤ 1−5≤ x2...n ≤ 5

f1 = x1

f2 = g[1−√

x1/g]

g = 1+10(n−1)+∑ni=2[x2

i −10cos(4πxi)]

non-convexx1 ∈ [0,1]x2...n = 0

ZDT6n = 100≤ xi ≤ 1

f1 = 1− exp(−4x1)sin6(4πx1)

f2 = g[1− ( f1/g)2]

g = 1+9[∑

ni=2 xi/(n−1)

]non-convexnon-uniformly spreadx1 ∈ [0,1]x2...n = 0


Accuracy - how close the generated non-dominated solutions are to the best known prediction,

(2) Coverage - how many different non-dominated solutions are generated and how well they

are distributed, (3) Variance for every objective - is the maximum range of non-dominated

front, covered by the generated solutions. Performance metrics are important performance

assessment measure, which also allow us to compare algorithms and to adjust their parameters

for better results. They are classified in three categories: metrics evaluating closeness to

the pareto optimal front (convergence), metrics evaluating distribution (diversity) amongst


3.6 MOO test problems and Performance measures 71

non-dominated solutions and, metrics evaluating convergence and diversity (Deb, 2001). Two

critical issues that are normally taken into consideration while evaluating the performance

of multi-objective optimization algorithms are: distance between obtained solutions and,

spread and uniformity among the obtained solutions.Here, generational distance (GD) metric

is used as a measure for convergence to true pareto front and the spread metric to represent

the distribution of solutions in the pareto front.

Generational distance is an average distance of the solutions to the true pareto front. For a

set Q of N solutions from a known set of the pareto optimal set P∗, the Generational Distance

(GD), γ is defined as follows:

γ =

(∑|Q|i=1 dp

i

)1/p

|Q|(3.2)

where Q is solution set containing |Q| members, p=2 and, di is the minimum distance between

the member in solution set and nearest member is true pareto set, given by:

di = min

√M

∑m=1

( f (i)m − f ∗(k)m )2

(3.3)

where M is number of objectives, i and k are member index in solution set and true pareto

set respectively.

f ∗(k)m is the mth objective function value of the kth member of P∗ and f (i)m is the correspond-

ing objective function value from the true pareto front. The concept of generational distance is

illustrated graphically in Fig. (3.5). When the objective function values are of different order

or magnitudes, they should be normalized by an appropriate weighing factor in defining the

distance, di.

The spread matrix is defined as follows:

∆ =∑

Mm=1 de

m +∑|Q|i=1 |di− d|

∑Mm=i de

m + |Q|d(3.4)



Figure 3.5. Generational Distance matrix for an obtained MOO solution set (Source: Deb et al. (2002))

where, dem is the distances between the extreme solutions and the boundary solutions of the

obtained non-dominated solution set Q from the known end solutions of known solution set

P∗. The parameter di is the distance measured between the neighbouring solutions and d is the

mean value of this distance measure. The concept of spread matrix is illustrated graphically

in Fig. (3.6). Note that the maximum value of ∆ can be greater than one. Though, a good

distribution would make all distances di equal to d and would make dem = 0. Thus, the most

widely and uniformly spread of the non-dominated solutions results in the zero value of ∆.

For any other distribution, the value of the metric would be greater than zero.

3.7 Sensitivity Analysis of Proposed Algorithm

The proposed algorithm uses both binary and real encoding for GA. Hence all the opera-

tor parameters of evolutionary scheme like size of chromosome, encoding and decoding of

chromosome, fitness assignment, fitness selection, crossover, mutation, elitism and survival

selection influences the performance of algorithm. In addition to the parameters of native

binary and real coded GA, there are two new important parameters of the proposed algorithm

that influences the performance of the proposed algorithm. One parameter is the fractional

distribution of binary and real encoding fixing the size of binary and real coded populations,


3.7 Sensitivity Analysis of Proposed Algorithm 73

Figure 3.6. Spread matrix for an obtained MOO solution set (Source: Deb et al. (2002))

defined as binary fraction. The second parameter is the number of aliens transferred from

binary to real population at every generation. These two parameters are the proposed algorithm

specific parameters. The sensitivity analysis for these two parameters is carried out using

SCH, FON, POL and KUR test functions. The binary population fraction values of 0.1 to

0.9 in increment of 0.1 and number of aliens transferring information from binary to real

population from 1 to 10 are used. All the other parameters in the algorithm are kept constant

while testing the sensitivity of the selected parameter. Tournament selection, simulated binary

crossover, SBX (with ηc = 20, crossover probability 0.90) and non-uniform mutation (with b =

4, mutation probability 1/n) for real population evolution is used. Single point crossover and

random single point mutation for binary population is used. Non-dominated sorting, crowding

distance calculation and binary tournament selection operators for both binary and real coded

GA as recommended in the NSGA-II are used(Deb et al., 2002).

The proposed PUALGA MOO algorithm is implemented in MATLAB 2011, which is used

in this work. The simulation results for ten runs with different initial population are presented

for all the four test functions. The Fig. (3.7) shows the generation wise convergence metric, γ

values for SCH1 test function for three different binary fractions, 0.2, 0.5 and 0.8. The values



plotted are a mean of the ten simulation runs. The plot is presented as semi-log scale to clearly

represent the difference towards the convergence. The generation wise mean values of spread

metric, ∆ for the same binary fractions, 0.2, 0.5 and 0.8 are presented in Fig. (3.8).

0 10 20 30 40 50

Generation No

10-4

10-2

100

102

104

106

108

me

an

BinFr=0.2

BinFr=0.5

BinFr=0.8

Figure 3.7. Effect of binary fraction on generation wise performance (Generational Distance metric) for SCH1test function



0 10 20 30 40 50

Generation No

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

me

an

BinFr=0.2

BinFr=0.5

BinFr=0.8

Figure 3.8. Effect of binary fraction on generation wise performance (Spread metric) for SCH1 test function

The effect of changing binary fraction values from 0.1 to 0.9 (increment of 0.1) is plot-

ted in Fig. (3.9). The binary fraction 0.6 gives the best results for convergence and binary

fraction 0.8 gives the best value for distribution. The binary fraction 0.3 and 0.8 had worst

convergence. Distribution metric values improved as the binary fraction increased from 0.1 to

0.6 and it got deteriorated beyond 0.6 till 0.9. The statistical results of mean and variance for

both performance metrics are presented in table (3.2) along with the computational time for

simulation runs. The effect of the second parameter which is number of alien addition keeping

the binary fraction value fixed at 0.5 is studied.



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Binary Fraction

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

me

an

10-4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Binary Fraction

0.02

0.03

0.04

0.05

0.06

0.07

me

an

(a) Generational Distance metric (b) Spread metric

Figure 3.9. Sensitivity of binary fraction on performance metric at 50th Generation for SCH1 test function

Table 3.2. Sensitivity analysis for binary fraction on performance metrics for SCH1 test function

Binary FractionGD metric Spread metric Time

Mean Variance Mean Variance Mean Variance0.1 0.0017 0.0001 0.4101 0.0398 5.5352 0.84060.2 0.0016 0.0001 0.3971 0.0356 5.6542 1.01640.3 0.0017 0.0002 0.4229 0.0376 5.4527 0.89960.4 0.0016 0.0001 0.4025 0.0389 5.7705 0.74990.5 0.0016 0.0001 0.4225 0.0326 5.9248 0.83940.6 0.0016 0.0001 0.4172 0.0251 8.0805 2.88660.7 0.0016 0.0001 0.4110 0.0277 7.0386 1.26390.8 0.0017 0.0001 0.4582 0.0210 8.5578 1.45660.9 0.0017 0.0001 0.4635 0.0662 8.6332 1.3064



The Fig. (3.10) shows the generation wise convergence metric, γ values (mean of ten

simulation runs) for SCH1 test function for three different number of alien transfer: 2, 5 and 9.

The generation wise mean values of spread metric, ∆ for the same number of alien transfer,

2, 5 and 9 are presented in Fig. (3.11). The effect of changing number of alien transfer

from 1 to 10 is plotted in Fig. (3.12). The 6 numbers of alien transfer gives the best results

for convergence and, 2 numbers of alien transfer gives the best value for distribution. The

statistical results of mean and variance for both performance metrics are presented at table

(3.3) along with computational time for simulation runs.

0 10 20 30 40 50

Generation No

10-4

10-2

100

102

104

106

108

me

an

nAl=1

nAl=5

nAl=9

Figure 3.10. Effect of number of alien transfer on generation wise performance (Generational Distance metric)for SCH1 test function



0 10 20 30 40 50

Generation No

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6m

ea

n

nAl=1

nAl=5

nAl=9

Figure 3.11. Effect of number of alien transfer on generation wise performance (Spread metric) for SCH1 testfunction

0 2 4 6 8 10

Number of Aliens transerred

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

me

an

10-4

0 2 4 6 8 10


0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

me

an


Figure 3.12. Sensitivity for number of alien transfer on performance metric at 50th Generation for SCH1 testfunction


simulation runs) for FON test function for three different binary fractions: 0.2, 0.5 and 0.8.

The generation wise mean values of spread metric, ∆ for the same binary fractions, 0.2, 0.5

and 0.8 are presented in Fig. (3.14). The trends of convergence and distribution clearly



Table 3.3. Sensitivity analysis for number of alien tranfer on performance metrics for SCH1 test function

No of Alien transferGD metric Spread metric Time

Mean Variance Mean Variance Mean Variance1 0.0016 0.0001 0.4321 0.0353 6.1061 0.87832 0.0016 0.0001 0.4350 0.0233 6.7611 1.04913 0.0016 0.0001 0.4217 0.0464 6.4894 0.93174 0.0016 0.0001 0.4240 0.0376 6.2737 0.80385 0.0016 0.0001 0.4225 0.0326 5.7188 0.83286 0.0016 0.0001 0.4411 0.0392 6.2276 1.03547 0.0017 0.0001 0.4229 0.0506 6.2793 1.23678 0.0017 0.0001 0.3951 0.0240 7.9084 1.93809 0.0017 0.0001 0.4233 0.0504 7.8511 1.8544

10 0.0016 0.0001 0.4127 0.0292 6.8706 0.8988

shows the effect of binary fraction during evolution process. The effect of binary fraction on

convergence and distribution at the end of 70 generations is plotted in Fig. (3.15). The binary

fraction 0.2 gives the best results for convergence and binary fraction 0.3 gives the best value

for distribution. The binary fraction 0.1 and 0.9 had worst convergence and distribution. The

statistical results of mean and variance for both performance metrics are presented at table

(3.4) along with computational time for simulation runs. Though binary fraction value of

0.2 and 0.3 performed best, the effect of the second parameter which is the number of alien

addition keeping the binary fraction value fixed at 0.5 is studied to maintain the consistency in

the sensitivity study.



0 10 20 30 40 50 60 70

Generation No

10-3

10-2

10-1

100

me

an

BinFr=0.2

BinFr=0.5

BinFr=0.8

Figure 3.13. Effect of binary fraction on generation wise performance (Generational Distance metric) for FONtest function

0 10 20 30 40 50 60 70

Generation No

0.2

0.4

0.6

0.8

1

1.2

1.4

me

an

BinFr=0.2

BinFr=0.5

BinFr=0.8

Figure 3.14. Effect of binary fraction on generation wise performance (Spread metric) for FON test function



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Binary Fraction

0

1

2

3

4

5

6

me

an

10-3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Binary Fraction

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

me

an


Figure 3.15. Sensitivity of binary fraction on performance metric at 70th Generation for FON test function

Table 3.4. Sensitivity analysis for binary fraction on performance metrics for FON test function






simulation runs) for FON test function for three different number of alien transfer: 2, 5 and 9.

The generation wise mean values of spread metric, ∆ for the same number of alien transfer,

2, 5 and 9 are presented in Fig. (3.17). The effect of changing number of alien transfer

from 1 to 10 is plotted in Fig. (3.18). The 2 numbers of alien transfer gives the best results

for convergence and, 3 numbers of alien transfer gives the best value for distribution. The

statistical results of mean and variance for both performance metrics are presented in table

(3.5) along with computational time for simulation runs.

0 10 20 30 40 50 60 70

Generation No

10-3

10-2

10-1

100

me

an

nAl=1

nAl=5

nAl=9

Figure 3.16. Effect of number of alien transfer on generation wise performance (Generational Distance metric)for FON test function



0 10 20 30 40 50 60 70

Generation No

0.2

0.4

0.6

0.8

1

1.2

1.4

me

an

nAl=1

nAl=5

nAl=9

Figure 3.17. Effect of number of alien transfer on generation wise performance (Spread metric) for FON testfunction

0 2 4 6 8 10


1.5

2

2.5

3

3.5

4

4.5

me

an

10-4

0 2 4 6 8 10


0.015

0.02

0.025

0.03

0.035

0.04

me

an


Figure 3.18. Sensitivity for number of alien transfer on performance metric at 70th Generation for FON testfunction

The Fig. (3.19) represents the generation wise convergence metric, γ values for POL test

function for three different binary fractions: 0.2, 0.5 and 0.8. The values plotted are mean

values for ten simulation runs. The generation wise mean values of spread metric, ∆ for the

same binary fractions: 0.2, 0.5 and 0.8 are presented in Fig. (3.20). The effect of binary



Table 3.5. Sensitivity analysis for number of alien transfer on performance metrics for FON test function


Mean Variance Mean Variance Mean Variance1 0.0021 0.0003 0.4155 0.0238 2.3936 0.03242 0.0021 0.0002 0.4269 0.0334 2.3992 0.02613 0.0021 0.0003 0.4288 0.0186 2.6426 0.43694 0.0022 0.0004 0.4080 0.0244 2.6148 0.47045 0.0021 0.0003 0.4096 0.0289 2.4938 0.02926 0.0022 0.0003 0.4276 0.0364 2.6176 0.34567 0.0022 0.0004 0.4284 0.0273 2.5539 0.03208 0.0022 0.0003 0.4206 0.0389 2.5707 0.03589 0.0024 0.0003 0.4156 0.0318 2.8786 0.550910 0.0023 0.0002 0.4180 0.0232 3.1450 0.8670

fraction on convergence and distribution at the end of 100 generations is plotted in Fig. (3.21).

The binary fraction 0.5 gives the best results for convergence and binary fraction 0.1 and 0.5

gives the best value for distribution. Similarly, in the trend of effect of binary fraction, charge

is observed for both, convergence and distribution. The statistical results of mean and variance

for both performance metrics are presented in table (3.6) along with computational time for

simulation runs. Additional studies of the effect of the second parameter which is the number

of alien addition keeping the binary fraction value fixed at 0.5 to maintain consistency in the

sensitivity study.



0 20 40 60 80 100

Generation No

10-2

10-1

100

101

me

an

BinFr=0.2

BinFr=0.5

BinFr=0.8

Figure 3.19. Effect of binary fraction on generation wise performance (Generational Distance metric) for POLtest function

0 20 40 60 80 100

Generation No

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

me

an

BinFr=0.2

BinFr=0.5

BinFr=0.8

Figure 3.20. Effect of binary fraction on generation wise performance (Spread metric) for POL test function



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Binary Fraction

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

me

an

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Binary Fraction

2

4

6

8

10

12

me

an

10-3


Figure 3.21. Sensitivity of binary fraction on performance metric at 100th Generation for POL test function

Table 3.6. Sensitivity analysis for binary fraction on performance metrics for POL test function






simulation runs) for POL test function for three different number of alien transfer: 2, 5 and 9.

The generation wise mean values of spread metric, ∆ for the same number of alien transfer: 2,

5 and 9 are presented in Fig. (3.23). The effect of changing number of alien transfer from 1 to

10 is plotted in Fig. (3.24). The 1 and 10 number of alien transfer gives the best results for

convergence and distribution both. The 4 numbers of alien transfer gives the worst value for

both metrics. The statistical results of mean and variance for both performance metrics are

presented at table (3.7) along with computational time for simulation runs.

0 20 40 60 80 100

Generation No

10-2

10-1

100

101

me

an

nAl=1

nAl=5

nAl=9

Figure 3.22. Effect of number of alien transfer on generation wise performance (Generational Distance metric)for POL test function



0 20 40 60 80 100

Generation No

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25m

ea

n

nAl=1

nAl=5

nAl=9

Figure 3.23. Effect of number of alien transfer on generation wise performance (Spread metric) for POL testfunction

0 2 4 6 8 10


0

0.005

0.01

0.015

0.02

me

an

0 2 4 6 8 10


2

3

4

5

6

7

8

9

me

an

10-3


Figure 3.24. Sensitivity for number of alien transfer on performance metric at 100th Generation for POL testfunction

The Fig. (3.25) represents the generation wise convergence metric, γ values for KUR test

function for three different binary fractions: 0.2, 0.5 and 0.8. The values plotted are mean

values for ten simulation runs. The generation wise mean values of spread metric, ∆ for the

same binary fractions: 0.2, 0.5 and 0.8 are presented in Fig. (3.26). The effect of binary



Table 3.7. Sensitivity analysis for number of alien transfer on performance metrics for POL test function


Mean Variance Mean Variance Mean Variance1 0.0078 0.0005 0.8139 0.0096 8.0456 0.82332 0.0085 0.0008 0.8141 0.0083 8.3446 0.90443 0.0078 0.0012 0.8170 0.0095 8.3918 0.86394 0.0081 0.0013 0.8209 0.0142 9.1694 1.01675 0.0078 0.0012 0.8234 0.0088 8.4940 0.89756 0.0080 0.0005 0.8252 0.0108 8.5960 0.85257 0.0088 0.0010 0.8196 0.0099 8.9015 0.98018 0.0085 0.0013 0.8178 0.0107 9.3009 0.97879 0.0085 0.0014 0.8214 0.0095 9.0292 0.9850

10 0.0081 0.0007 0.8201 0.0097 8.7204 0.5071

fraction on convergence and distribution at the end of 250 generations is plotted in Fig. (3.27).

The binary fraction 0.4 and 0.6 gives the best results for convergence and binary fraction

0.5 and 0.7 gives the best value for distribution. The statistical results of mean and variance

for both performance metrics are presented in table (3.8) along with computational time for

simulation runs. We study the effect of the second parameter, number of alien addition keeping

the binary fraction value fixed at 0.5 to maintain consistency in the sensitivity study.



0 50 100 150 200 250

Generation No

10-2

10-1

100

101

me

an

BinFr=0.2

BinFr=0.5

BinFr=0.8

Figure 3.25. Effect of binary fraction on generation wise performance (Generational Distance metric) for KURtest function

0 50 100 150 200 250

Generation No

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

me

an

BinFr=0.2

BinFr=0.5

BinFr=0.8

Figure 3.26. Effect of binary fraction on generation wise performance (Spread metric) for KUR test function



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Binary Fraction

4

5

6

7

8

9

10

11

12

13

me

an

10-4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Binary Fraction

0.008

0.009

0.01

0.011

0.012

0.013

me

an


Figure 3.27. Sensitivity of binary fraction on performance metric at 250th Generation for KUR test function

Table 3.8. Sensitivity analysis for binary fraction on performance metrics for KUR test function






simulation runs) for KUR test function for three different number of alien transfer: 2, 5 and 9.

The generation wise mean values of spread metric, ∆ for the same number of alien transfer: 2,

5 and 9 are presented in Fig. (3.29). The effect of changing number of alien transfer from 1 to

10 is plotted in Fig. (3.30). The 1 and 6 numbers of alien transfer gives the best results for

convergence metric. The 2 and 5 numbers of alien transfer gives the best value for distribution

metric. The statistical results of mean and variance for both performance metrics are presented

in table (3.9) along with computational time for simulation runs.

