Hybrid Multi-Gradient Explorer Algorithm for Global Multi-Objective Optimization
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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
© DESAI RUPANDE NITNBHAI
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
under my supervision and (iii) The thesis represents independent research work on the part of
the Research Scholar.
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Name of Supervisor: Dr. S. A. Puranik
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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
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Wang. "Dimension-Independent HarnackInequalities for Subordinated Semigroups",Potential Analysis, 2010Publicat ion
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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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Signature of Research scholar: ...................................................
Name of Research Scholar: Desai Rupande Nitinbhai
Date:December ,2019. Place: Ahmedabad
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The viva-voce of the PhD Thesis submitted by Ms. Desai Rupande Nitinbhai (Enrollment
No. 139997105010) entitled “Multi-Objective Optimization: Algorithm Development
and Applications in Rubber Technology" was conducted on ............................... at Gujarat
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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.
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
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.
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
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].
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 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
Ph.D. thesis Desai Rupande Nitinbhai
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)
Desai Rupande Nitinbhai Ph.D. thesis
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-
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Problem binGA realGA jgGA PUALGA
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.
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
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:
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Helix Angle (degree)
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.
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
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.
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 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.
Desai Rupande Nitinbhai Ph.D. thesis
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.
Ph.D. thesis Desai Rupande Nitinbhai
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
List of Figures lv
3.32 Generation wise convergence metric γ (average of 20 runs) for ZDT2 testfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.33 Generation wise convergence metric γ (average of 20 runs) for ZDT3 testfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.34 Generation wise convergence metric γ (average of 20 runs) for ZDT4 testfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.35 Generation wise convergence metric γ (average of 20 runs) for ZDT6 testfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.36 Distribution and coverage of pareto front as spread metric ∆ (average of 20runs) for ZDT1 test function . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.37 Distribution and coverage of pareto front as spread metric ∆ (average of 20runs) for ZDT2 test function . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.38 Distribution and coverage of pareto front as spread metric ∆ (average of 20runs) for ZDT3 test function . . . . . . . . . . . . . . . . . . . . . . . . . . 100
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
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
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
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.7 Sensitivity analysis for number of alien transfer on performance metrics for
POL test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.8 Sensitivity analysis for binary fraction on performance metrics for KUR test
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.9 Sensitivity analysis for number of alien transfer on performance metrics for
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
Ph.D. thesis Desai Rupande Nitinbhai
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.
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
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.
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 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
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
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
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 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.
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
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.
Ph.D. thesis Desai Rupande Nitinbhai
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
14 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.1 Multi-Objective Optimization 15
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
Desai Rupande Nitinbhai Ph.D. thesis
16 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.1 Multi-Objective Optimization 17
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-
Desai Rupande Nitinbhai Ph.D. thesis
18 2. Literature Review
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.
Ph.D. thesis Desai Rupande Nitinbhai
2.1 Multi-Objective Optimization 19
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)
Desai Rupande Nitinbhai Ph.D. thesis
20 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
22 2. Literature Review
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.
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
24 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
26 2. Literature Review
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.
Ph.D. thesis Desai Rupande Nitinbhai
2.3 Evolutionary Optimization 27
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
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28 2. Literature Review
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.
Ph.D. thesis Desai Rupande Nitinbhai
2.3 Evolutionary Optimization 29
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
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30 2. Literature Review
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:
Ph.D. thesis Desai Rupande Nitinbhai
2.3 Evolutionary Optimization 31
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
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32 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
34 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.4 Rubber Extruder Design Optimization 35
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,
Desai Rupande Nitinbhai Ph.D. thesis
36 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.4 Rubber Extruder Design Optimization 37
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
Desai Rupande Nitinbhai Ph.D. thesis
38 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.4 Rubber Extruder Design Optimization 39
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-
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40 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.4 Rubber Extruder Design Optimization 41
(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
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42 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.4 Rubber Extruder Design Optimization 43
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
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44 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
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:
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46 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.5 Rheology of Rubber 47
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.
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48 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.5 Rheology of Rubber 49
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
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50 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
2.5 Rheology of Rubber 51
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
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52 2. Literature Review
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)
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2.5 Rheology of Rubber 53
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.
Desai Rupande Nitinbhai Ph.D. thesis
54 2. Literature Review
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
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
60 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
62 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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,
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
64 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
66 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
(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
Ph.D. thesis Desai Rupande Nitinbhai
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))
Desai Rupande Nitinbhai Ph.D. thesis
68 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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).
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
70 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
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 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
Ph.D. thesis Desai Rupande Nitinbhai
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)
Desai Rupande Nitinbhai Ph.D. thesis
72 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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,
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
74 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 75
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.
Desai Rupande Nitinbhai Ph.D. thesis
76 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 77
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
Desai Rupande Nitinbhai Ph.D. thesis
78 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
Number of Aliens transerred
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
me
an
(a) Generational Distance metric (b) Spread metric
Figure 3.12. Sensitivity for number of alien transfer on performance metric at 50th Generation for SCH1 testfunction
The Fig. (3.13) shows the generation wise convergence metric, γ values (mean of ten
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
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 79
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.