0 50 100 150 200 250

Generation No

10-2

10-1

100

101

me

an

nAl=1

nAl=5

nAl=9

Figure 3.28. Effect of number of alien transfer on generation wise performance (Generational Distance metric)for KUR test function



0 50 100 150 200 250

Generation No

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

me

an

nAl=1

nAl=5

nAl=9

Figure 3.29. Effect of number of alien transfer on generation wise performance (Spread metric) for KUR testfunction

0 2 4 6 8 10


5

6

7

8

9

10

11

12

13

14

me

an

10-4

0 2 4 6 8 10


0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

me

an


Figure 3.30. Sensitivity for number of alien transfer on performance metric at 250th Generation for KUR testfunction

The sensitivity analysis for binary fraction and number of alien transfer clearly reflects

the influence of these parameters on convergence and diversity in the solutions obtained. Too

small values do not serve the purpose and very high values hamper the process. Based on

analysis, 5 numbers of alien transfer and 0.5 binary fraction value is recommended to be



Table 3.9. Sensitivity analysis for number of alien transfer on performance metrics for KUR test function


Mean Variance Mean Variance Mean Variance1 0.0217 0.0064 0.8703 0.0288 2.7489 0.06902 0.0234 0.0076 0.8703 0.0374 2.7951 0.08743 0.0277 0.0102 0.8689 0.0321 2.9033 0.37964 0.0259 0.0093 0.8568 0.0295 3.1323 0.67375 0.0207 0.0050 0.8444 0.0234 2.8320 0.02876 0.0209 0.0076 0.8439 0.0358 3.2600 0.84397 0.0201 0.0063 0.8314 0.0257 3.1938 0.71378 0.0272 0.0189 0.8326 0.0388 3.1353 0.48179 0.0206 0.0069 0.8317 0.0357 3.2370 0.723010 0.0254 0.0208 0.8246 0.0329 2.9578 0.0398

used for performance analysis and comparison with other benchmark algorithms. ZDT test

functions are used for this purpose.

3.8 Performance Evaluation Results and Discussion

Multi-Obejective Optimization program developed in MATLAB 2011 implementing the

PUALGA algorithm is used in this work. The performance is compared with two native:

real and binary coded GA, which are hybirdized, as well as, Jumping Gene GA. The MOO

program uses single point crossover and binary mutation for binary population evolution.

It uses tournament selection, simulated binary crossover, SBX (with ηc = 20, crossover

probability 0.90) and non-uniform mutation (with b = 4, mutation probability 1/n) for real

population evolution. It uses non-dominated sorting along with elitism survival selection

operators for both binary and real coded GA. The PUALGA uses same binary and real

coded GA operators along with alien operator to exchange information between populations.

All MOO programs use non-dominated sorting, crowding distance calculation and binary

tournament selection operators as recommended in the NSGA-II Deb et al. (2002). The

jumping gene GA uses randomly created five bit chromosome with probability of 0.2 Guria

et al. (2005). The decision variables, their upper and lower limits for all the problems are

taken as used in Deb et al. (2002). Population size is 100 for all test problems. Test problems

selected are well known benchmark test problems, hence, skipping its definitions here. Since

the techniques used are stochastic optimization technique, it does not produce the same


3.8 Performance Evaluation Results and Discussion 95

solution each time even with the same starting population. Hence, twenty simulation runs

were carried out for every test problem with different initial population and, average results

are presented for the comparison of the algorithms.

3.8.1 Convergence to true pareto front

Since, the identified test problems have known true pareto fronts, it is feasible to calculate

the convergence. Convergence metric γ for ZDT1 function is shown in Fig. 3.31. The metric

value is a measure of average distance of the obtained solution from true front, hence, smaller

the value, better is the convergence. The average generation wise convergence rate for the

PUALGA algorithm is observed to be the best among all four algorithms. PUALGA achieves

γ metric value very close to zero in only 15 generations, where as, real coded GA required

100 generations to attain the same convergence. aJGGA took around 250 generations and

binGA coluld not even converge within 250 generations. PUALGA is just the combination

of real GA and binary GA which outperforms both native algorithms. The average values of

convergence metric γ and its standard deviation at the end of 250 generations are given in

Table 3.10 for all the four algorithms.

Convergence metric γ for ZDT2, ZDT3, ZDT4 and ZDT6 test functions are presented

in Figs. 3.32-3.35. Similar to ZDT1 test function, drastic improvement in convergence rate

for PUALGA is observed for all test functions. Three to ten times faster convergence is

observed for PUALGA. The statistical analysis in Table 3.10 also conforms consistent better

performance of PUALGA compared to other all algorithms. The statistical analysis presented

is at the end of 250 generations for ZDT1, ZDT2, ZDT3 and 250 generations for ZDT4 and

ZDT6 test functions.



0 50 100 150 200 250

0

0.5

1

1.5

Generation No

mean γ

rNSGA−II

bNSGA−II

aJGGA

PUALGA


0 50 100 150 200 250

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Generation No

mean γ

rNSGA−II

bNSGA−II

aJGGA

PUALGA




0 50 100 150 200 250

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Generation No

mean γ

rNSGA−II

bNSGA−II

aJGGA

PUALGA


0 50 100 150 200 250

0

2

4

6

8

10

12

14

16

18

20

Generation No

mean γ

rNSGA−II

bNSGA−II

aJGGA

PUALGA




0 50 100 150 200 250

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Generation No

mean γ

rNSGA−II

bNSGA−II

aJGGA

PUALGA


Table 3.10. Statastical analysis of convergence metric γ for 20 simulation runs


ZDT1mean 0.0021 0.1148 0.0105 0.0010std 0.0005 0.0323 0.0033 0.0001

ZDT2mean 0.0016 0.2342 0.0136 0.0009std 0.0003 0.0683 0.0039 0.0001

ZDT3mean 0.0026 0.0525 0.0042 0.0025std 0.0002 0.0228 0.0006 0.0002

ZDT4mean 0.0025 4.8790 0.1295 0.0007std 0.0004 2.3012 0.1144 0.0001

ZDT6mean 0.0029 0.0027 0.0027 0.0068std 0.0002 0.0001 0.0001 0.0097

3.8.2 Distribution of solutions within pareto front

The diversity metric ∆, represents the spread of solutions. It is a measure of distribution

of solution along Pareto front. Zero value of the diversity metric indicates solutions are

uniformly distributed covering full range of true front; smaller the value, better the spread.

The generation wise progress of diversity metric ∆ is presented in Fig. (3.36). The figure

clearly indicates that distribution is also observed to be the best for PUALGA compared to all



other algorithms. Similar trends in distribution metric ∆ are observed in Fig. (3.37) for ZDT2

and Fig. (3.38) for ZDT3 test functions respectively.

0 50 100 150 200 2500.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Generation No

mean ∆

rNSGA−II

bNSGA−II

aJGGA

PUALGA


The statistical analysis of distribution and coverage of pareto front are presented in Table

3.11. The statistical analysis presented is at the end of 250 generations for ZDT1, ZDT2, ZDT3

and 250 generations for ZDT4 and ZDT6 test functions. The results indicate that PUALGA

performance is observed to be better for all the three criteria: convergence, distribution and

coverage. To represent the same information on pareto front, an intermediate pareto front for

one of the run is presented for ZDT4 test function in Fig. (3.39). The figure clearly shows that

PUALGA population has already converged at 200 generations, where as, other algorithms

are away from true pareto front. Similar information is represented for ZDT6 test function in

Fig. (3.40).



0 50 100 150 200 250

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Generation No

mean ∆

rNSGA−II

bNSGA−II

aJGGA

PUALGA


0 50 100 150 200 250

0.6

0.8

1

1.2

1.4

1.6

Generation No

mean ∆

rNSGA−II

bNSGA−II

aJGGA

PUALGA




Table 3.11. Statasical analysis of distribution and coverage as spread metric, ∆ for 20 simulation runs


ZDT1mean 0.3733 1.0443 0.5670 0.3827std 0.0329 0.1360 0.0807 0.0262

ZDT2mean 0.3599 1.3100 0.8082 0.3766std 0.0322 0.1246 0.1110 0.0277

ZDT3mean 0.5516 1.1660 0.7838 0.5514std 0.0231 0.1118 0.1053 0.0335

ZDT4mean 0.3845 0.7346 0.5856 0.3892std 0.0328 0.0467 0.3691 0.0380

ZDT6mean 0.3492 0.2878 0.2817 0.4836std 0.0318 0.0300 0.0330 0.2748

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

f1

f 2

rNSGA−II

bNSGA−II

aJGGA

PUALGA




0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

f1

f 2

rNSGA−II

bNSGA−II

aJGGA

PUALGA


3.9 Summary

Hybridization of binary and real coded GA is explored to enhanced convergence rate. The

focus of the hybridization is to combine the strengths of both algorithms. Binary encoding has

the flexibility of adjusting accuracy of decision variables by adjusting binary chromosome

size. The mechanism of binary encoding gives better exploration of search space using small

chromosome size. Use of small chromosome size supports very good initial convergence

but can not converge to true solutions at later stage of evolution. Real coded GA takes up

that responsibility of convergence at that stage. The algorithm uses the concept of parallel

population and combined binary and real GA. Non-dominated sorting is used in all algorithms

for survival selection. The advantage of using two sub populations reduces the complexity

of sorting and achieves better results with same computational efforts (Number of function

Evaluations).

The concept of alien transferring information from one sub-population to another sub-

population is used to transfer the information between parallel evolving populations. The


3.9 Summary 103

information transfer is one way, from binary population to real population. This approach is

adopted to preserve the binary genetic information free from the influence of real population.

Real encoded population gets the benefits of exportation capacity of binary encoding, but does

not guide or influence it to maintain unbiased exploration. The investigation of the sensitivity

of binary fraction and number of alien transfers from binary to real population at every gener-

ation are explored in detail. Another parameter which can influence the performance of the

binary encoding is the chromosome size. The sensitivity of that parameter is well established

in the literature. The purpose of binary population in current evolution is to explore search

space, hence, smaller size of chromosomes are used.

The proposed PUALGA algorithm has two tuning parameters, the binary fraction of

population and number of alien transfer from binary to real population. Both these parameter

influence the performance of the proposed algorithm. The sensitivity analysis for these two

parameters was carried out using four different test problems and recommend 0.5 binary

fraction and 5 numbers of alien addition, which will serve most cases. For complex problems

with larger population size, fine tuning may be needed for these parameter for better perfor-

mance. Though, this concept can be applied for any population based MOEAs, the results

are shown under GA framework in this study. The proposed PUALGA has been compared

with its native binary and real coded GAs and Jumping Gene adaptation of GA. The proposed

PUALGA algorithm drastically enhances the initial convergence rate for all bench mark MOO

test problems taking the benefit of exploration capacity of binary encoding.


105

Chapter 4

Boundary Inspection Approach for

Constrained handling in Evolutionary

Optimization Algorithms

Constraint handling is always a critical part in the performance of optimization method. There

exists conventional approaches like, substitution, penalty function, slack variable, Lagrangian

multiplier and, ignorance infeasible for constraint handling. Some of them are convenient

to use with evolutionary algorithms. There exists some hybrid methods, algorithm and/or

problem specific approaches for constraint handling in optimization. There has been a lack of

efficient and generic constraint handling techniques. A new generalized boundary inspection

approach based constraint handling mechanism for population based evolutionary algorithms

(EAs)is being proposed here. The concept is general and can be used with any population

based EAs. A demonstration of its implemented for Multi-Objective Optimization (MOO)

is shown in this work. A comparative study of the proposed algorithm with the augmented

penalty function method and ignorance infeasible are presented in this work. Parallel universe

Alien Genetic Algorithm (PUALGA) with non-dominated sorting as basic MOO algorithms

is used and, evaluation of the proposed constraint handling mechanism is carried out. Three

benchmark test problems are considered for evaluating the proposed mechanism. Though the

proposed constraint handling method is demonstrated for PUALGA, it is very generalized

and can be used with any evolutionary algorithm easily. The method proposed converts all

106 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms

infeasible solutions into feasible solutions maintaining diversity in search space.

4.1 Introduction

During the last two decades, Evolutionary Algorithms (EAs) have proved to become an

important tool for solving complex engineering optimization problems. Most real-world

problems are however constrained and a possible criticism of the EAs has been the lack of

efficient and generic constraint handling techniques. It should be noted that the evolutionary

optimization algorithms are unconstrained by nature and hence need additional mechanisms to

handle the constraints. Three excellent review articles on existing constraint handling methods

for EAs are presented by Coello and Efre’n (2002); Kramer (2010); Mezura-Montes and

Coello Coello (2011). Some of the popular constraint handling approaches for the EAs are:

penalty method, preservation of feasible solutions method, augmented lagrangian method and,

feasibility based rule. In the penalty method, penalty parameter is multiplied with the extent

of constraint violation and is augmented with the objective function. While it is the simplest

method of handling constraints, finding the appropriate penalty values is a challenging task.

Preservation of feasible solutions method does not distinguish the extent of constraint violation

and requires large number of generations to converge. This may not necessarily increase the

extra objective function evaluations, but it certainly requires computing constraint functions

for the infeasible members. When the constraint function is computationally expensive, this

method becomes very slow in convergence. All these methods address the issue of guiding

the solution candidates from infeasible to feasible region. Moreover, the constraint handling

mechanisms were not explicitly intended for enhancing the convergence property.

Constraint handling becomes even more crucial and complex in multi-objective EAs.

Singh et al. (2010) extended simulated annealing for multi-objective constrained optimization

problems. Yang and Deb (2014) used constrained method and adaptive operator selection

in Multi-objective evolutionary algorithm based on decomposition (MOEAD). Yang and

Deb (2013) proposed a new cuckoo search for multi-objective optimization under complex

non-linear constraints. A study on the constrained multi-objective optimization has been

presented by Qu and Suganthan (2011b). They have investigated three constraint handling


4.2 Boundary Inspection Approach for Constraint Handling 107

methods along with their ensemble of constraint handling methods (Mallipeddi and Suganthan,

2010). They ensemble self-adaptive penalty, superiority of feasible solution and, ε-constraint

methods. While the fitness values are calculated for both, the feasible and infeasible members

in the self-adaptive penalty method, only feasible members are evaluated for their fitness val-

ues in the method of the superiority of feasible solution method. ε-constraint method employs

fitness assignment process similar to the superiority of feasible solutions method, but with an

adaptive relaxation in constraint violation for initial few generations. Here, augmented penalty

function and ignore infeasible methods for comparison with the new proposed algorithm is

used in this work.

A generalized constraint handling approach for population based EAs using Boundary

Inspection (BI) approach is presented in this work. The BI approach converts every infeasible

member to a feasible one during the evolution process. The algorithm attempts to move

infeasible point in a direction joining an infeasible point and a feasible point such that we

reach within feasible area. At every generation using this approach, all infeasible members

are converted to feasible members by moving towards randomly selected feasible point. The

parameter deciding the location of the new point is used from a predefined pool of values

based on its success history.

The BI approach for constraint handling is discussed in the next section. The BI approach

for constraint handling is tested with a multi-objective evolutionary algorithm : Parallel

Universe Alien Genetic Algorithm (PUALGA) proposed by Desai et al. (2018). The PUALGA

algorithm with the BI approach for constraint handling is discussed in section 4.2. Performance

measures for MOO is discussed in section-4.5 followed by test problem summary in section

4.6. The results are presented in section 4.7 and concluding remarks are drawn in section 4.8.

4.2 Boundary Inspection Approach for Constraint Handling

A randomly created population is classified in two groups: feasible and infeasible. For every

member from the infeasible group, one member from feasible group is selected randomly.

The BI approach can be applied using half moves as demonstrated in the Fig. (4.1). Point R is



the worst point selected from infeasible group and point S is the corresponding point selected

from feasible group. Point N1 is located moving R towards S in the direction joining R and

S, half the distance between point R and S. The point N1 is not feasible, hence further half

distance move from N1 is carried out, reaching to N2. That point is also not feasible, hence,

we move to point N3 moving half distance towards S, which is a feasible point. This procedure

is applied to all infeasible points and convert them to feasible points at every generation of

evolution.

Figure 4.1. Boundary Inspection Approach for Constraint Handling

A predefined ensemble of parameter λ is proposed to be used to locate the new point on

the line joining an infeasible point and the corresponding feasible point selected as shown in

Fig. (4.2). Each value in the ensemble is given equal opportunity during initial learning period.

The success count by each value in the learning period is converted to success probability,

which is used in the next learning period. During the learning period, the success probability is

kept constant. The value of parameter λ to locate the new point is selected based on its success

probability. Thus, the value of parameter λ generating feasible point will be automatically

preferred over the failing value. This will avoid the parameter tuning during evolution and



problem specific tuning to the algorithm.

Figure 4.2. Boundary Inspection Approach for Constraint Handling

For each infeasible member R, one member S from feasible population is selected randomly.

A new point N, dividing the line, joining point S and infeasible point R, in the λ : 1 ratio is

obtained such that, it is feasible. The division ratio is selected from a predefined pool of λ

values based on the past performance history. An ensemble of possible values of ratio λ used

are [-0.6, -0.3, 0.3, 0.6, 1, 1.5, 2]. The Proposed algorithm for BI approach is as follows:

1. Classify the population in two groups: feasible and infeasible.

2. For every member from the infeasible group, apply BI treatment described in Algorithm

(1) to generate the corresponding feasible solution. [The algorithm employs an ensemble

of various λ values ENSλ and the corresponding success probability pk, for every λ

value.]

3. Each infeasible member R, is projected through the randomly selected one point from



feasible population.

4. kth value of λ from the ENSλ is randomly selected with a success probability Pk and

the corresponding attempt counter for the kth value of λ is updated .

5. The infeasible member is then projected through the selected feasible point S, using the

BI approach. The new point N is calculated using equation 4.1.

6. If the new member N, violates any bound constraints, it is placed at the appropriate

boundary value.

7. If N becomes feasible, it replaces the member R and the success count is updated for the

corresponding kth value of λ .

8. If N violates any other constraints, a new point S is secluded and process is continued

from 4 till all the members become feasible.

The BI algorithm is applied to initial randomly created population and the child populations

at every generation during evolution to convert all infeasible members to feasible members.

The BI treatment algorithm applied to a selected member S to make it feasible is presented in

Algorithm (1). The new point N, located on a line joining R and S, dividing the line in the

ration of λ : 1 can be calculated using the equation (4.1):

x(i,N) = (1+λ )x(i,S)−λx(i,R) (4.1)

Since, the optimum value of λ is likely to be problem specific, we use the ensemble of

λ values and, select λ based on its success probability, Pk. Note that this mechanism of

selection of λ is adaptive to the evolution process in addition to being adaptive to the problem

characteristics. The implementation of this Ensemble λ concept is discussed in the next

subsection.

4.2.1 Ensemble of the projection parameter λ in BI approach

An ensemble of possible values of λ is used and a value is selected from this ensemble, which

is guided based on the past performance history. This adaptive mechanism of selecting the



Algorithm 1 BI assisted constraint handling strategy for an infeasible population member, R

Require: define ENSλ , Pk;while M is infeasible do

Randomly select one feasible member from feasible population;select λ from ENSλ using probability, Pk;update attempt count for Ensemble k;compute the new member, N = (1+λ )S−λR;if N is outside the bounds then

reset it to the bound limit;end ifEvaluate constraints for N;if N is feasible then

update success count for Ensemble k;end if

end while

parameter provides freedom from the parameter tuning. Thus, with this ensemble approach,

the parameter tuning for a specific problem or its tuning during the evolution process can be

avoided. Zhao et al. (2012) used the concept of ensemble for neighbourhood size selection

in multi-objective EA, MOEA/D. Ensemble of different constraint handling techniques were

also explored by Mallipeddi and Suganthan (2010).

A set of K fixed values of λ are used as the ensemble of parameter λ . In the present work,

the ensemble consists of seven λ values, [-0.6, -0.3, 0.3, 0.6, 1, 1.5, 2]. A λ value from this

ensemble is selected from the pool based on its previous performance history. The previous

performance history is accounted in the success probability pk,G, which is updated after every

fixed number of generations G, during the evolution as learning period LP. LP = 20 is used in

this work. Thus, probability of selecting a λ value from the ensemble pool is constant during

the learning period of 20 generations and updated at the end of every 20 generations (i.e. G =

20, 40, 60,...). The probability of selecting the kth λ is updated by the following equation:

pk,G =Rk,G

∑Kk=1 Rk,G

(4.2)

where,

Rk,G =∑

G−1g=G−LP SCk,g

∑G−1g=G−LP ACk,g

+ ε (4.3)



Rk,G is the fraction of the success achieved with kth λ value in previous LP generations.

Success count (SC) is measured as the number of successful attempts of obtaining a feasible

solution out of total number of attempts count (AC) while projecting an infeasible solution

by box-complex operation. Thus, the SCk,g and ACk,g represent the success and total attempt

counts of the kth λ value at the gth generation, respectively. The ratio of the success count

to the total attempts count represents the success probability Rk,G, of the kth λ value at the

Gth generation. A small constant value of ε = 0.025 is used to avoid possible zero selection

probability. Equal probability is used for all the members of ensemble for the first learning

period, which is afterwards updated according to the performance of the individual members

of the ensemble. At the end of every learning period, the success and attempt counts are reset

to zero.

The proposed method of constraint handling using BI approach with an ensemble of λ is

a generalised mechanism which can be applied to any single or multi-objective population

based EA. However, this approach is implemented for multi-objective PUALGA algorithm

discussed in previous chapter. The PUALGA algorithm with BI constraint handling approach

is presented in the next section.

4.3 Parallel Universe Alien Genetic Algorithm (PUALGA) with BI Ap-

proach

Hypothesis to use two sub-populations(Parallel Universe), one real coded and another binary

coded is proposed in Genetic Algorithm. One or more best members from binary coded

population known as Alien members are allowed to go to real coded population and take part

in evolution. It will transfer the information from one sub-population to the another. This

approach provide robustness without any additional computational burden. In fact, dividing

the population in sub-population will reduce the calculations needed for sorting and selection

and hence will increase the overall efficiency of the algorithm.

The Hypothesis to use two sub-populations, one real coded and another binary coded,


4.3 Parallel Universe Alien Genetic Algorithm (PUALGA) with BI Approach 113

Parallel Universe proposed as PUALGA improved convergence Desai et al. (2018). The

algorithm is then tested for constraint handling feature. A BI approach for constraint handling

along with Ensemble method for automated parameter selection is being proposed. Commonly

used constraint handling mechanisms, such as, Feasibility rules (Ignore Infeasible) and Penalty

functions (Deb, 2001) under PUALGA framework are tested and, their performance with

proposed BI approach is compared. Constraint handling method to both real coded and binary

populations is applied. The overall PUALGA with BI approach for constraint handling is as

summarised follows:

1. Initialization of GA Parameters.

2. Generation of binary and real coded population, fitness calculation and movement of

infeasible points towards feasible members using BI approach applying the Algorithm

(1).

3. Selection for nPopul members for binary and (nPopul – nAl) members for real popula-

tion.

4. Add nAl members from binary to real population.

5. Carry out Crossover and Mutation for each population .

6. Check constraint and move infeasible points towards feasible members using BI ap-

proach applying the algorithm (1).

7. Do Elitism selection for each binary and real population.

8. Alien member addition from binary to real coded population replacing the worst member

in real coded population.

9. Continuation of loop if appropriate convergence criteria has not been met or maximum

number of generations are not reached otherwise continue the loop; go to 5.

The two methods selected for constraint handling: Feasibility rules (Ignore Infeasible)

and Penalty functions are well studied, hence, it is directly implement under the PUALGA



framework. The proposed concept of BI approach is new, hence, sensitivity study needs to be

carried out before implementing it under the PUALGA framework. Sensitivity analysis of the

proposed BI approach is discussed in details in next section.

4.4 Sensitivity analysis of propose BI approach

The concept of moving infeasible member towards feasible member till it crosses the boundary

separating feasible and infeasible region is explored in the proposed algorithm as Boundary

Inspection(BI) approach. The BI approach is sensitive to the ratio of feasible to infeasible

region and nature of feasible region (convex or concave). The parameter λ is used to decide

the location of the new point on the line joining the feasible point and infeasible point. The

value of this parameter is to identify the feasible boundary and move infeasible point to a

feasible region. Four case studies to investigate the sensitivity of the proposed algorithm have

been considered.

To study the effectiveness of this concept, a hypothetical two dimensional objective space

is used for this study, with 0≤ x1,x2 ≤ 10. Four different test cases studied are (A) FIcircle -

Feasible area inside the circle; (B) FOcircle - Feasible area outside the circle; (C) FIsquare -

Feasible area inside the square; and (D) FOsquare - Feasible area outside the square. The four

test cases are shown in Fig. (4.3).

Figure 4.3. Different test cases of feasible regions for study in two dimensional space, feasible area inside or ourside of circle or square

A population of 250 members is created randomly which is, uniformly distributed in

variable space. The infeasible members are converted to feasible members using the proposed


4.4 Sensitivity analysis of propose BI approach 115

BI approach. Crossover and mutations are carried out at every generation using simulated

binary crossover and non-uniform mutation. The simulated binary crossover, SBX uses

parameter ηc = 20 and crossover probability of 0.90, where as, the non-uniform mutation uses

parameter b = 4 and mutation probability of 0.1 for population evolution. The population is

randomly shuffled before every crossover and pairs of two populations are selected serially

from it for crossover. The evolution and proposed constraint handling mechanism process

being stochastic in nature, the results presented are an average of ten simulation runs using

randomly created ten different initial populations. The details of four case studies selected are

presented in table (4.1).

The BI approach for constraint handling in Multi-Objective Optimization is implemented

MATLAB. The sensitivity analysis results for all the four test cases designed are discussed in

subsequent subsections.

4.4.1 Feasible circular area inside square

The effect of % feasible region on total number of members requiring BI treatment and total

number of function evaluations needed for circular feasible area inside a square are presented

in fig. (4.4). The count of members requiring BI treatment is equal to the count of infeasible

members, as all the infeasible members at each generation are converted to feasible members

using BI treatment. Both the NCEs and BI treatment decreases as feasible area increases. The

NCEs required per BI treatment decreases as the feasible region % increases.

The generation wise members requiring BI treatment and NCEs are plotted for three

different cases in Fig. (4.5). The initial randomly created population has large number of

infeasible members which require BI treatment. As the populations evolves, new members are

created through crossover among feasible members resulting to more feasible members. This

reflects as reduction in BI treatment count as population evolves. The three cases clearly reflect

the effect of feasible area and generation wise evolution on number of members requiring BI

treatment. When feasible area is large, the members requiring BI treatment is lower. The three

cases compared in the plot have 13%, 27% and, 50% feasible region. After 100 generations of

evolution, the members requiring BI treatment are negligible and close to each other for all



Table 4.1. Details of test cases of feasible regions for study in two dimensional space, feasible area inside andour side of circle or square

Case Parameter Value / RelationsA) Feasible area inside the circle FIcircle

Constraint on Decision Variable 0≤ x1,x2 ≤ 10

Feasible Area constraint x21 + x2

2 ≤ r2

Feasible Area calculation r2

Values of r [ 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 ]

Feasible Area [ 12.6 15.2 18.1 21.2 24.6 28.3 32.2 36.3 40.7 45.4 50.3 ]

B) Feasible area outside the circle FOcircle


Feasible Area constraint x21 + x2

2 ≥ r2

Feasible Area: 100− r2

Values of r [ 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 ]

Feasible Area [ 99.2 98.5 97.5 96.2 94.7 92.9 90.9 88.7 86.1 83.4 80.4 ]

C) Feasible area inside the square FIsquare


Feasible Area constraint |x1− l/2| ≤ l/2 & |x2− l/2| ≤ l/2

Feasible Area: l2

Values of r [ 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 ]

Feasible Area [ 16.0 19.4 23.0 27.0 31.4 36.0 41.0 46.2 51.8 57.8 64.0 ]

D) Feasible area outside the square FOsquare


Feasible Area constraint |x1− l/2| ≥ l/2 & |x2− l/2| ≥ l/2

Feasible Area: 100− l2

Values of r [ 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 ]

Feasible Area [ 91.0 88.4 85.6 82.4 78.8 75.0 70.8 66.4 61.6 56.4 51.0 ]

the three cases.

The generation wise NCEs required for BI treatment are plotted for the same three cases

in Fig. (4.6). The BI treatment algorithm uses one value to locate a point on the line joining



10 15 20 25 30 35 40 45 50

feasible area, %

1000

2000

3000

4000

5000

6000

7000

8000

9000

tota

l co

un

t

BI treated members

NCEs

Figure 4.4. Effect of % feasible are on BI treatment count and NCEs for FIcircle case

0 50 100 150 200

Generation No.

0

10

20

30

40

50

60

BI

tre

ate

d m

em

be

rs

small feasible area

medium feasible areaa

large feasible area

Figure 4.5. Generation wise infeasible members requiring BI treatment FIcircle case



infeasible point and selected feasible point. The one value is selected randomly using a proba-

bility distribution from the Ensemble of predefined pool of values. The probability distribution

gets updated at the end of every learning period. 20 generations are used as learning period in

this study. The BI treatment requirement also reduces as an effect of learning of probability.

The attempt required to convert all infeasible solutions to feasible solution is represented in

terms of number of constraints evaluations (NCEs). The NCEs per member requiring the BI

treatment increases as infeasible region increases. For smaller feasible regions, the NCEs

increase drastically for initial few generations.

0 50 100 150 200

Generation No.

0

50

100

150

200

250

NC

Es

small feasible area


large feasible area

Figure 4.6. Generation wise NCEs required in BI treatment for FIcircle case

The plot of performance probability for all the selected three cases of different feasible

area (13%, 27% and, 50%) are plotted as bar chart in Fig. (4.7). The learning period used in

this study is 20 generations. The ensemble used in this sensitivity analysis has seven different

values of parameter λ . One member is to be selected from this based on the performance

probability. Initially, all the values are given equal probability of 1/7. The first bar chart repre-

sent that distribution. Based on the total attempts available, and success received in converting

infeasible member into a feasible member for those trials is converted to performance proba-



bility for each member of ensemble. The λ =−0.6 and λ =−0.3 does not succeed, hence, its

probability reduces and the λ = 2 has better performance, hence, its probability is increased.

This updated values are used in the next cycle of learning period of 20 generations. The three

plots clearly reflect the effect of feasible region on success probability. Success probabilities

has 10 cycles for 200 evolution generations. As populations evolves, the success probability

for lambda = 0.3 improves, where as, success probability for lambda = 2 decreases, second

learning cycle onward. It can be noticed that, this concept of selecting a value of parameter

from an Ensemble of predefined pool based on success probability eliminates the need for

problem specific parameter tuning.

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Pe

rfo

rma

nce

pro

ba

bili

ty,

pK

g

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Perf

orm

ance p

robabili

ty, pK

g

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Pe

rfo

rma

nce

pro

ba

bili

ty,

pK

g

= -0.6

= -0.3

= 0.3

= 0.6

= 1

= 1.5

= 2

Figure 4.7. Effect of feasible area on adoptive learning probability distribution for FIcircle case

4.4.2 Feasible circular area outside the circle within a square

The second case study is considering feasible area outside the circle within a square. It is a

more difficult case of dealing with constraint. The feasible region ranges from 80% to 98%.

The members requiring BI treatment decreases, as feasible part increases. The NCEs also

linearly decrease as the feasible region increases. Both rates are linear, but the rate of decrease

in members requiring BI treatment is much more larger than the NCEs, which is reflected as

different slopes of two lines. The total count of number requiring BI treatment and NCEs

are much higher for FOcircle case compared to FIcircle case, which is an indication of the

difficult type of constant.



80 85 90 95 100

feasible area, %

1000

2000

3000

4000

5000

6000

7000

8000

tota

l co

un

t

BI treated members

NCEs

Figure 4.8. Effect of % feasible are on BI treatment count and NCEs for FOcircle case

The generation wise total count for BI treatment and NCEs are plotted in Fig. (4.9). The

small feasible area corresponds to 80%, medium feasible region to 90% and, large feasible

region corresponds to 98%. The generation wise trends of all the three cases are identical to

FIcircle case. Initial few generations have larger members requiring BI treatment and, hence,

the NCEs are also large.


area (80%, 90% and, 98%) are plotted as bar chart in Fig. (4.10). The learning period used

in this case study is kept constant, which is 20 generations. The ensemble used is also same

having values [-0.6 -0.3 0.3 0.6 1 1.5 2] for parameter λ . Initially, all the values are given equal

probability of 1/7 as there are total 7 members. The first bar chart represent that distribution

in all the three plots. The dramatic changes in the success probability can be observed from

the plot. For small feasible area, λ = 1.5 and λ = 2 covers the whole bar chart in last learning

cycle. The three plots clearly reflect the effect of change of feasible region % on success

probability. The bar charts clearly conforms the sensitivity of automated selection process



0 50 100 150 200

Generation No.

0

5

10

15

20

25

30

BI

tre

ate

d m

em

be

rssmall feasible area


large feasible area

0 50 100 150 200

Generation No.

0

50

100

150

NC

Es

small feasible area


large feasible area

(a) Members requiring BI treatment (b) NCEs required for BI treatment

Figure 4.9. Generation wise infeasible members requiring BI treatment and NCEs required for FOcircle case

using ensemble approach with success probability based selection.

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Perf

orm

ance p

robabili

ty, pK

g

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Perf

orm

ance p

robabili

ty, pK

g

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Perf

orm

ance p

robabili

ty, pK

g

= -0.6

= -0.3

= 0.3

= 0.6

= 1

= 1.5

= 2

Figure 4.10. Effect of feasible area on adoptive learning probability distribution of selecting a division rationvalue from an Ensemble pool

4.4.3 Feasible square area inside a square

The third case study is considering feasible area inside a small square within a larger square.

Outer square decides the bounds on decision variable, where as, the inner square decides

the constraint boundary. The size of inner square is varied to change the feasible region

% area. It is a little difficult case compared to FIcircle case, as the feasible boundaries are



straight lines. The feasible region ranges from from 15% to 65%. The members requiring BI

treatment decreases exponentially as feasible part increases. The NCEs decrease linearly as

the feasible region increases. Both are decreasing rates, but the rate of decrease in members

requiring BI treatment is much more larger than the NCEs and, is exponentially decreasing.

This causes large differences for small feasible % area and the gap decreases as the % feasible

area increases. The total count of number requiring BI treatment and NCEs are little higher for

FIsquare case compared to FIcircle case, which is an indication of little increase in constraint

difficulty.

10 20 30 40 50 60 70

feasible area, %

1000

2000

3000

4000

5000

6000

7000

tota

l co

un

t

BI treated members

NCEs

Figure 4.11. Effect of % feasible are on BI treatment count and NCEs for FIsquare case



region corresponds to 65%. The generation wise trends of all the three cases of FIsquare

case are identical to FIcircle case. Initial few generations of small % feasible area have larger

members requiring BI treatment and, hence, the NCEs are also large.

The plot of performance probability for all the selected three cases of different feasible area



0 50 100 150 200

Generation No.

0

10

20

30

40

50

BI

tre

ate

d m

em

be

rssmall feasible area


large feasible area

0 50 100 150 200

Generation No.

0

50

100

150

200

NC

Es

small feasible area


large feasible area


Figure 4.12. Generation wise infeasible members requiring BI treatment and NCEs required for FIsquare case

(15%, 25% and, 65%) are plotted as bar chart in Fig. (4.13). The learning period used in this

case study is also kept constant, which is 20 generations. The same pool of ensemble is having

values [-0.6 -0.3 0.3 0.6 1 1.5 2] for parameter λ . Like all other cases, initially, performance

probability values for all the members are equal, which is taken 1/7 for 7 members. The first

bar chart represents equal probability distribution in all the three plots. The dramatic changes

in the success probability is also observed from the plot in this case. The three plots clearly

reflect the effect of change of feasible region % on success probability. For the case of larger

% feasible region area, all the probability values for positive λ are performing consistently. In

other two cases, dominating λ values are changing every learning cycle. The bar charts also

conforms the sensitivity of automated selection process for FIsquare case.

4.4.4 Feasible area outside the small square within a square

The fourth case study is considering feasible area outside a small square within a larger

square, named as FOsquare. Outer square decides the bounds on decision variable, where

as, the inner square decides the constraint boundary. The size of inner square is varied to

change the feasible region % area. It is the most difficult case compared to all the three

previous cases. The feasible region ranges from from 51% to 92%. The members requiring BI

treatment decreases linearly as % feasible area increases. The NCEs also decrease linearly as



1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Perf

orm

ance p

robabili

ty, pK

g

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Perf

orm

ance p

robabili

ty, pK

g

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Perf

orm

ance p

robabili

ty, pK

g

= -0.6

= -0.3

= 0.3

= 0.6

= 1

= 1.5

= 2


the % feasible area increases. Both are decreasing rates, but the rate of decrease in members

requiring BI treatment is much more larger than the NCEs resulting to a larger difference of

BI treatment count. The NCEs per BI treatment decreases as % feasible area increases.

50 60 70 80 90 100

feasible area, %

1000

2000

3000

4000

5000

6000

7000

8000

9000

tota

l co

un

t

BI treated members

NCEs

Figure 4.14. Effect of % feasible are on BI treatment count and NCEs





region corresponds to 92%. The generation wise trends of all the three cases of FOsquare case

are identical to FOcircle case. Initial few generations for small % feasible area have larger

members requiring BI treatment compared to the NCEs count. The BI treatment requirement

reduces as an effect of increased % feasible area and improvement in performance probability.

The attempt required to convert all infeasible solutions to feasible solution is represented in

terms of number of constraints evaluations (NCEs). The NCEs per member requiring the BI

treatment increases, as infeasible region increases. For smaller feasible regions, the NCEs

increase drastically for initial few generations.

0 50 100 150 200

Generation No.

0

5

10

15

20

25

30

BI

tre

ate

d m

em

be

rs

small feasible area


large feasible area

0 50 100 150 200

Generation No.

0

50

100

150

NC

Es

small feasible area


large feasible area


Figure 4.15. Generation wise infeasible members requiring BI treatment and NCEs required for FOsquare case


area (51%, 75% and, 92%) are plotted as bar chart in Fig. (4.16). The learning period used is

kept constant, which is 20 generations. The ensemble of BI treatment parameter λ used is

having values [-0.6 -0.3 0.3 0.6 1 1.5 2]. Initially, as there is no information available about

which value should be preferred: all the values are given equal probability, which is 1/7 for

each of the 7 members. The first strip of bar chart represents the distribution in all the three

plots. The different patterns of changes in the success probability can be observed from the

three different bar plots with different % feasible area. Negative values of the parameter λ

represents extrapolation of line joining infeasible point and selected feasible point. For this



case of FOsquare and FOcircle, these parameter values could get success. For other two cases

of FIcircle and FIsquare, they get vanished. The reason for that is when feasible area is inside

the infeasible zone, negative value of parameter λ can not crest feasible point. The domination

of negative values for parameter λ over the other values can be observed in this case.

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Perf

orm

ance p

robabili

ty, pK

g

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1P

erf

orm

ance p

robabili

ty, pK

g

1 2 3 4 5 6 7 8 9 10

Learning cycles

0

0.2

0.4

0.6

0.8

1

Perf

orm

ance p

robabili

ty, pK

g

= -0.6

= -0.3

= 0.3

= 0.6

= 1

= 1.5

= 2


The sensitivity analysis clearly represents the effectiveness of automated selection of

parameter for BI treatment from the Ensemble of pre defined pool of probable values. The

selection is based on success probability based on performance history. The automated

selection process proposed takes care of the nature of the problem and, changes during

evolution process. The bar charts for all the four cases conforms the sensitivity of automated

selection process.

4.5 Performance Measure for MOO

The aim of all multi-objective optimization algorithms is to find as many different solutions as

possible in the Pareto optimal set. A multi-objective optimization algorithm has to perform

two tasks: (i) to guide the search towards the global Pareto optimal region and (ii) to maintain

the population diversity (in the objective space, in the parameters space or in both of them)

in the current non-dominated front. The general performance criteria for the multi-objective

optimization algorithms are:


4.5 Performance Measure for MOO 127

• Accuracy - how close the generated non-dominated solutions are to the best known

prediction.

• Coverage - how many different non-dominated solutions are generated and how well

they are distributed.

• Variance for every objective - which is the maximum range of non-dominated front,

covered by the generated solutions (fraction of the maximum range of the objective in

the non-dominated region, covered by a non-dominated set).

The performance of the search algorithm is difficult to evaluate when, true Pareto optimal

set is not known. Those results are generally presented using various performance measures

for the search algorithms. Some tools for visual representations of non-dominated solutions

are: scatter-plot matrix, value path, bar chart, star coordinate and, visual methods. Visual

descriptions are now inadequate as the area of multi-objective optimization has become much

popular and number of different algorithms and modifications are coming up. Performance

metrics are important performance assessment measure, which also allow us to compare

algorithms and to adjust their parameters for better results. Deb (2001) categorised them in

three groups: metrics calculating closeness to the Pareto optimal front, metrics calculating

diversity amongst non-dominated solutions and, metrics calculating closeness and diversity.

4.5.1 Convergence to true pareto front

The commonly used metrics for evaluating closeness to the true pareto optimal front are error

ratio Veldhuizen (1999), generational distance (GD) Veldhuizen (1999), maximum pareto

optimal front error Veldhuizen (1999) and, set convergence metric Zitzler (1999). GD being

simple to evaluate and one of the widely used parameter. GD is an average distance of the

solutions fond by the algorithm to the true pareto front. For a set Q of N solutions from a

known set of the pareto optimal set P∗. Veldhuizen (1999) has defined average distance of Q

from P∗, the generational distance γ as:

γ =

(∑|Q|i=1 dp

i

)1/p

|Q|(4.4)



where, Q is solution set having |Q|members. We use p=2 and di is minimum distance between

the member in solution set and nearest member is true pareto set, which is defined as:

di = min

√M

∑m=1

( f (i)m − f ∗(k)m )2

(4.5)

where, M is number of objectives, i and k are member index in solution set and true pareto set

respectively.

f ∗(k)m is the mth objective function value of the kth member of P∗ and, f (i)m is the corre-

sponding objective function value from the true pareto front. When the objective function

values are of different order or magnitudes, they should be normalized by an appropriate

weighing factor in defining the distance, di. A large number of solutions uniformly distributed

in the true pareto should be used to calculate the γ matrix. The γ matrix measures the extent

of convergence to a known set of pareto optimal solutions. Since, multi-objective algorithms

would be tested on problems having a known set of Pareto-optimal set, the calculation of this

metric is possible. But, realize that such a metric cannot be used for any arbitrary problem.

Even when all solutions converge to the Pareto-optimal front, the above convergence metric

does not have a value zero. The metric will be zero only when each obtained solution lies

exactly on each of the chosen solutions. Although this metric alone can provide some infor-

mation about the spread in obtained solutions, we need to define another metric to measure

the spread in solutions obtained by an algorithm.

4.5.2 Matrix to measure distribution of solutions

The purpose of distribution metric is to represent the span of true pareto front covered by

the obtained solutions and its uniformity in the span covered. There exists many metrics to

find diversity amongst the obtained non-dominated solutions. Few popular amongst them,

are spacing matrix (Schott, 1995), Chi-square like deviation measure matrix (Deb, 1989),

maximum spread matrix (Zitzler, 1999) and, spread matrix (Deb et al., 2002). The commonly

used spread for performance measure representing the distribution of solutions in the pareto

front, which can be defined as follows:


4.6 Test Problems and Engineering Design Applications 129

∆ =∑

Mm=1 de

m +∑|Q|i=1 |di− d|

∑Mm=i de

m + |Q|d(4.6)

where, dem is the distance between the extreme solutions and the boundary solutions of the

obtained non-dominated solution set Q from the known end solutions of known solution set

P∗. The parameter di is the distance measured between the neighbouring solutions and d is

the mean value of this distance measure. Note that the maximum value of ∆ can be greater

than one. Though, a good distribution would make all distances di equal to d and would make

dem = 0. Thus, the most widely and uniformly spread of the non-dominated solutions result to

the zero value of ∆. For any other distribution, the value of the metric would be greater than

zero. Note that the above diversity metric can be used on any non-dominated set of solutions,

including one which is not the Pareto-optimal set.

4.5.3 Matrix evaluating closeness and diversity

There are some metrics which combinedly evaluates closeness and diversity. They are:

hypervolume, attainable surface based statistical metric, weighted metric, non-dominated

evaluation metric and, Inverted Generational Distance (IGD). IGD is a well known and widely

accepted performance measure, which accounts convergence and distribution both Zhao et al.

(2012). Let P∗ be a set of uniformly distributed true pareto optimal solutions and A is the

obtained solution set, then IGD value is the average distance from P∗ to A. Note that the

smaller the IGD value, better is the performance of the MOO algorithm. We use IGD metric

in this work for performance comparison of results obtained using different MOO algorithms.

4.6 Test Problems and Engineering Design Applications

For testing the efficiency and effectiveness of the proposed BI approach for constraint handling

with EAs, we use three two-objective constrained optimization test problems with known

pareto optimal solutions. We also use two well studied design applications of disk brake and

welded beam to test the performance of proposed algorithm. The three test problems are

namely: Constr-Ex , BNH (from Binh and Korn 1997 study) , OSY (from Osyczka and Kundu

1995 study). All the problems have two objective functions, which are to be reduced. Every



test function have certain difficulties for constrained multi-objective optimisation. We use test

problems with known sets of constrained Pareto-optimal solutions. The detailed discussion of

the problem and its solution are available in Deb (2001). For the convenience of the reader we

briefly define the test problems here.

4.6.1 Test problem-1: Constr-Ex

A well studied two variable two objective constrained optimization problem, Constr-Ex is an

extension of Max-Ex unconstrained problem.

Minimize

f1(x) = x1,

f2(x) =1+ x2

x1,

Subject to

g1(x)≡ x2 +9x1 ≥ 6,

g2(x)≡−x2 +9x1 ≥ 1,

0.1≤ x1 ≤ 1,

0≤ x2 ≤ 5.

(4.7)

The constrained pareto optimal set comprises of two regions: region A with 0.39≤ x∗1 ≤

0.67 and, x∗2 = 6−9x∗1 while, region B with 0.67≤ x∗1 ≤ 1.00 and x∗2 = 0.

4.6.2 Test problem-2: BNH

We use another well studied constrained optimization two variable problem, BNH defined as

follows:



Minimize

f1(x) = 4x21 +4x2

2,

f2(x) = (x1−5)2 +(5− x2)2,

Subject to

g1(x)≡ (x1−5)2 + x22 ≥ 25,

g2(x)≡ (x1−8)2 +(x2 +3)2 ≥ 7.7,

0≤ x1 ≤ 5,

0≤ x2 ≤ 3.

(4.8)

The constrained pareto optimal set for this test problem also consists of two regions: region

A with x∗1 = x∗2 ∈ [0,3] and, region B with x∗1 ∈ [3.5],x∗2 = 3.

4.6.3 Test problem -3: OSY

OSY is a six variable, six constrained test problem, which has only 3.25% feasibility ratio as

compared to 52.52% for Constr-Ex and 93.61% for BNH. The OSY problem is defined as

follows:



Table 4.2. Pareto optimal solutions for the OSY problem

Region x∗1 x∗2 x∗3 x∗51 5 1 [1-5] 52 5 1 [1-5] 13 [4.056-5] (x∗1−2)/3 1 14 0 2 [1-3.732] 15 [0-1] (2− x∗1) 1 1

Minimize

f1(x) =−[25(x1−2)2 +(x2−2)2

+(x3−1)2 +(x4−4)2 +(x5−1)2],

f2(x) = x21 + x2

2 + x23 + x2

4 + x25 + x2

6,

Subject to

g1(x)≡ x1 + x2−2≥ 0,

g2(x)≡ 6− x1− x2 ≥ 0,

g3(x)≡ 2+ x1− x2 ≥ 0,

g4(x)≡ 2− x1 +3x2 ≥ 0,

g5(x)≡ 4− (x3−3)2− x4 ≥ 0,

g6(x)≡ (x5−3)2 + x6−4≥ 0,

0≤ x1,x2,x6 ≤ 10,

1≤ x3,x5 ≤ 5,

0≤ x4 ≤ 6.

(4.9)

The pareto optimal set for this test problem consists of five regions, where every region

lies on one of the constraints. The pareto optimal set solutions are obtained at x∗4 = x∗6 = 0

while, the remaining variables are shown in table (4.2).

4.6.4 Engineering application-1: Design of welded beam

Design of a welded beam is a classical benchmark test application, which has been solved

by many researchers. The problem has four design variables: the width, w and the length

of the welded area, L; the depth, d and the thickness of the main beam, h. The objective is

to minimize both, the overall fabrication cost and the end deflection. The multi-objective



problem formulation is described below:

Minimize

f1(x) = 1.10471w2L+0.04811dh(14+L),

f2(x) = δ ,

Subject to

g1(x)≡ w−h≤ 0,

g2(x)≡ δ (x)−0.25≤ 0,

g3(x)≡ τ(x)−13,600≤ 0,

g4(x)≡ σ(x)−30,000≤ 0,

g5(x)≡ 0.10471w2

+0.04811dh(14+L)−5≤ 0,

g6(x)≡ 0.125−w≤ 0,

g7(x)≡ 6000−P(x)≤ 0,

0.1≤ L,d ≤ 10,

0.125≤ w,h≤ 2.0,

where,

σ(x) =504,000

hd2 , δ (x) =6000√

2wL,

J =√

2wL(

L2

6+

(w+d)2

2

),

D = 0.5√

L2 +(w+d)2,

Q = 6000(14+0.5L),

τ(x) =

√λ 2 +

λβLD

+β 2,

P(x) = 614230dh3

6

(1−

d√

30/4828

).

(4.10)

4.6.5 Engineering application-2: Design of disk brake

Design of a multiple disc brake is another benchmark application for multi-objective con-

strained optimization described as follows:



Minimize

f1(x) = 4.9×10−5(R2− r2)(s−1),

f2(x) =9.82×106(R2− r2)

Fs(R3− r3),

Subject to

g1(x)≡ 20− (R− r)≤ 0,

g2(x)≡ σ(x)−30,000≤ 0,

g3(x)≡F

3.14(R2− r2)−0.4≤ 0,

g4(x)≡2.22×10−3F(R3− r3)

(R2− r2)2 −1≤ 0,

g5(x)≡ 900− 0.0266Fs(R3− r3)

(R2− r2)≤ 0,

55≤ r ≤ 80,

75≤ R≤ 110,

1000≤ F ≤ 3000,

2≤ s≤ 20.

(4.11)

The two objectives are minimizing the overall mass and the braking time. The decision

variables are the discs inner radius, r; discs outer radius, R; the engaging force, F and, the

number of friction surface, s. The design constraints are applied on the torque, pressure,

temperature and, length of the brake.

4.7 Results and Discussion

PUALGA algorithm implemented in MATLAB using non-dominated sorting and elite survival

selection operator for MOO is used to evaluate three constraint handling approaches. The

population size is kept as 100 for all the test problems. Twenty simulation runs were carried

out for every test problem with distinct initial populations and a statistical analysis is presented

for the comparison study of various algorithms. Number of function evaluations (NFEs) and

number of constraint evaluations (NCEs) are the two important measures for evaluating the

computational expense of any constrained optimization algorithm. Performance metric IGD

values are presented as the functions of Run time, NFEs and NCEs for the augmented penalty,


4.7 Results and Discussion 135

ignore infeasible and boundary inspection to compare the computational performance. The

results obtained for three test problems and two design applications are discussed in separate

subsections.

4.7.1 Test problems

The average of twenty runs in terms of IGD convergence profiles for the test problem Constr-

Ex are presented in Fig. (4.17). The figure clearly indicates that the convergence of the

proposed BCA constraint is better than the other two algorithms. The BI approach IGD

values continues to decrease at a higher rate than the other two algorithm, which indicates

its better convergence capability. The BI approach converts infeasible members to feasible

ones by projecting them through the feasible solutions. This mechanism creates possibilities

of exploring guided search, which in turn improves the convergence.

0 2 4 6 8 10

Run time (s)

0.02

0.04

0.06

0.08

0.1

me

an

IG

D

Augmented Penalty

Ignorance Infeasible

Bounday Inspection

Figure 4.17. Average IGD values against Run time for ConstrEx test function

Since the BI implementation needs to evaluate constraints for all trial points, its conver-

gence is also evaluated in terms of NCEs. The two IGD profiles, with respect to the NFEs and

NCEs have similar trends among the three algorithms. The average IGD value convergence



for Constr-Ex function is presented in terms of NFEs in Fig. (4.18) and NCEs in Fig. (4.19).

The convergence profile became stagnant after 5,000 NFEs for BI approach and 10,000 NFEs

for ignorance infeasible .

0 0.5 1 1.5 2 2.5

NFEs 104

0.02

0.04

0.06

0.08

0.1

me

an

IG

D

Augmented Penalty


Bounday Inspection

Figure 4.18. Average IGD values against NFEs for ConstrEx test function

Pareto front obtained at the end of 25 generations and, 100 generations for Constr-Ex test

problem are presented in Fig. (4.20). The pareto plot at 25 generations clearly indicate that

the BI approach has uniform and better converged pareto.

The convergence plots for the BNH test problem is shown in Fig. (4.21, 4.22 and 4.23).

The obtained results with this test problem are similar to the Constr-Ex problem. Since this

test problem has high (93.61%) feasibility ratio, the nature of convergence plots with respect

to NFEs and NCEs are quite similar.

Pareto front obtained at the end of 50 generations for BNH test problem is presented in

Fig. (4.24). Though, all algorithms converge very close to true pareto front, better uniformity

of distribution of pareto optimal solutions is observed in BI approach.

OSY test problem convergence plots are shown in Fig. (4.25,4.26, and 4.27). As this test

problem have very low feasibility ratio of 3.25%, the nature of the convergence plots with

respect to NFEs and NCEs are expected to be different. The other two algorithms show good



0 0.5 1 1.5 2 2.5

NCEs 104

0.02

0.04

0.06

0.08

0.1

me

an

IG

D

Augmented Penalty


Bounday Inspection

Figure 4.19. Average IGD values against NCEs for ConstrEx test function

0.3 0.4 0.5 0.6 0.7 0.8 0.9

f1

1

2

3

4

5

6

7

8

9

f 2

Augmented Penalty


Bounday Inspection

0.3 0.4 0.5 0.6 0.7 0.8 0.9

f1

1

2

3

4

5

6

7

8

9

f 2

Augmented Penalty


Bounday Inspection

(a) at 25 Generations (b) at 100 Generations

Figure 4.20. Pareto Front for ConstrEx test function

performance in terms of NCEs compared to BI approach due to low feasibility ratio.

Pareto front obtained at the end of 100 generations and, 250 generations for OSY test

problem are presented in Fig. (4.28). Though, all algorithms converge very close to true

pareto front, better uniformity of distribution of pareto optimal solutions is observed in BI

approach. Augmented penalty approach obtained best coverage of pareto front covering both



0.1 0.2 0.3 0.4 0.5 0.6 0.7

Run time (s)

0.4

0.5

0.6

0.7

0.8m

ea

n I

GD

Augmented Penalty


Bounday Inspection

Figure 4.21. Average IGD values against Run time for BNH test function

500 1000 1500 2000 2500

NFEs

0.4

0.5

0.6

0.7

0.8

me

an

IG

D

Augmented Penalty


Bounday Inspection

Figure 4.22. Average IGD values against NFEs for BNH test function



500 1000 1500 2000 2500

NCEs

0.4

0.5

0.6

0.7

0.8

me

an

IG

D

Augmented Penalty


Bounday Inspection

Figure 4.23. Average IGD values against NCEs for BNH test function

0 20 40 60 80 100 120 140

f1

0

10

20

30

40

50

f 2

Augmented Penalty


Bounday Inspection

Figure 4.24. Pareto Front for BNH test function at 50 Generations



0 5 10 15 20

Run time (s)

0

5

10

15

20

25m

ea

n I

GD

Augmented Penalty


Bounday Inspection

Figure 4.25. Average IGD values against Run time for OSY test function

0 0.5 1 1.5 2 2.5 3

NFEs 104

0

5

10

15

20

25

me

an

IG

D

Augmented Penalty


Bounday Inspection

Figure 4.26. Average IGD values against NFEs for OSY test function



0 0.5 1 1.5 2 2.5 3

NCEs 104

0

5

10

15

20

25

me

an

IG

D

Augmented Penalty


Bounday Inspection

Figure 4.27. Average IGD values against NCEs for OSY test function

the end of the pareto front.

-300 -250 -200 -150 -100 -50 0

f1

0

20

40

60

80

f 2

Augmented Penalty


Bounday Inspection

-300 -250 -200 -150 -100 -50 0

f1

0

20

40

60

80

f 2

Augmented Penalty


Bounday Inspection


Figure 4.28. Pareto Front for OSY test function



4.7.2 Design applications

The efficacy of the proposed algorithm is further validated by two engineering design applica-

tions: the welded beam design and, disc break design. The true theoretical pareto front for the

design applications are not available, hence, we use filtered pareto set from a large solution

set converged after very long evolution as true front is used for both the applications. This

true pareto front is used to calculate IGD values and compare performance of all the algorithms.

The average convergence plots as function of run time for the welded beam design

application is presented in Fig. (4.29). The performance of augmented penalty constraint

method is observed to be very good at start but it became slow later on. The performance of

BI approach is the best as compared to the other two algorithm.

0 5 10 15

Run time (s)

0

0.2

0.4

0.6

0.8

1

mean IG

D

Augmented Penalty


Bounday Inspection

Figure 4.29. Average IGD values against Run time for Welded Beam design application

The average IGD values are also plotted as a function of NCEs and NFEs in Fig. (4.30). It

can be observed from the plot that, the performance of ignore infeasible and BI approach with

respect to NCEs and NFEs are identical. The performance of augmented penalty improves

when plotted it against NCEs. The design application has tough constraints to satisfy, which



lower the initial convergence for BI approach algorithm. The BI approach lags the other two

algorithms with respect to the NCEs for initial few generations.

0 0.5 1 1.5 2

NFEs 104

0

0.2

0.4

0.6

0.8

1

me

an

IG

D

Augmented Penalty


Bounday Inspection

0 1 2 3 4

NCEs 104

0

0.2

0.4

0.6

0.8

1

me

an

IG

D

Augmented Penalty


Bounday Inspection

Figure 4.30. IGD convergence profiles for Disk Welded Beam design application

Pareto optimal solutions obtained at the end of 100 and 250 generations by all the three

algorithms are presented in Fig. (4.31) for the welded beam design application. The solution

distribution within pareto front is observed to have good uniformity for BI approach. The

coverage, convergence and, distribution all improves as the algorithm converges.

1.5 2 2.5 3 3.5 4 4.5 5 5.5

f1

2

4

6

8

10

12

14

f 2

10-3

Augmented Penalty


Bounday Inspection

1.5 2 2.5 3 3.5 4 4.5 5 5.5

f1

2

4

6

8

10

12

14

16

f 2

10-3

Augmented Penalty


Bounday Inspection


Figure 4.31. Pareto Front for Welded Beam design design application

The average IGD value plots for the disc brake design application is presented in Fig.

(4.32). Since the theoretical(true) pareto front is unknown for this problem, the optimal

solutions obtained after 2000 generations are used as the reference pareto optimal set for the



IGD value calculations.

0 2 4 6 8

Run time (s)

0.04

0.05

0.06

0.07

0.08

0.09

0.1

mean IG

D

Augmented Penalty


Bounday Inspection

Figure 4.32. Average IGD values against Run time for Disk Break design application

The average IGD values are also plotted as a function of NCEs and NFEs in Fig. (4.33). It

can be observed from the plot that the performance of BI approach is best when we plot IGD

against NFEs, where as, the same convergence plot against NCEs looks little inferior to other

two algorithms for initial few generations. This can be attributed to the difficulty to satisfy the

constraint. Lower initial convergence rate for BI approach algorithm trend is observed for all

tough constraints. This lag increases as the constraints become more tough.

Pareto optimal solutions obtained for the welded beam design application at the end of

25 and 250 generations by all the three algorithms are presented in Fig. (4.34). The solution

distribution within pareto front is observed to have good uniformity for BI approach. The

coverage is also better with BI approach.

The simulation results of all the test problems and design applications indicate that BI

constraint is a very efficient algorithm for constraint handling for multi-objective optimization

compared to augmented penalty function and ignore infeasible constraint handling algorithms.


4.8 Summary 145

0 1 2 3 4

NFEs 104

0.04

0.05

0.06

0.07

0.08

0.09

0.1m

ea

n I

GD

Augmented Penalty


Bounday Inspection

0 1 2 3 4 5

NCEs 104

0.04

0.05

0.06

0.07

0.08

0.09

0.1

me

an

IG

D

Augmented Penalty


Bounday Inspection

Figure 4.33. IGD convergence profiles for Disk Break design application

0 0.5 1 1.5 2 2.5 3

f1

2

4

6

8

10

12

14

16

18

f 2

Augmented Penalty


Bounday Inspection

0 0.5 1 1.5 2 2.5 3

f1

2

4

6

8

10

12

14

16

18

f 2

Augmented Penalty


Bounday Inspection


Figure 4.34. Pareto Front for Disk Break design application

4.8 Summary

Multi-objective constrained optimization problems are typically very difficult to solve. Evolu-

tionary Algorithms (EAs) perform very well for multi-objective optimization problems due to

their capacity to evolve multiple conflicting solutions simultaneously. The EAs are criticised

for their constraint handling capacity. EAs can handle bounds on decision variable without

any modification or attention as it is the part of their natural design. Relationship among

the decision variables result into constraints, which are to be satisfied by members evolving.

There are different constraint handling mechanisms proposed and tested in literature for EAs,

but still they lack in their capacity and efficiency. In this chapter, the constraint handling



by BI approach in evolutionary algorithms was presented and compared with existing well

known approaches. The BI approach with automated parameter selection using performance

based probability was also presented. This concept of using parameter Ensemble and selecting

parameter at run time based on adaptive performance probability adds additional capacity to

the proposed BI approach.

In the proposed BI algorithm, every infeasible member is projected through the randomly

selected feasible member . This approach uses the original objective function values without

any modification. The selection of parameter which locates the new point on the line joining

infeasible and feasible point selected from an Ensemble based on success probability history.

This approach of automated selection of parameters from pool helps in adaptive tuning during

the evolution process. The sensitivity of the algorithm is tested using four test cases. They

include feasible area inside or outside of square or circle. The four test cases are designed

to test the sensitivity of algorithm to % feasible area and nature of boundary separating the

feasible region. The sensitivity study clearly reflected the effect of BI treatment and automated

selection process. The bar charts presented for performance probability based on which the

selection reflected the sensitivity of the ensemble approach.

The efficacy of the BI approach is presented using the multi-objective PUALGA algo-

rithm and has been tested with three bench mark test functions and two engineering design

applications. Though, the algorithm proposed is very general and can be implemented for any

population based evolutionary method for single or multi-objective optimization, we demon-

strated the performance under genetic algorithm framework using the PUALGA algorithm

developed by us. Statistical analysis of the performance measure, IGD is presented using 20

simulation runs for all the test problems and design applications. Further, the performance of

the BI approach is compared with two popular constraint handling algorithms namely: aug-

mented penalty function and, ignore infeasible. Converge plot in terms of IGD are presented

against run time, NFEs and NCEs to evaluate the comparative performance.


147

Chapter 5

Rubber Extruder Modelling and

Simulation

Efficient design of extruder that produces defect free product with stable flow is an impotent

concern of the experts in polymer processing industries. Modelling and simulation is exhaus-

tively explored to assist engineers in enhancing designs, but complex processing requirements

along with complex behaviours of polymers has limited its application in polymer processing

area. With recent advancements in computational powers and modelling simulation tools,

computations for polymer processing has become feasible. Extrusion is an important polymer

processing equipment for rubber, plastic and food industries. A mathematical model for rubber

extrusion is developed using finite difference technique considering temperature dependent

viscosity modelled using Carreau-Yasuda model. The model solution algorithm is also pro-

posed and tested to converge velocity and temperature profiles within the extruder channel.

This validated model is used for optimization of screw design parameters and temperature

profile simultaneously to maximize throughput while minimizing power consumption. The

temperatures of the material under process within the extruder and residence time distribution

of product are also tracked for assured quality of product. The screw helix angle, channel depth

and, screw speed are used as manipulated design parameters along with barrel temperature

profile. Best screw geometry, screw speed and, barrel temperature profile are obtained using

multi-objective optimization algorithm. These multiple optimum solutions assist the decision

maker in selecting an appropriate design which is the best according to his needs.

148 5. Rubber Extruder Modelling and Simulation

5.1 Introduction

Extrusion is one of the important production methods in the polymer processing industry.

It is an energy intensive process, hence, the energy efficiency is one of the major concerns.

Selection of the most energy efficient design and processing conditions are explored to reduce

operating costs. Extruder consumes energy for the screw drive motor, barrel heaters, cooling

fans, cooling water pumps and, gear pumps. The screw drive motor is the largest energy

consuming device in an extruder and, barrel/die heaters are the second largest energy demand-

ing components. Screw design and barrel temperature plays significant role in these energy

demands and both are inter connected. Extruders frequently run at non-optimised conditions

and can account for 15–20 % of overall process energy losses (Deng et al., 2014).

Extruder is used by rubber, plastic and food industries for different purposes. Each appli-

cation has different characteristics of material processed, different demands and performance

expectations, hence, the understanding of extruder process become very important (Rauwen-

daal, 2009; Campbell and Spalding, 2013). Modelling and simulation of extruder helps to

expand the understanding and gives flexibility in testing modifications before implementation.

Optimization using experimentally validated model of extruder gives much confidence to the

designer in implementation of modification to improve performance saving time. The current

study focuses on extruder application for rubber industry. The traditional design of rubber

extruder screws is a costly procedure, both in terms of actual screw modification and, its

evaluation. If a proposed modification does not give the desired improvement, that design is

scrapped and further modification is undertaken with a new screw. Such trials are considerably

costly and time consuming. Hence, it is becoming important to use modelling techniques to

speed up such screw design modifications. Modelling the extrusion process which resemble

the real process very closely and trying design modification followed by experimental valida-

tion is currently adopted practice. Different modelling approaches are reviewed to develop

an extruder model to optimize rubber extruder screw design (Azhari et al., 1998; Desai and

Patel, 2005; Ghoreishy et al., 2005; Ha et al., 2008; Trifkovic et al., 2012; Rauwendaal, 2014b).

Modelling and simulation of flow in single screw extruder has been the subject of many


5.1 Introduction 149

studies due to the importance of screw extrusion in different manufacturing operations(Li

and Hsieh, 1996; Ghoreishy et al., 2000; Wood and Rasid, 2003; Vera-Sorroche et al., 2013;

Chaturvedi et al., 2017; Orisaleye et al., 2018). Initially, models studied used two-dimensional

flow of Newtonian fluids with temperature independent viscosity which could not closely

represent the practical results. Polymers are strongly non-Newtonian and, the assumption of

Newtonian behaviour resulted in substantial errors. The extruder models were extended to

deal with non-Newtonian, temperature independent fluids, which complicated the problem to

the extent that analytical solution for two or three dimensional velocity profiles was no longer

possible. It made it compulsory to switch over to numerical simulations using discrete compu-

tational domain Karwe and Jaluria (1990). There are very few public literature claiming robust

algorithm for extrusion simulation capable to converge velocity, temperature and pressure

profiles considering temperature dependent non-Newtonian viscosity under non-isothermal

operating conditions Nejad and Javaherdeh (2014); Ke et al. (2014); Abeykoon et al. (2016).

Syrjälä (1997) studied screw extrusion process and developed the flow and heat transfer

characteristics in a rectangular channel covered by an isothermally heated moving wall for

a non-Newtonian fluid under fully developed creeping flow conditions. They solved the

partial differential equations in model using the finite element method together with a penalty

formulation. They noticed that the diagonally moving top wall created circulatory motion

within the channel which influenced heat transfer. Ferretti and Montanari (2007) presented a

finite-difference approach for solving the down channel velocity in a single screw extruder for

Newtonian fluids. The authors developed a tool in MS Excel to obtain the velocity field which

is very easy to apply and efficient also. Vignol et al. (2005) presented a simplified model

relating material properties and extruder operating conditions to predict mass flow rate and

pressure at the exit of a single screw extruder.

Abeykoon et al. (2011) developed a static non-linear polynomial model to predict the die

melt temperature profile and used the model to predict optimum process settings to achieve

the desired average die melt temperature while minimising melt temperature variance across

the melt flow. Abeykoon et al. (2014) experimentally observed the mass throughput, total



energy consumption and power factor of an extruder over different processing conditions.

They developed an empirical model for total extruder energy demand using a commercially

available extrusion simulation software. They noticed that extruder energy demand is heavily

coupled between the machine, material and process parameters. Marschik et al. (2017) pre-

sented a heuristic approach for predicting the three-dimensional flow of fluids in single-screw

extruders without using computationally costly numerical simulations. They obtained the

output-pressure gradient relationship depending on four independent parameters: (i) height-to-

width ratio, (ii) pitch-to-diameter ratio, (iii) power-law index and, (iv) dimensionless pressure

gradient in the down-channel direction. The proposed heuristic approach is capable to provide

close approximation to numerical solutions.

Modelling, simulation and optimization for extruder are explored in open literature using

analytical, numerical and empirical approaches. Finite difference technique coupled with

numerical techniques and multi-objective evolutionary optimization algorithm is used in this

work. The current study is focused on maximization of throughput, while minimizing energy

demand maintaining quality of product and performance of operation by simultaneous manipu-

lation of screw design parameters and barrel temperature profile. The mathematical modelling

of single screw extruder is discussed in section(5.2) followed by Finite Element Analysis

(FEA) model development process in the section(5.3). The section of FEA model development

includes discretization of space, converting differential model equation to algebraic equations

in discrete variables and solution algorithm. Sensitivity of parameters influencing extruder

throughput are reviewed, followed by FEA simulation in the next section. Conclusions are

summarised based on the discussions.

5.2 Mathematical modelling of single screw extruder

Rubber extruder has three distinct zones: the feed zone, metering zone and, delivery zone.

The metering zone is the most critical part contributing to the overall performance of extruder

(Crowther, 1998). Hence, the current focus of model considers metering section of single

screw extruder to demonstrate the scope of simultaneous optimization of screw geometry

and temperature profile at heating surface. The simplified geometry of single screw rubber


5.2 Mathematical modelling of single screw extruder 151

extruder is shown in Fig. (5.1). The material passes through a very long, but shallow helical

channel formed by the flight of the screw. The channel boundaries are barrel root, screw root

and flight. For the analysis purpose, it is assumed that the screw is stationary and the barrel

rotates in the opposite direction of the screw thread. This simplification is adopted in extruder

analysis since it is easier to visualise and study the extrusion physical phenomena. This simpli-

fication also supports the computational implementation convenience without adding any error.

Figure 5.1. Rubber Extruder schematic diagram

Figure 5.2. Rubber Extruder screw and barrel

The channel between two flights, screw root and, barrel is considered in very small seg-

ments. As the finite difference technique is adopted for solution by dividing the channel in

very small segments, flat surfaces of barrel and screw root for each segment can be assumed.

This assumption of flat barrel converts the flow channel as straight long rectangular screw

channel of constant cross section. The flat channel system is described by means of a Cartesian



coordinate system. The channel configurations in Cartesian coordinates are given in Fig. 5.3.

The clearance between the screw flights and the barrel surface is expected to be very small,

and hence the effect of leakage flow on the flow rate is also neglected.

In the flat plate approximation with moving barrel, the screw root is stationary and the

barrel moves at a constant rotational speed, Vb = πDN. Velocity component in two principal

directions(x and z) considered are:

1. Flow parallel to the flight axis, caused by a barrel velocity of Vbz =Vb cosφ relative to

the stationery flights and screw.

2. Flow normal to the flight axis, caused by a barrel velocity of Vbx =Vb sinφ relative to

the stationary flights and and screw.

Figure 5.3. Rubber Extruder screw channel

The mathematical model of the system comprises of mass, momentum and energy balance

along with viscous and thermal behaviour of polymer compounds. Mass conservation law

says that the fluid mass is conserved; it means all fluid particles that flow into any fluid region

must flow out. Momentum continuity equations for Cartesian coordinate system considering

creeping flow in x and z direction (Bird et al., 2006) for the screw channel defined are written

as:

∂ p∂x

=τyx

∂y;

∂ p∂y

= 0;∂ p∂ z

=τyz

∂y(5.1)

where, p is pressure and τ is the shear stress. Temperature and velocity may change



along the screw channel length (i.e. z direction). If the temperature of all surfaces remain

constant along z direction, the velocity and temperature of the fluid within channel also reach

to steady state conditions, known as fully developed velocity and temperature profile. When

there is addition of heat from barrel or screw surface along with viscous heating due to shear,

temperature along z direction changes. The energy equation for such condition is represented

as:

ρCw∂T∂ z

= K∂ 2T∂y2 + τyx

u∂y

+ τyzw∂y

(5.2)

where, T is temperature, K is thermal conductivity, C specific heat of polymer, ρ is density,

τ is the shear stress and u and w are velocity in x and z directions respectively. The shear

stresses are described as:

τyx = η∂u∂y

τyz = η∂w∂y

(5.3)

where, η is viscosity of polymer.

No screw can be designed prior having all the thermal and rheological properties of the

rubber material to be processed. For the screw design to be aceeptable, the physical properties

of the polymer needs to be known accurately. The data which is essential for a acceptable screw

design is material properties, operating conditions, and screw geometry. Modelling the viscous

behaviours of the polymers processed in extruder plays a very important role in simulation

for getting the results which resembles closely to the experimental results. Polymers may

be observed as liquid when it is above the glass transition or melting temperatures, or solid

when the temperature is lower than the glass transition temperature. Polymers are actually

not liquid or solid, but they are viscoelastic. A polymer can be either a liquid or a solid,

depending on the speed at which its molecules are being deformed. In the current model

for single screw rubber extruder, Rubber is considered as incompressible non-Newtonian

fluid following Carreau–Yasuda model. The Power Law model is the straight forward model

generally used for high shear rate. The Cross-WLF model is widely used model in numerical

simulations because of its capacity to fit wide range of viscosity data of polymer materials.



Rubber compounds for practical applications are closely followed by the Carreau-Yasuda

model, which are being considered in this model. The Carreau-Yasuda model is capable to

accommodate the behaviour of fluids that have a yield stress, like a Bingham fluid, but that

otherwise dispaly shear thinning behaviour. The viscosity of elastomer using the Carreau-

Yasuda model along with temperature dependence by Arrhenious approach (Osswald and

Menges, 2012) is given as:

η(γ,T ) =σ0

γ+η0e−b(T−Tre f )

[1+(λe−b(T−Tre f )γ)

a]m−1a (5.4)

where, η = molecular viscosity, γ = shear stress, m = Power law index, a = Yasuda pa-

rameter, λ = relaxation time, η0 = zero shear viscosity, σ0 = Yield stress, b = temperature

coefficient of viscosity and Tre f = reference temperature.

The overall mass conservation equation for the screw channel considered results into two

constraints for velocity fields represented as Eq. (5.5). The total net flow in x direction is zero

and in y direction it is the extruder throughput Q.

∫ H

0udy = 0

∫ H

0wdy =

QW

(5.5)

where, H is channel height, W is channel width and, Q is throughput. The above all

model equations are to be solved considering the boundary conditions defined to the system.

Assuming no slip condition at barrel wall and screw surface, velocity of the first polymer layer

near barrel wall and screw root are barrel velocity and screw velocity. This will result into

boundary conditions as follows:

u = 0; w = 0; at y = 0 (screw root)

u =Vbx; w =Vbz; at y = H (barrel)(5.6)

The temperature of layer adjacent to the heating surface(barrel) is considered same as

the surface temperature neglecting heat transfer resistance. It is also considered that the non

heating surface (screw root)is at the temperature equal to the polymer in contact and no heat

gets transferred to the cold surface. This will result into boundary conditions defined as



follows:

∂T∂y

= 0 at y = 0 (screw root)

T = Tb(z) at y = H (barrel)(5.7)

It is also considered that the polymer temperature at inlet is uniform and velocity fields are

fully developed at inlet(z = 0). This will result into the boundary condition at inlet as follows:

T = Ti; u = udev; w = wdev; at z = 0 (screw channel inlet) (5.8)

where, udev and wdev are fully developed velocity profiles at temperature Ti. Thus, the

consolidated model equations and boundary conditions describing the extruder screw channel

system for pressure, velocity and temperature profiles can be represented as follows:

Model Equations:

∂ p∂x

= η∂ 2u∂y2

∂ p∂ z

= η∂ 2w∂y2

ρCw∂T∂ z

= K∂ 2T∂y2 +η

[∂ 2u∂y2 +

∂ 2w∂y2

]∫ H

0 udy = 0∫ H

0 wdy =QW

η(γ,T ) =σ0

γ+η0e−b(T−Tre f )

[1+(λe−b(T−Tre f )γ)

a]m−1

a

(5.9)

Boundary conditions:

u = 0; w = 0;∂T∂y

= 0; at y = 0 (screw root)

u =Vbx; w =Vbz; T = Tb(z); at y = H (barrel)

T = Ti; u = udev; w = wdev; at z = 0 (screw channel inlet)

(5.10)



5.3 FEA Implementation

The model equations described in Eq.(5.9) are solved under the boundary conditions given in

Eq.(5.10) using finite difference technique. The channel dimensions H×L along the Cartesian

coordinates y and z are discretized in h× l point mesh. The discrete computational domain is

presented at Fig. (5.4).

Figure 5.4. Computational Domain

The height of screw channel H in y coordinate is divided in h equal points number as 1

to h, 1 at screw root and h for barrel inner surface. The height of each discrete segment in

y direction is ∆y =Hh

. Similarly the length L in z coordinate is divided in l segments each

of length ∆z =Ll

. Model equations written in terms of pressure, temperature and velocity

variable values at discrete points results into conversion of partial differential equations to

algebraic equations which are represented in next subsection.

5.3.1 Finite Difference implementation for FEA model

Discretizing the model momentum balance equations described in Eq.(5.9), the following

equations are obtained:

ui−1−2ui +ui+1 =(∆y)2

η

∂ p∂x|i (5.11)

wi−1−2wi +wi+1 =(∆y)2

η

∂ p∂ z|i (5.12)

A total of 2(h-2) equations correlating the unknown velocities u and w at points 2 to h-1 are

achieved. The velocities at screw root and barrel surface are known from boundary conditions


5.3 FEA Implementation 157

given at Eq.(5.10). Applying these boundary conditions to the discrete form of momentum

balance equations, they can be represented in matrix form as follows:

Au = B (5.13)

Aw = D (5.14)

Where the matrix A,B and D are defined as,

A =

−2 1 0 0 ... 0 0 0

1 −2 1 0 ... 0 0 0

0 1 −2 1 ... 0 0 0

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

0 0 0 0 ... 1 −2 1

0 0 0 0 ... 0 1 −2

(5.15)

B =

(∆y)2

η2

∂ p∂x

(∆y)2

η3

∂ p∂x

(∆y)2

η4

∂ p∂x

...

(∆y)2

ηn−2

∂ p∂x

(∆y)2

ηn−1

∂ p∂x −Vbx

(5.16)

D =

(∆y)2

η2

∂ p∂ z

(∆y)2

η3

∂ p∂ z

(∆y)2

η4

∂ p∂ z

...

(∆y)2

ηn−2

∂ p∂ z

(∆y)2

ηn−1

∂ p∂ z −Vbz

(5.17)



The integration in total flow equations are evaluated using trapezoidal rule at the discrete

points. The total net flow along x and z direction applying trapezoidal rule take the following

form of equations:

(u1

2+u2 +u3 + ...un−2 +un−1 +

un

2)∆y = 0 (5.18)

(w1

2+w2 +w3 + ...wn−2 +wn−1 +

wn

2)∆y =

QW

(5.19)

Energy balance equation along z direction is also written in discrete form as follows,

ρCWTi+1−Ti

∆z= K

∂ 2T∂y2 +η(γ)2 (5.20)

Second order temperature derivatives in y direction are evaluated using the Crank-Nicholson

scheme. Crank-Nicholson scheme is capable to handle sudden fluctuation in temperature

profile, which are expected in the model proposed due to optimization of temperature profile

along heating barrel. Using the temperature values at zi−1 and zi location, the second order

derivative as presented at Eq.(5.21) is calculated as follows:

∂ 2T∂y2

∣∣∣∣yi,zi

=12

{Tyi−1−2Tyi +Tyi+1

∆y2

∣∣∣∣zi−1

+Tyi−1−2Tyi +Tyi+1

∆y2

∣∣∣∣zi

}(5.21)

Knowing the T values at zi−1, T at zi can be evaluated using following Eq. (5.22).

Ti+1 = Ti +∆z

ρCw

{K

∂ 2T∂ z2 +η(γ)2

}(5.22)

The algorithm implemented to solve these set of discretized model equations is discussed

in next subsection.

5.3.2 Numerical Solution Algorithm

The discretized algebraic model equations discussed in previous section are solved in Matlab

R2018a. The use of ’fsolve’ along with Levenberg-Marquardt algorithm for solving non-linear

algebraic equations. The linear algebraic equations (5.13) and (5.14) are solved taking inverse


5.4 Sensitivity of parameters influencing Extruder Throughput 159

of the matrix. Algorithm (2) represented the solution method implementation for single screw

extruder model. Input variable for this part of program which comes from outer optimizer loop

are the screw helix angle φ , screw channel height H, screw rotation speed N, feed temperature

Tf and, barrel temperature profile Tb. Barrel temperature profile includes a set of temperature

values equally distributed along the length.

Algorithm 2 Extruder screw channel velocity-temperature-pressure profile and throughput calculationalgorithm

Require: Input φ ,H,N,Tf ,Tb;specify parameters, P,D,Z,e;calculate parameters, Vbx,Vbz,W,L,qv;initialise variables h, l,u,w,T,η ;calculate grid distance ∆h,∆l;define matrix A,B and D;for j = 1 to l do

initialise guess for d pdx ,

d pdz ;

while ∆

(d pdx ,

d pdz

)≥ tolerance [fsolve loop] do

calculate, η ,u,w,B,D at grid j;solve Eq.(5.18) for u = A−1B;solve Eq.(5.19) for w = A−1D;

end whilecalculate, η ,u,w at grid i for converged d p

dx ,d pdz ;

initialise T at grid j+1;while all ∆(Ti+1)≥ tolerance [fsolve loop] do

calculate,∂T 2

∂yat grid j+1 using Eq. (5.21);

solve energy Eq.(5.20) at grid j+1;end whilecalculate T at grid j+1 using Eq. (5.22);store results for u,w, d p

dz ;end forcalculate Q;return u,w,T,P profiles and Q to calling function;

5.4 Sensitivity of parameters influencing Extruder Throughput

The FEA model and solution algorithm are implemented in MATLAB to develop a program.

The velocity and temperature profiles within the channel of single screw extruder are obtained

using the program. Power-law model and Carreau-Yasuda model are used as viscosity models.

Though, Carreau-Yasuda model will be used for rubber extruder simulations, the power law



model of viscosity is implemented to validate the model by comparing it with analytical and

empirical results.

The analytical solution of extruder model with non-Newtonian viscosity is difficult to

achieve without simplifying assumption. The simplifying assumptions causes a compromise in

the accuracy of solutions. There are very rare analytical solutions available for non-Newtonian

fluid extrusions due to complexity of the process and rheology. Recently, Orisaleye et al.

(2018) presented an analytical model suitable to design a screw extruder for non-Newtonian

(pseudoplastic) materials. Orisaleye et al. (2018) considered non-Newtonian power law model

for throughput prediction, but did not included temperature dependency of viscosity. They

used the model to predict the performance of the screw extruder for processing materials with

power law indices in the range of 0.5 to 1. They carried out analysis for the effects of design

and operational parameters and determined the optimum channel depth and helix angle. This

analytical solution was used to validate the numerical solution procedure and implementation.

Orisaleye et al. (2018) obtained non-dimensional volumetric throughput as:

Q =n

2(n+1)πH sinφ cosφ − n3

(n+1)2(2n+1)

(∂P∂Zs

sinφ

) 1n

πH2n+1

n sinφ (5.23)

Where, n - power law index; φ - helix angle in deg; H - dimensionless channel depth,

h/D; h - channel depth in mm; D - screw diameter in mm; P - dimensionless pressure,

p/m0Nn; p - pressure Pa; m0 consistency index in Pa/s; N - screw speed in RPM; Zs is

dimensionless distance along screw length, Zs = Z sin(φ); Z - dimensionless distance along

screw channel, z/D; z - distance on coordinate axis down the screw channel in mm. The

relation of dimensionless throughput with helix angle, φ and dimensionless channel depth, H

is presented in Fig. (5.5) keeping all other parameter constant. The increase of helix angle

beyond 45 °resulted to decrease in throughput. This is also reflected in the plot of effect

of dimensionless channel height. The two very close lines corresponding for helix angle

of 35 °and 60 °represent that both have very close throughput channel height relationships.

This plot reflects the interconnectivity of throughput, helix angle and, channel height. The



pressure gradient and viscosity index also influence the throughput in addition to helix angle

and channel height.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Dimentionless Channnel Height, H

0

0.02

0.04

0.06

0.08

0.1

0.12

Dim

entionle

ss T

hro

ughput, Q

=5 deg

=35 dec

=60 deg

10 20 30 40 50 60

Helix Angle, deg

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Dim

entionle

ss T

hro

ughput, Q

H=1 mm

H=8 mm

H=15 mm

Figure 5.5. Effect of helix angle and channel height on throughput

The optimum channel depth for maximum throughput is obtained by differentiating the

volumetric throughput Eq.(5.23), with respect to H as∂Q∂H

= 0. The relationship of optimum

H with helix angle and viscosity index is represented as:

Hopt =

[(n+1)

n

(∂P∂Zs

sinφ

)−1n

cosφ

] nn+1

(5.24)

The optimum helix angle for maximum throughput of the screw extruder is obtained by

differentiating Eq. (5.23), with respect to φ as∂Q∂φ

= 0. The Eq. (5.25) is relationship of

optimum φ with channel height, H and consistence index, n.

tan2φopt =2n+1

n

Hn+1

n(

∂P∂Zs

) 1n

−1

(5.25)

The relation ship for optimum helix angle for the optimum channel depth can be obtained

combining Eq. (5.24) and (5.25) as:

tan2φopt

tanφopt=

2(2n+1)n+1

(5.26)

The extruder throughput can also be calculated by empirical equation (5.27) suggested by

Rauwendaal (2014b) which is used to validate the developed FEA based model.



Q =

(4+n

10

)WhπDN cosφ −

(1

1+2n

)Wh3

4η

(d pdz

)(5.27)

Where, n - power law index; W - channel width in mm; φ - helix angle in deg; h - channel

depth in mm; D - screw diameter in mm; p - pressure Pa; N - screw speed in RPM; η - viscosity

in Pa s. The relation of throughput with helix angle, φ and channel depth, H and consistency

index, n is presented keeping all other parameter constant to understand the interconnectivity

of different extruder screw design parameters. The parameters of the extruder used in the

analysis are summarised in table (5.1).

Table 5.1. Parameteres of single screw extruded used in simulation

Parameter ValueScrew Diameter, D 36 mmLength of Metering Section, L 800 mmScrew Thread Thickness, e 6 mmClearance, δ 0.05 mmChannel Height, H, 8 mmRotational Speed, N 30 RPMPressure output, P 2×105 PasHelix Angle, φ 20 deg

The relationship of single screw extruder throughput with helix angle for different channel

height is presented in Fig. (5.6). The effect of change of helix angle from 5 °to 45 °is repre-

sented for three different values of channel height keeping all other parameters constant. Effect

of helix angle becomes more dominant as channel height increases. For smaller channel height,

the relationship of throughput helix angle is near linear, which becomes little exponential as

channel height increase.

The effect of changing channel height on throughput for different helix angle is presented

in Fig. (5.7). The effect is presented for three different values of helix angle: 5 °, 25 °and, 45

°. The effect of channel height on throughput is almost linear for all cases, the slope of line

increases as the helix angle increases. Throughput increases increasing channel height, but

the effect of channel height becomes more dominant with higher helix angle.



5 10 15 20 25 30 35 40 45

Helix Angle, deg

0

10

20

30

40

50

Th

rou

gh

pu

t, Q

L

/min

H=1 mm

H=5 mm

H=10 mm

Figure 5.6. Effect of helix angle on throughput for different channel height

2 3 4 5 6 7 8 9 10

Channnel Height, h mm

0

10

20

30

40

50

Th

rou

gh

pu

t, Q

L

/min

=5 deg

=25 dec

=45 deg

Figure 5.7. Effect of channel height on throughput for different helix angle



The effect of changing the value of channel height on extruder throughput is presented for

three different polymers, highly non-Newtonian having power law index n = 0.2, moderate

non-Newtonian having power law index n = 0.6 and Newtonian fluid with n = 1 is presented

in Fig. (5.8). The relationship of throughput with channel height is linear and the effect

of change in viscosity index is also linear. This is reflected as three straight line plots with

different slopes. The effect of viscosity index is small when channel height has smaller values,

the influence of viscosity index increases as channel height increases, which is reflected as

three lines diverging apart as channel height increases.

2 3 4 5 6 7 8 9 10

Channnel Height, h mm

5

10

15

20

25

30

35

40

Th

rou

gh

pu

t, Q

L

/min

n=0.2 deg

n=0.6 dec

n=1 deg

Figure 5.8. Effect of channel height on throughput for different viscosity index

The extruder throughput sensitivity to helix angle for the same three different types of

materials; highly non-Newtonian having power law index n = 0.2, moderate non-Newtonian

having power law index n = 0.6 and Newtonian fluid with n = 1 is presented in Fig. (5.9).

As already discussed, the effect of helix angle on throughput is non-linear, the same trend

is observed here. The larger values of helix angle has more influence of viscosity index on

extruder throughput.



5 10 15 20 25 30 35 40 45

Helix Angle, deg

0

5

10

15

20

25

30

Th

rou

gh

pu

t, Q

L

/min

n=0.2 deg

n=0.6 dec

n=1 deg

Figure 5.9. Effect of helix angle on throughput for different viscosity index

The viscosity plays significant role in deciding the relationship of throughput with helix

angle and channel height, hence, its effect is also reviewed. The effect of viscosity index on

throughput for different values of channel height and helix angle are presented in Fig. (5.10).

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Consitency Index, n deg

0

5

10

15

20

25

30

35

40

Thro

ughput, Q

L/m

in

H=1 mm

H=5 mm

H=10 mm

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Consitency Index, n deg

0

5

10

15

20

25

30

Thro

ughput, Q

L/m

in

=5 deg

=25 dec

=45 deg

Figure 5.10. Effect of polymer viscosity on throughput for different channel height and helix angle

The above analytical and empirical result analysis shows good consistency in extruder

parameter relations. In the current work, power law model is used to study the sensitivity of



different extruder parameters. Simulations using both, power law model and Carreau-Yasuda

model are presented in the subsequent sections to demonstrate compatibility of both models.

For optimization of rubber extruder screw design, Carreau-Yasuda model will be used.

5.5 Simulation using the FEA model

Rubber extruder simulation is carried out using the FEA model developed. The model gives

detailed velocity, pressure and temperature profiles across extruder channel. The throughput

by FEA model is calculated using velocity profile obtained at exit of the extruder channel. The

throughput is calculated using Eq. (5.28). The properties of materials and model parameters

used for simulation are summarised in table (5.2) and (5.1).

Q =W∫ H

0wdy (5.28)

Table 5.2. Matrial Properties used for simulation of single screw extruded

Property/Parameter ValuePower law index, n 0.49Zero shear viscosity, η0 20614 Pa-sThermal coefficient of viscosity, b 0.01Yasuda parameter, a 1.11Relaxation time, λ 15.48 sYield stress, σ0, 44.34 PaDensity, ρ 0.9 kg/LSpecific Heat, C 2500 J/kg KThermal Conductivity, K 0.30 w/ m K

The velocity in two directions, x and z are denoted as u and w respectively. The net

velocity in x direction is zero, where as the net velocity in z direction produces movement of

polymer through screw channel resulting to extruder throughput. The fully developed velocity

profiles, u and w are presented in Fig. (5.11). The velocity near barrel is maximum as we

have assumed screw stationery and barrel rotating at N RPM. The maximum velocity in x

direction is Vbx = πDN sin(φ) and in z direction is Vbz = πDN cos(φ). The velocity in the

lower part of the screw channel is in the opposite direction. Both the velocity profiles u and w

are jointly presented in a 3D plot at Fig. (5.12) for better understanding of velocity flow in


5.5 Simulation using the FEA model 167

screw channel.

-10 0 10 20 30 40

Velocity u mm/sec

0

1

2

3

4

5

6

7

8

Channel H

eig

ht (H

) m

m

-10 0 10 20 30 40

Velocity w mm/sec

0

1

2

3

4

5

6

7

8

Channel H

eig

ht (H

) m

m

Figure 5.11. Velocity Profile at exit along x and z direction

020

2

60

4

Ch

an

ne

l H

eig

ht

(H)

mm

10 40

6

u mm/sec w mm/sec

8

2000

-20

Velocity in x direction (u)

Velocity in z direction (w)

Figure 5.12. Three dimensional view of u and w velocity profile at screw channel exit

The pressure profile along the screw channel is presented in Fig. (5.13). The pressure at

inlet of extruder screw channel is 1×105 N/m2 which increases up to 2×106 N/m2 at exit.

Though the rate of increase looks linear in the pressure profile plot, the change in magnitude of

pressure difference across screw channel length is more clearly visible in the pressure gradient

plot in Fig. (5.14).



0 200 400 600 800 1000 1200

Channel Length (Z) mm

0

0.5

1

1.5

2P

ressu

re (

P)

N/m

2106

Figure 5.13. Pressure profile along extruder screw channel length

0 200 400 600 800 1000 1200

Channel Length (Z) mm

1.8

1.82

1.84

1.86

1.88

1.9

1.92

Pre

ssu

re c

ha

ng

e (

dP

) N

/m2

104

Figure 5.14. Pressure gradient profile along extruder screw channel length


5.6 Summary 169

The temperature profile along screw channel height and length is presented as three

dimensional plot at Fig. (5.15). The inlet temperature is 40 °C, which increases due to heating

at barrel surface and viscous dissipation heat generation. The barrel temperature is assumed

to be constant at 80 °C across the length. The temperature profile presented in the Fig. (5.15)

is the result of conductive and convective movements of energy within extruder channel. The

model has provision to maintain different temperatures in different sections of barrel, which

will be useful for process optimization.

0100

20

40

40

Te

mp

era

ture

, d

eg

C

30

60

Channel Lenght

50

Channel Height

80

2010

0 0

Figure 5.15. Three dimensional view of Temperature profile along screw channel height and length

5.6 Summary

Rubber Extruder is a machine for pre-forming unvulcanized rubber compounds by pushing

material through the die to get definite shapes and sizes. The process involves several complex

phenomena: complex rheology, fluid flow and, heat transfer. The critical part of extruder is

designing a screw. It is noticeably one of the most important part of the extruder. The aim of

extrusion process design is to find the screw - die design and the process parameters (flow

rate, temperature ) which allow a stable flow of high accuracy and good quality profile.



The objective of the design is to deliver the largest amount of output of good quality.

Unfortunately, high output & mixing quality are, to some extent, conflicting requirements.

The Helix angle is the most important parameter affecting the performance of the extruder

screw. It affects throughput, mixing, discharge pressure and power consumption. Increasing

the screw speed, throughput increases. Too high speed will result in greater temperature

variation and poor mixing, and thus, deteriorates quality of products. The computer program

developed using the model can simulate the effect of different design parameters, operating

parameters and material properties on performance of a single screw extruder. The program

can be used for both simulation and design optimization. Extruder response to changes in the

operating conditions, or in the geometry can be studied using design equations and correlations.

Currently, the modelling and simulation study is carried out to validate the model developed

which will be used further for optimization.

The extruder model comprises of momentum and energy balance equations along with

rheological properties of material. The geometry and rheology together results in to a model,

which is difficult to solve. Carreau-Yasuda model is used to represent rheology of rubber com-

pounds. FEA model and solution algorithm for extruder screw channel velocity, pressure and

temperature profile is developed. The throughput, energy and residence time can be calculated

using these profiles. The parameters which influence throughput, power consumption and

residence time distribution of extruder are screw length, channel depth, flight width, clearance,

helix angle, screw speed, pressure, and viscosity. Helix angle, channel height and viscos-

ity relationships with throughput are reviewed to find design parameters which maximize

throughput. Response of extruder to changes in the operating conditions, and the geometry is

studied using the design equations and correlations along with FEA model. The FEA model

developed can be further enhanced to be used for multi-objective optimization to generate

pareto optimal solutions for throughput maximization - power consumption minimization.


171

Chapter 6

Multi-Objective Optimization:

Application to Rubber Extruder Screw

Design

Multi-objective optimization (MOO) is a group of optimization, which can handle multiple

and conflicting objectives concurrently. MOO problems with conflicting objectives have

a set of solutions, which are named pareto optimal solutions. Evolutionary algorithms are

predominantly used for solving MOO problems. The pareto optimal solutions represent

trade-offs among all the objectives, which are all are non-dominated solutions. None can be

said to be better than the others, with respect to all objectives, and hence, all are important.

The pareto set is used by the decision maker to choose the best out of all possibilities deciding

trade-offs. The optimization problems when solved considering all the conflicting criteria,

becomes MOO problems. All design problems are multi objective, but conventionally solved

as singe objective optimization problems.

Rubber extrusion design optimization problem is formulated as a MOO problem. The

extrusion process is very complex and the design of screw is the most complex task. It

influences capacity of the extruder along with performance. The extruder throughput, power

consumption, mixing in extruder, Residence Time Distribution(RTD) of material being pro-

cessed are all important design objectives, which are conflicting. All these objectives are

172 6. Multi-Objective Optimization: Application to Rubber Extruder Screw Design

influenced by screw design, operating parameters and material properties. The objective of the

screw design is to deliver the largest amount of output at minimum energy needs, which are

conflicting. The FEA model developed is used to generate performance response of extruder.

The PUALGA algorithm along with BI approach for constraint handling is used to solve the

MOO problem formulated.

6.1 Introduction

Multi-Objective is a class of optimization, which deals with multiple conflicting objectives si-

multaneously. The multi-objective optimization problems with conflicting objectives will have

a set of solutions, which are called pareto optimal solutions. The pareto optimal solutions are

representing trade-offs among all the objectives. All the members of pareto optimal solution

set are non-dominated, hence, none can be said to be better than the others, with respect to

all objectives. Usually, the decision makers wants a small set of solutions to make a choice

among them. The challenge to generate a pareto set, as small as possible, that represents the

whole set of choices. The computational efficiency and robustness are important aspects for

choosing the method to generate this pareto set. Evolutionary optimization algorithms are one

of the preferred choice to solve MOO problem. All the optimization problems are naturally

multi-objective, but generally, they are not solved as multi objective problems.

One of the mechanism of solving MOO problems is to augment all the conflicting objec-

tives using different weights and, solve the formulated single objective optimization problem

(SOO). The critical challenge with this technique is to identify the appropriate weights to

the individual objectives. Further, this formulated SOO is to be solved multiple times using

different weights to obtain the entire pareto front, which does not assure unique solution with

each different weight set. Moreover, this technique has limitation of missing the concave

portions of a pareto front (Das and Dennis, 1997). The final population of the properly

designed and implemented population based evolutionary algorithms (EAs) converges to

the pareto front in a single run. Another classical mechanism for solving MOO problems

is to minimize one objective considering the others as constraints. The limitation of this

approach is the choice of the function to be minimized and specifying the constraint limits.


6.1 Introduction 173

Non-dominated sorting, rank based sorting (Qu and Suganthan, 2010) and, evolution with

decomposition (Jiao et al., 2013; Zhao et al., 2012) are recently evolving approaches for

MOO. The dominance based classification of populations (Deb et al., 2002) needs multiple

comparisons of the members for sorting, and hence, are computationally costly. Logist et al.

(2013) used a well organised scalarization technique using ACADO multi-objective toolkit for

dynamic optimization problems. The limitations of these technique is that, one SOO problem

is to be worked out for every pareto solution point. Thus, large number of SOO problems

need to be solved to cover the entire pareto front. However, the count of SOO problems to

be solved can be minimized using interactive tools and visualization approaches (Sindhya

et al., 2014; Vallerio et al., 2015). Constraint handling in EAs becomes a limitation, as they

are naturally designed for unconstrained optimization. Bounds on decision variable are part of

natural design feature of EAs, but they require an additional mechanism for other types of

constraints.

The design problems are naturally multi-objective, but rarely solved as multi-objective

problem. Formulation and solution of MOO problems is difficult compared to SOO problems,

hence, most design optimization applications use SOO. All the decisions by designer fixing the

importance of different criteria is converting the MOO problem to SOO problem. Compromise

in economy, safety and, environment impacts on the design is an example of MOO problem

getting converted to SOO problem. The rubber etrusion process involves several complex

phenomena: complex rheological behaviour, fluid flow and heat transfer. The critical part

of extruder is a screw. Optimization of extrusion includes selection of the operating and

geometrical variables that maximize output maintaining quality and minimizes the remaining

in order to save energy, increase efficiency and avoid polymer degradation (Rauwendaal,

2014a).

Rubber Extruder design optimization is explored as multi-objective design optimization

problem in this work. Conventionally, extruder is designed for maximum throughput with

acceptable quality. Power consumption minimization is solved as a sub-optimization problem

to throughput maximization. The acceptable quality of the product is assured as a constraint.



The multi-objective formulation gives all the possible solutions to the designer without fix-

ing any preferences. The designer will choose one solution as best among all the pareto

solutions, knowing the compromise he is making by his decision in all the objectives. The

helix angle and channel height are the most important parameters affecting the performance

of screw. Helix angle and channel height both, affects throughput, power consumption and

discharge pressure. Increasing the screw speed, throughput increases. Too high speed will

result in greater temperature variation and poor mixing and thus, deteriorates the quality of

products. Effect of different design parameters, operating parameters and material properties

on conflicting performance parameters: throughput and power for a single screw extruder

are investigated formulating MOO problem. The eutopia point is obtained as the best point

balancing the compromise among the two objectives.

6.2 Multi-Objective optimization of extruder screw design

Optimization is a process of finding the best solution satisfying the conditions imposed. The

best solution is decided by the criteria of selecting the best. If there are multiple criteria

which are conflicting with each other under which best solutions are to be selected, then there

would be multiple optimal solutions. This class of optimization is defined as Multi-Objective

Optimization(MOO), which deals with multiple conflicting objectives simultaneously. MOO

problems with conflicting objectives will have a set of solutions (representing trade-offs

among the objectives), which are called pareto optimal solutions, of which none can be said

to be better than the others with respect to all objectives (Steuer, 1989). The relevance and

importance of MOO is increasing due to increasing complexities in the design and operation

of processes. MOO solutions assists decision maker in selecting the best solution according to

the need of the time giving an entire spectrum of best solutions. Usually, the decision makers

want a small set of solutions to make a choice among them. The challenge is to provide them

with a set, as small as possible, that represents the whole set of choices. Population based

EAs have become significantly popular for MOO solutions finding an edge over the classical

methods owing to their ability to converge the entire population to the optimal pareto front in

a single run (Deb, 2001; Coello et al., 2006; Rangaiah and Bonilla-Petriciolet, 2013). Parallel


6.2 Multi-Objective optimization of extruder screw design 175

Universe Alien Genetic Algorithm(PUALGA) along with BI approach for constraint handling

developed by the authors is used to solve MOO problem formulated in this work. The detailed

algorithm is discussed in previous chapters, hence, discussion of MOO algorithm is skipped

here.

There are different types of extruders in the market to deal with different applications and

viscoelastic fluids to be handled. The simplest amongst them is a single screw pin-barrel

extruder, where a screw rotates inside a barrel moving the viscoelastic product. Heat may

be applied to meet the process requirements. Rubber extrusion process consists of pushing

compound by means of screw through feeding channels and die. The channels are used to

condition the rubber flow parameters (velocity, temperature) and, to distribute the flow rate

of different blends in the case of co-extrusion. The critical part of extruder is designing a

screw, which lies at the heart of the extruder. Optimization of extrusion includes selection

of the operating and geometrical variables that maximize mass output maintaining quality

with minimum energy demand. All these objectives are conflicting with each other hence it is

a good MOO problem to investigate. MOO problem is formulated considering, throughput

maximization and energy consumption minimization as two objectives. The design parameters

considered are the screw helix angle φ , screw channel height H, screw rotation speed N and,

barrel temperature profile T b. The three objective MOO problem formulated is represented as

follows:

max f1 = Q(Throughput)

min f2 = E(Energy Consumption)

φ ,H,Tb

sub jectto T ≤ 90

(6.1)

The two objectives are discussed in details along with residence time distribution in the

next subsections.



6.2.1 Throughput Maximization

The objective of screw design is to deliver the largest amount of output of acceptable quality.

The helix angle is the most important parameter affecting the performance of screw. It affects

throughput, power consumption, mixing and discharge pressure. Increasing the screw speed,

throughput increases. Too high speed will result in greater temperature variation and poor

mixing, and thus, deteriorates the quality of products. Increasing channel depth of a screw,

throughput increases. Shallow channel depth screw can be operated at higher speed than a

deep screw, giving better throughput. Flight width and clearance are also affecting the through-

put. Increasing the radial clearance in an extruder, mixing efficiency decreases. Standard

clearance value is 0.001D, where D is the screw diameter. If you double the clearance, the

mixing goes down by 25%. If you triple it, the mixing rate is reduced by 35%. This shows

that there is heavy wear in the mixing zone of the extruder; it has serious bad effect on the

mixing performance. On the other hand slight wear in the mixing zone helps to reduce the

power consumption. Increases in land length of screw is found to have significant effects on

increasing linear output, but only marginal effects on increases in mass output. There also

exists an interactive effect between land length and die temperature on head pressure. Head

pressure increases associated with increased land length and can be minimized with increased

die temperatures.

Throughput can be calculated by empirical equation (6.2) suggested by Rauwendaal

(2014b) which is used to calculate the estimated u,w in iterative numerical solution procedure.

Q =

(4+n

10

)WHπDN cosφ −

(1

1+2n

)WH3

4η

(d pdz

)(6.2)

The throughput is calculated using velocity profile obtained at exit of the extruder channel

using Eq. (6.3) for MOO solution in this work.

f1 = Q = f (φ ,H,N,Tb) =W∫ H

0wdy (6.3)


6.2 Multi-Objective optimization of extruder screw design 177

6.2.2 Energy Consumption Minimization

Runtime energy consumption of extruder is becoming more and more important as the cost

of energy is increasing. Hence, it is very important to design an extruder which is energy

efficient. Effects of screw design parameters like the screw helix angle φ , screw channel

height H, screw rotation speed N and, barrel temperature profile Tb plays an important role in

the energy demand of the extruder. The power consumption of the extruder increases linearly

with both screw speed and viscosity of compound. Energy consumption in an extruder is

inversely proportional to the channel depth and flight radial clearance. Thus, greater the

channel depth and larger the flight clearance, lesser is the power consumption. Larger flight

clearance drastically reduces mixing efficiency and overall extruder performance. If you want

to achieve a reduction in mechanical power you may decrease the screw speed. Output Q and

pressure P would decrease approximately in proportion to speed N. But, when we reduce the

speed, the mechanical power would reduce more than in proportion to speed.

In general the pumping efficiency of a screw extruder is 10% or less. This means that

energy consumed in actual pumping of the polymer material is less than 10% of the input

energy. The rest 90% or more goes into the power consumed in viscous heating of the polymer.

Viscous heat generation is the dissipation of mechanical energy in a viscous fluid which occurs

throughout the fluid. The local rate of heat generation depends on the local shear rate. If the

shear rate is constant throughout the entire volume of a fluid, the viscous heat generation will

be uniform throughout the fluid. Since, viscous heat generation occurs throughout in a fluid,

it is an effective way of heating a polymer compound because it will result in a relatively

uniform temperature increase. Uniform temperature distribution in product is one of the

crucial parameters in extruder design. The temperature of extrudate increases with the speed

and viscosity. Extruder barrel is water cooled to remove the heat generated. Temperature

rises due to poor thermal conductivity of rubber. When temperature rises above a desired

level, scorch formation takes place and creates a dead spot in the extruder, resulting into

chocking. Chocking may cause screw flight erosion and deteriorates the extrudate quality. If

the temperature is too low, material will resist to flow. Thus, if temperature is not properly

controlled the efficiency and effectiveness of extruder may decline.



The energy consumed by the extruder screw is the subtotal of energy consumed for viscous

heating, for increase in pressure and kinetic energy (Zuilichem et al., 2011). Kinetic energy is

very very small compared to the other energy components, hence, the total energy E consumed

is considered as sum total of energy consumed for viscous energy dissipation in screw channel

Evsc, in Screw tip Evst , and increasing pressure E p. The total energy consumed by extruder is

calculated using Eq. (6.4) for objective f2 in MOO problem solution.

f2 = E = f (φ ,H,N,Tb,e) = ρCW∫ H

0(Ti−Tf )wdy (6.4)

Evst =(πDN)2eL

δ sinφ(6.5)

6.3 Residence Time Distribution

Residence time distribution(RTD) is a very important performance parameter of extruder for

producing a good quality product. It is a parameter indicating the amount of time a polymer

spends in the extruder. RTD is generally investigated using tracer study experiments followed

by analysis and modelling. RTD prediction is important in designing extruder when chemical

reactions take place during the extrusion process. Scorch formation takes place if the material

remains at high temperature for a longer time within the extruder channel. Rubber extruder

needs to be designed considering RTD in account for good quality consistent product. The

extruder parameters like channel height, width, helix angle and, screw speed influences RTD

strongly. Different combinations of these parameters can influence RTD from ideal plug flow

to perfect mixing. The relationship of RTD with extruder design parameters is very complex,

hence, the RTD in screw extruder is generally investigated by tracer experiments (Kemblowski

and Sek, 1981; Joo and Kwon, 1993; Reitz et al., 2013; Sievers and Stickel, 2018).

Analytical solutions for RTD calculation considering non-Newtonian flow are not feasible.

There are very few articles (Karwe and Jaluria, 1990; Joo and Kwon, 1993) predicting RTD

using numerical simulation data. A method to determine RTD using numerical simulation data


6.3 Residence Time Distribution 179

is presented in this work. The RTD data is then used to find extruder deviation from ideal plug

flow reactor, which is used as one of the objectives for extruder screw design optimization.

Maximum deviation from ideal plug flow (closer to perfect mixing) is preferred when extruder

is aimed to provide mixing, whereas, minimum deviation from plug flow is a favourable

condition if it is expected to deliver polymer at required pressure.

Tanks-in-series (TIS) model is used to analyse non-ideal flow in the extruder. The TIS

model is a one parameter model used for reactor analysis and modelling. The RTD is analysed

to determine the number of ideal tanks n, in series that will give approximately the same

RTD. n=1 represents perfect mixing and very large value of n indicates ideal plug flow. The

generalized form of RTD using TIS model for a series of n CSTRs (Fogler, 2005) is given as:

E(t) =tn−1

(n−1)! τni

e−t/τ (6.6)

The total reactor volume is nVi, τi = τ/n, where τ is the total reactor volume divided by

the total flow rate. The parameter n of TIS model is calculated as:

n =τ2

σ2 (6.7)

where, τ is residence time (first moment of RTD ) and σ is variance (second moment

of RTD). The First and second moment of RTD can calculated using Eq. (6.8) and Eq.(6.9)

respectively.

τ =∫

∞

0t E(t) dt (6.8)

σ2 =

∫∞

0(t− τ)2 E(t) dt (6.9)

The average residence time τ for an extruder can be calculated using Eq. (6.10) from the

ratio between the volume and the volumetric throughput.

τ =πNDHW

Q(6.10)



E(t) is the internal age distribution defined as the fraction of the material that has residence

time between t and t +dt, so∫

∞

0 E(t)dt = 1. E(t) data for extruder can be calculated from

local residence time t(y). The local residence time t(y) is calculated from velocity as:

E(t) =W∫ H

0wydy (6.11)

t(y) =∫ Z

0

cosφ

wsinφ −ucosφdz (6.12)

The TIS model parameter ntank calculated using E(t) represents the number of ideal

mixing tanks connected in series. A very large value represents plug flow and value near unity

represents mixing tank behaviour of the channel. The value of this parameter is observed and

restricted to a minimum value to assure proper mixing in the extruder screw channel. The Eq.

(6.13) represents the parameter ntank for system.

ntank =τ2

σ2 (6.13)

6.4 Results and Discussion

Extruder throughput corelations with different design parameters of extruder screw were

reviewed in detail in chapter 5. Helix angle and Channel height are considered here as

manipulated design parameters for extruder screw design. The values of other parameters are

as mentioned in table (6.1). The energy consumed by the extruder screw is the subtotal of the

energy consumed for viscous heating, for increase in pressure and kinetic energy (Zuilichem

et al., 2011). Kinetic energy is very very small compared to the other energy components,

hence, the total energy E consumed is considered as sum total of energy consumed for viscous

energy dissipation in screw channel Evsc, in Screw tip Evst , and increasing pressure E p.

The extruder FEA model and its solution approach discussed in chapter 5 are used along

with PUALGA algorithm developed for MOO. The design parameters and properties of

materials used in simulation are summarised in table (6.1). Natural rubber is considered as

the material being processed in the extruder. The cold feed single screw rubber extruder

optimization is carried out to simultaneously maximize throughput and minimize power



Table 6.1. Parameteres and matreial properties of single screw extruded used in optimizing screw design

Parameters/Properties ValueScrew Diameter, D 36 mmLength of Metering Section, L 800 mmScrew Thread Thickness, e 6 mmRotational Speed, N 30 RPMPressure output, P 2×105 PasThermal coefficient of viscosity, b 0.01Yasuda parameter, a 1.11Relaxation time, λ 15.48 sYield stress, σ0, 44.34 PaDensity, ρ 0.9 kg/LSpecific Heat, C 2500 J/kg KThermal Conductivity, K 0.30 w/ m K

consumption. The barrel is provided with heating and cooling arrangements in five sections.

Each section is assumed to have constant temperature at the barrel wall. The residence

time distribution and temperature profiles across extruder are monitored to assure the quality

of product. They are implemented as constraints in the model. If the compound remains

at high temperature for longer time, there is a possibility of scorch formation in compound

or chocking in the screw channel. To avoid this, temperature exposure above 90 °C is restricted.

The MOO problem formulated is solved using PUALGA algorithm to get pareto solutions

for maximization of throughput, while minimizing energy demand. The resultant pareto

front obtained at the end of 300 generations for a population size of 100 is plotted in Fig.

6.1. The conflicting nature of two objectives, throughput and power requirement for single

screw extruder are clearly reflected in the pareto plot. Point A corresponds to the maximum

throughput and point B corresponds to minimum power consumption. If the formulated

problem is solved using any SOO technique and throughput as the only objective to be

maximized, we get a single solution corresponding to point A. Similarly, SOO solution for

minimization of power consumption as objective will result to point B. By using MOO

techniques, all the solution points on line joining A and B are obtained as pareto optimal set.

They represent the solutions with all possible combinations of the two objectives.

Each point on pareto front between the points A and B represents some compromise in



36 38 40 42 44 46

Throughput (m3/h)

7.5

8

8.5

9

9.5

10P

ow

er

(kW

)10-4

A

B

R

CE

Figure 6.1. Throughput-Power pareto front for single screw extruder

either throughput or power consumption to improve upon the other objective value. Near

point A, very small compromise in throughput value will result into large savings in power

consumption. Similarly, near point B a small compromise in power consumption will result in

great improvements in throughput. Hence, it is very important to choose an operating point

appropriate to the need. All the points on the pareto front represent optimal solutions. The

Point C is the Eutopia point for throughput-power Pareto front. Eutopia point is the best point,

which is at minimum distance from the reference point R. The helix angle corresponding

to Eutopia point is 35 °. A screw was tested with help of Pioneer Rubber Industries for this

configuration. The experimental results were very close to the simulation results as shown in

Fig. 6.1, marked as point E.

The effect of helix angle on optimum throughput and power consumption is presented

in Fig. 6.2. As helix angle increases from 25 °, initially throughput increases at a fast rate,

but near 45 °the influence of helix angle on throughput reduces. The reverse phenomena is

observed in case of power consumption. This two plots clearly show the conflicting nature of

throughput and power requirements and presents influence of helix angle as a manipulated



variable.

25 30 35 40 45


1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

Pow

er

(kW

)

10-3

25 30 35 40 45


34

36

38

40

42

44

46

Thro

ughput (m

3/h

)

Figure 6.2. Influence of Helix Angle on optimum Throughput and Power consumption for single screw extruder

The effect of Channel height on optimum throughput and power consumption is presented

in Fig. 6.3. The relationship of channel height with throughput and power consumption

is observed to be almost linear. Both objective function values increases as channel height

increases. Our objective is to maximize throughput and minimize power consumption. Chan-

nel height influences both objective function values. The optimum value of channel height

corresponding to the Eutopia point is 8 mm. The two plots show the influence of channel

height on throughput and power consumption.

6 7 8 9 10 11 12

Channel Height (mm)

5

10

15

20

25

30

Thro

ughput (L

/h)

6 7 8 9 10 11 12

Channel Height (mm)

1

1.5

2

2.5

3

3.5

4

4.5

Pow

er

(kW

)

10-4

Figure 6.3. Influence of Channel Height on optimum Throughput and Power consumption for single screwextruder

The fully developed velocity profile at the end of the screw metering section is also

presented here. The velocity profile is used to calculate RTD of the extruder. RTD values

represents the quality of mixing in extruder. The profile of u representing velocity component



in x direction along channel height H is shown in Fig. (6.4). The velocity in the x direction

clearly reflects that, the net flow in x direction is zero.

-20 -10 0 10 20 30 40 50

Velocity u mm/sec

0

1

2

3

4

5

6

Ch

an

ne

l H

eig

ht

(H)

mm

Figure 6.4. Velocity profile in x direction

The velocity component in z direction along channel height H is shown in Fig. (6.5).

Velocity in z direction is denoted as w profile. The distribution of velocity in the z direction

shows the contribution to the net flow in z direction, the total net flow at any location along z

direction is always equal to throughput Q of the extruder.

Temperature profile along extruder metering section length and channel height is plotted

as a surface plot. The surface plot of optimum temperature profile along channel length and

channel height is shown in Fig. (6.6). The effect of optimum barrel temperature profile can be

observed at channel height node 40. Node 1 represents root of the screw. The viscous heating

is dominant near screw root. The temperature at any location in the channel do not exceed the

limit of 90 °C imposed to avoid scorch formation. The temperature profile conforms that the

constraint is observed while optimization.

Residence time distribution (RTD)for extruder channel is plotted in Fig. (6.7). The RTD



-20 0 20 40 60 80 100 120

Velocity w mm/sec

0

1

2

3

4

5

6

Ch

an

ne

l H

eig

ht

(H)

mm

Figure 6.5. Velocity profile in z direction

0100

20

40

40

Te

mp

era

ture

, d

eg

C

60

30

Channel Lenght

80

50

Channel Height

100

2010

0 0

Figure 6.6. Temperature profile



is a characteristic of an extruder indicating the average amount of time the material spends

within an extruder. If the material spends an excessive amount of time, it may have adverse

effects on extruder product properties. In actual practice, the RTD is obtained by injecting dye

near the inlet and then measuring the flow rate of the dye material as it comes out at the outlet.

Numerically, RTD computation is done from the time distribution plot presented in Fig. (6.7).

The plot presents the time taken by the martial to travel distance Z−D mm and Z mm. There

are two lines: magenta colour line represents the time required to travel Z−D mm distance

and red colour line represents the time required to travel Z mm. The difference between these

two lines represents the time the material spent in extruder channel. This information of the

curve is converted to E curve for RTD calculation.

0 2 4 6 8 10 12 14

Residence time (ti) sec 105

0

1

2

3

4

5

6

7

8

Ch

an

ne

l H

eig

ht

(H)

mm

Figure 6.7. Time distribution for RTD calculation

The E curve is an important component of RTD study. The E curve for the extruder

channel is plotted in Fig. (6.8). E curve graphically represents RTD. Area under the curve

represents average time the material spent in the extruder.

The F curve also represents RTD. The F curve for extruder obtained from the E curve

is plotted in Fig. (6.9). The first moment of RTD represents average residence time of the

extruder channel, where as, the second moment represents the mixing effect. The ratio of the


6.5 Summary 187

0 2 4 6 8 10

Residence time (t) sec 105

0

0.5

1

1.5

2

E(t

)

10-5

Figure 6.8. The E(t) curve presentation of RTD

first and second moment represents the nature of flow. Large values represents stirred tank

reactor representing intense mixing in channel, where as, values near unity represents plug

flow.

6.5 Summary

The objective of screw design is to deliver the maximum amount of output at acceptable quality.

The output is to be maximized maintaining quality and simultaneously consuming minimum

power. The throughput and power consumption are two conflicting objectives for an extruder,

hence, they generate a set of pareto optimal solutions. The helix angle along with the channel

height are the most important parameters affecting the throughput and power consumption.

Discharge pressure also depends upon helix angle. Increasing the screw speed; throughput

increases, but high speed will result in greater temperature variation and poor mixing, and

thus, deteriorates the quality of products. Increasing channel depth of a screw, throughput

increases. It is connected with screw speed. Shallow channel depth screw can be operated

at higher speed than a deep screw giving better throughput. Flight width and clearance are



0 1 2 3 4 5 6

Residence time (t/ ) sec

0

0.2

0.4

0.6

0.8

1C

um

pu

lative

E(t

)

Figure 6.9. The F(t) curve presentation of RTD

also affecting the throughput. Increasing the radial clearance in an extruder, mixing efficiency

decreases. Standard clearance value is 0.001D, where D is the screw diameter. If we double

the clearance the mixing goes down by 25%. If we triple it, the mixing rate is reduced by 35%.

This shows that there is a heavy wear in the mixing zone of the extruder; having a serious

effect on the mixing performance. On the other hand, slight wear in the mixing zone helps to

reduce the power consumption.

The power consumption of the extruder increases linearly with both screw speed and

viscosity of compound. Power consumption depends on Material characteristics, Screw ge-

ometry, Screw speed, Cooling arrangement and Extruder type. The pumping efficiency of

a screw extruder is 10% or less. This means that energy consumed in actual pumping of

the polymer material is less than 10% of the input energy. The rest 90% or more goes into

the power consumed in channel and flight clearance and, in viscous heating of the polymer.

Energy consumption in an extruder is inversely proportional to the channel depth and flight

radial clearance. Thus, greater the channel depth and larger the flight clearance, lesser is

the power consumption. Larger flight clearance drastically reduces mixing efficiency and


6.5 Summary 189

overall extruder performance. If we want to achieve a reduction in mechanical power, the

screw speed may be decreased. Throughput and Pressure would decrease approximately in

proportion to screw speed. But when we reduce the speed, the mechanical power would be

reduced more than in proportion to speed. That is, if we have reduced speed by 10%, the

power would reduce by more than 10% and torque required to drive the extruder would be less.

The relations among all the design parameters along with the rheological properties

of material are complex, and hence, MOO solutions can guide decision maker in taking

appropriate design decisions for rubber extruder. The velocity, pressure and temperature

profiles obtained using FEA models are converted to throughput and energy consumption.

The information is also utilised to restrict scorch formation and proper mixing monitoring

temperature and RTD.


191

Chapter 7

Conclusions and Scope of Future Work

Genetic Algorithm(GA) has proved its capacity as an evolutionary computation method for

Global optimization of complex problems. Its popular for its robustness, flexibility and

efficiency, but it is computationally expensive compared to the classical methods. It is

designed for unconstrained problems, and hence, require additional mechanism for constraint

handling. It can handle multi-objective optimization problems easily and effectively due to its

evolutionary nature capable of handling conflicting objectives. Solving the complex multi-

objective problems requires very long time. The research focused on upgrading the GA to

enhance the convergence and constraint handling capabilities for multi-objective optimization.

The proposed approaches are tested by benchmark test functions and further validated using

rubber extruder screw design application. The conclusions of research work are summarised

in the next section.

7.1 Conclusions

Hybridization of binary and real coded GA is explored to enhance the convergence rate.

Binary encoding and real parameter encoding of GA has their strengths and limitations.

Along with encoding benefits, each algorithm will contribute the benefits of their selection,

crossover and mutation operators. The focus of the hybridization is to combine the strengths

of both the algorithms. Binary encoding has the flexibility of adjusting accuracy of decision

variables by adjusting binary chromosome size. The mechanism of binary encoding gives

192 7. Conclusions and Scope of Future Work

better exploration of search space using small chromosome size. Use of small chromosome

size supports very good initial convergence, but convergence slows down as it reaches near

the optimum solution. Real coded GA takes up that responsibility of convergence at that

stage. The algorithm uses the concept of parallel population and combined binary and real GA.

The developed algorithm using hybridization of binary and real coded GA is presented as

Parallel Universe Alien Genetic Algorithm (PUALGA). The information from binary popula-

tion is transported to real coded population by Aliens. The concept can be applied for any

population based evolutionary algorithm; however, the results are shown under GA framework

in this study. Though, the concept can also be used for single or multi-objective optimization,

it is explored for multi-objective optimization application in the current work. Non-dominated

sorting is used for survival selection being a benchmark in multi-objective optimization. The

advantage of using two sub populations reduces the complexity of sorting and achieves better

results with same number of function evaluations. The proposed PUALGA algorithm has

two tuning parameters, binary fraction of population and, number of Aliens transporting

information after every generation. Sensitivity analysis is carried out using benchmark test

problems for both of the tuning parameters. Binary fraction of 0.4 and 5 Alien numbers are

the recommended parameters based on the analysis. The performance of PUALGA algorithm

has been compared with its native, binary and real coded GAs and, Jumping Gene Adaptation

of GA. The proposed PUALGA algorithm drastically enhances the initial convergence rate

for all bench mark MOO test problems taking the benefit of exploration capacity of binary

encoding.

Constraint handling is always a critical part in performance of optimization method. Ana-

lytical and numerical methods of constraint handling needs to be interfaced with optimization

method to handle constant. Constraint handling is more critical in case of evolutionary op-

timization algorithms, as they are naturally designed for unconstrained optimization. Even

constraint handling becomes more crucial and typical for multi-objective optimization prob-

lems. They are very difficult to solve. A new constraint handling mechanism for population

based evolutionary algorithms(EAs) using generalized Boundary Inspection (BI) approach is


7.1 Conclusions 193

proposed here. The concept is general and can be used with any population based EAs. Its

implementation for multi-objective optimization has been demonstrated in this work.

The PUALGA algorithm is enhanced for constraint handling using BI approach. The

proposed BI algorithm converts all infeasible members to feasible members at every gen-

eration of evolution. Every infeasible member is projected through the randomly selected

feasible member. A parameter λ is selected which decides the location of the new point on

the line joining the selected infeasible and feasible points. The value of this parameter λ is

to be selected such that the new point is inside the feasible region. It is very difficult to tune

this parameter for different types of problems. Its value depends upon nature of infeasible

region, as well as, distribution of feasible and infeasible members within their region. An

automated selection of this tuning parameter is implemented based on success probability

history. The parameter λ is selected from an Ensemble of predefined values. The selection

process has self learning with automatic tuning, avoiding adaptive selection or tuning during

the evolution process. The efficacy of the BI approach is presented using multi-objective

PUALGA algorithm. The PUALGA with BI approach has been tested with three bench mark

constrained optimization test functions and two design applications. Computation efforts are

evaluated in terms of total computational time, number of function evaluations and number

of constraint evaluation. Performance is evaluated using Inverted Generational Distance

(IGD) metric value representing convergence to true pareto front, uniformity of distribution

within the front and coverage. The performance is compared with two popular constraint

handling algorithms namely, augmented penalty function and ignore infeasible. The proposed

algorithm performed very well improving IGD value of obtained optimal solutions for same

computational efforts.

The extrusion process involves several complex phenomena: complex rheology, fluid flow

and heal transfer. The most critical part of rubber extruder design is designing a screw, which

is the most crucial part of the extruder. The Helix angle is the most important parameter affect-

ing the performance of the extruder screw. It affects throughput, mixing, discharge pressure

and power consumption. By increasing the screw speed, the throughput increases. Too high


194 7. Conclusions and Scope of Future Work

speed will result in greater temperature variation and poor mixing, and thus, deteriorates the

quality of products. The objective of the design is to deliver the largest amount of output

of good quality. A mathematical model for rubber extruder is developed using finite differ-

ence technique considering temperature dependent viscosity modelled using Carreau-Yasuda

model. The extruder model consists of momentum and energy balance equations along with

rheological properties of material. The complexity of geometry and rheology together results

in to a model, which is difficult to solve.

The FEA model and solution algorithm for extruder screw channel velocity, pressure and

temperature profile is developed. The throughput, energy and residence time are calculated

using these profiles. The parameters which influence throughput, power consumption and

residence time distribution of extruder are screw length, channel depth, flight width, clearance,

helix angle, screw speed, pressure, and viscosity. Relationships of helix angle, channel height

and viscosity are reviewed to find design parameter that maximizes throughput. Response

of extruder to changes in the operating conditions, and the geometry is studied using the

design equations and correlations along with FEA model. The developed model is validated

comparing it with analytical and empirical model results. Simulation study is also carried

out to evaluate sensitivity of different model parameters. The FEA model developed can

be used for multi objective optimization to generate pareto optimal solutions for throughput

maximization - power consumption minimization.

Multi objective optimization of rubber extruder screw design is explored for throughput

maximization and power consumption minimization. The FEA model developed is used for

multi objective optimization of rubber extruder screw design. The velocity, pressure and

temperature profiles obtained using FEA models are converted to throughput and energy con-

sumption. The screw design parameters and temperature profile are obtained for simultaneous

maximization of throughput and minimization of power consumption. The temperatures of

the material under process within the extruder and residence time distribution of product are

also tracked for maintaining the quality of product. The screw helix angle, channel depth and,

screw speed are used as manipulated design parameters along with barrel temperature profile.


7.2 Scope of Future Work 195

Best screw geometry, screw speed and barrel temperature profile are obtained using proposed

the PUALGA algorithm with BI approach for multi-objective optimization. These multiple

optimum solutions assist the decision maker in selecting an appropriate design which is the

best according to his needs. Eutopia point with helix angle of 35 deg and channel height of 8

mm is selected as the best point and, its design is experimentally verified. The experimental

results obtained, were very close to the optimum design.

7.2 Scope of Future Work

The objectives of the current research are fully satisfied. While working in the area doing

research, it was felt that the work can further be explored in following areas:

• The FEA modelling technique can be used for Reactive extrusion modelling and simula-

tion. The model can be used used to design extruder such that no vulcanization takes

place during extrusion. If the cross-linking starts during extrusion, then scorch formation

takes place. Scorch formation deteriorates the quality of product. This model can be

used for design reactive extruder.

• The Model can be further enhanced for different types of geometry of screw. The current

model is for single screw extruder. The model can be further enhanced to incorporate

co-rotating twin screw extruder.

• Ensemble approach of automated parameter tuning used for BI approach can be used

for any parameter of evolutionary optimization. It can also be explored for PUALGA

algorithm to dynamically adopt tuning parameters.

• Co-operative Co-evolution can be explored for parallel universe alien genetic algorithm

in place of survival fittest selection. The concept can be explored to enhance evolutionary

optimization algorithms.


197

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217

Appendix A

Non-dominated sorting GeneticAlgorithm (NSGA)-II

Non-dominated sorting Genetic Algorithm is...

Figure A.1. Non-dominated sorting algorithm (NSGA) pseudo code

218 A. Non-dominated sorting Genetic Algorithm (NSGA)-II

Figure A.2. Non-dominated sorting Genetic Algorithm-I (NSGA-I)

Figure A.3. Non-dominated sorting Genetic Algorithm-II (NSGA-II)


219

Figure A.4. Illustrative example of Pareto optimality in objective space (left) and the possible relations ofsolutions in objective space (right).


221

Appendix B

NSGA-II-JG

Elitist non-dominated sorting genetic algorithm with jumping genes, NSGA-II-JG (see

flowchart in Fig. (B.1)).

Source: Kasat and Gupta (2003)

Note: The algorithm assumes that we are minimizing all the objective functions, fq

1. Generate box, P, of Np parent chromosomes using a random-number code to generate

the several binaries. These chromosomes are given a sequence (position) number as

generated.

2. Classify these chromosomes into fronts based on non-domination (Deb (2001)), as

follows:

a) Create new (empty) box, P′, of size, Np

b) Transfer ith chromosome from P to P′, starting with i = 1

c) Compare chromosome i with each member, say, j , already present in P′, one at a

time.

d) If i dominates (Deb (2001)) over j (i.e. i is superior to or better than j in terms of

all objective functions), remove the jth chromosome from P′ and put it back in its

original location in P.

e) If i is dominated over by j , remove i from P′ and put it back in its position in P.

f) If i and j are non-dominating (i.e. there is at least one objective function associated

with i that is superior to/better than that of j ), keep both i and j in P′ (in sequence).

222 B. NSGA-II-JG

Test for all j present in P′.

g) Repeat for next chromosome (in the sequence, without going back) in P till all Np

are tested. P′ now contains a sub-box (of size ≤ Np) of non-dominated chromosomes

(a subset of P), referred to as the first front or sub-box. Assign it a rank number, Irank,

of 1.

h) Create subsequent fronts in (lower) sub-boxes of P′, using Step 2b above (with the

chromosomes remaining in P). Compare these members only with members present

in the current sub-box, and not with those in earlier (better) sub-boxes. Assign these

Irank = 2, 3,. . . Finally, we have all Np chromosomes in P′, boxed into one or more

fronts.

3. Spreading out: Evaluate the crowding distance, Ii,dist , for the ith chromosome in any

front, j, of P′ using the following procedure:

a) Rearrange all chromosomes in front j in ascending order of the values of any one

(say, the qth) of their several objective functions (fitness functions). This provides a

sequence, and, thus, defines the nearest neighbours of any chromosome in front j.

b) Find the largest cuboid (rectangle for two fitness functions) enclosing chromosome i

that just touches its nearest neighbours in the f -space.

c) Ii,dist = 1/2×(sum of all sides of this cuboid).

d) Assign large values of Ii,dist to solutions at the boundaries (the convergence charac-

teristics would be influenced by this choice).

4. Make Np copies randomly (duplication permissible),of the better chromosomes from P′

into a new box, P′′ using:

a) Select any pair, i and j , from P′ (randomly,irrespective of fronts).

b) Identify the better of these two chromosomes. Chromosome i is better than chromo-

some j if:

Ii,rank 6= I j,rank; Ii,rank < I j,rank

Ii,rank = I j,rank; Ii,dist < I j,dist


223

c) Copy (without removing from P′) the better of these two chromosomes in a new box,

P′′.

d) Repeat till P′′ has Np members. Not all of P′ need be in P′′. By this method, the

better members of P′ are copied into P′′ stochastically.

5. Copy all of P′′ in a new box, D, of size Np. Carry out crossover (using the stochastic re-

mainder roulette wheel selection procedure and mutation (Deb (2012)) of chromosomes

in D. This gives a box of Np daughter chromosomes.

6. JG Operation: Select a chromosome (sequentially) from D. Check if JG operation is

needed, using Pjump. If yes:

a) Generate a random number (between 0 and 1).

b) Multiply this by Ichr, the total number of binaries in the chromosome. Round-off to

convert into an integer. This represents the position of one end (either beginning or

end) of a transposon.

c) Repeat steps 6a and 6b to identify the second end of the transposon.

d) Invert or replace the set of binaries between these locations (use random numbers to

generate the transposon for the case of replacement).

7. Elitism: Copy all the Np best parents (P′′) and all the Np daughters with transposons (D)

into box PD. Box PD has 2Np chromosomes.

a) Reclassify these 2Np chromosomes into fronts (box PD′) using only non-domination

(as described in Step 2 above).

b) Take the best Np from box PD′ and put into box P′′′.

8. This completes one generation. Stop if appropriate criteria are met, e.g., the generation

number/maximum number of generations (user specified).

9. Copy P′′′ into starting box, P. Go to Step 2 above.


224 B. NSGA-II-JG

Figure B.1. Flowchart of NSGA-II-JG


225

Appendix C

List of Publications

JOURNAL

Published

1. Rupande Desai, Narendra Patel, Dr. S A Puranik., 2018. Multi- Objective Optimizationusing Binary-Real Multi Population Hybridization: Parallel Universe Alien GeneticAlgorithm (PUALGA)International Journal for Research in Engineering Application &Management (IJREAM), 04(6): 397 - 405, 2018.

2. Rupande Desai, Narendra Patel, Dr. S A Puranik., 2018. Boundary Inspection Approachfor Constrained Multi-Objective Optimization. International Journal for Research inEngineering Application & Management (IJREAM), 04(7): 224 - 231, 2018.

Submitted

1. Rupande Desai, Narendra Patel, Dr. S A Puranik. Parallel Universe Alien GA forMulti-Objective Optimization International Journal of Advances in Soft Computing andits Applications (IJASCA). [Submitted with ID : IJASCA-2018-188 and under review].

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