Desai Rupande Nitinbhai Ph.D. thesis
80 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 81
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
(a) Generational Distance metric (b) Spread metric
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
Binary FractionGD metric Spread metric Time
Mean Variance Mean Variance Mean Variance0.1 0.0035 0.0054 0.4635 0.1407 2.8165 0.58310.2 0.0017 0.0003 0.4212 0.0318 2.7880 0.68770.3 0.0019 0.0002 0.4196 0.0247 2.8394 0.71790.4 0.0019 0.0002 0.4176 0.0246 2.7419 0.69380.5 0.0021 0.0003 0.4096 0.0289 2.8109 0.66880.6 0.0023 0.0003 0.4155 0.0481 3.0887 0.65580.7 0.0024 0.0005 0.4129 0.0308 3.5446 0.93670.8 0.0031 0.0003 0.3953 0.0304 4.2125 1.38830.9 0.0035 0.0006 0.3809 0.0451 3.4466 0.6636
Desai Rupande Nitinbhai Ph.D. thesis
82 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
The Fig. (3.16) shows the generation wise convergence metric, γ values (mean of ten
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
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 83
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
Number of Aliens transerred
1.5
2
2.5
3
3.5
4
4.5
me
an
10-4
0 2 4 6 8 10
Number of Aliens transerred
0.015
0.02
0.025
0.03
0.035
0.04
me
an
(a) Generational Distance metric (b) Spread metric
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
Desai Rupande Nitinbhai Ph.D. thesis
84 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
Table 3.5. Sensitivity analysis for number of alien transfer on performance metrics for FON test function
No of Alien transferGD metric Spread metric Time
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.
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 85
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
Desai Rupande Nitinbhai Ph.D. thesis
86 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
(a) Generational Distance metric (b) Spread metric
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
Binary FractionGD metric Spread metric Time
Mean Variance Mean Variance Mean Variance0.1 0.0089 0.0004 0.8114 0.0092 8.5336 1.01000.2 0.0080 0.0008 0.8196 0.0101 7.8356 0.69680.3 0.0079 0.0011 0.8225 0.0110 7.8060 0.52080.4 0.0082 0.0006 0.8119 0.0130 7.9552 0.60370.5 0.0078 0.0012 0.8234 0.0088 8.1657 0.52450.6 0.0082 0.0006 0.8243 0.0101 9.5522 0.94760.7 0.0079 0.0009 0.8114 0.0087 8.6429 0.08040.8 0.0077 0.0008 0.8198 0.0119 9.1725 0.60300.9 0.0084 0.0009 0.8177 0.0082 10.1994 1.0971
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 87
The Fig. (3.22) shows the generation wise convergence metric, γ values (mean of ten
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
Desai Rupande Nitinbhai Ph.D. thesis
88 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
Number of Aliens transerred
0
0.005
0.01
0.015
0.02
me
an
0 2 4 6 8 10
Number of Aliens transerred
2
3
4
5
6
7
8
9
me
an
10-3
(a) Generational Distance metric (b) Spread metric
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
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 89
Table 3.7. Sensitivity analysis for number of alien transfer on performance metrics for POL test function
No of Alien transferGD metric Spread metric Time
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.
Desai Rupande Nitinbhai Ph.D. thesis
90 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
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
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 91
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
(a) Generational Distance metric (b) Spread metric
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
Binary FractionGD metric Spread metric Time
Mean Variance Mean Variance Mean Variance0.1 0.0245 0.0069 0.8649 0.0242 3.2536 0.91110.2 0.0207 0.0030 0.8657 0.0375 2.7041 0.02110.3 0.0273 0.0179 0.8571 0.0277 2.7347 0.04410.4 0.0249 0.0074 0.8503 0.0219 2.8870 0.42840.5 0.0207 0.0050 0.8444 0.0234 2.9862 0.52210.6 0.0278 0.0170 0.8657 0.0356 3.4936 0.88410.7 0.0224 0.0054 0.8698 0.0152 2.9501 0.04420.8 0.0220 0.0051 0.9008 0.0364 3.0330 0.04070.9 0.0176 0.0061 0.9302 0.0361 3.1113 0.0718
Desai Rupande Nitinbhai Ph.D. thesis
92 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
The Fig. (3.28) shows the generation wise convergence metric, γ values (mean of ten
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
Ph.D. thesis Desai Rupande Nitinbhai
3.7 Sensitivity Analysis of Proposed Algorithm 93
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
Number of Aliens transerred
5
6
7
8
9
10
11
12
13
14
me
an
10-4
0 2 4 6 8 10
Number of Aliens transerred
0.008
0.009
0.01
0.011
0.012
0.013
0.014
0.015
me
an
(a) Generational Distance metric (b) Spread metric
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
Desai Rupande Nitinbhai Ph.D. thesis
94 3. Parallel Universe Alien Genetic Algorithm (PUALGA) for Multi-Objective Optimization
Table 3.9. Sensitivity analysis for number of alien transfer on performance metrics for KUR test function
No of Alien transferGD metric Spread metric Time
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
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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.
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0 50 100 150 200 250
0
0.5
1
1.5
Generation No
mean γ
rNSGA−II
bNSGA−II
aJGGA
PUALGA
Figure 3.31. Generation wise convergence metric γ (average of 20 runs) for ZDT1 test function
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
Figure 3.32. Generation wise convergence metric γ (average of 20 runs) for ZDT2 test function
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3.8 Performance Evaluation Results and Discussion 97
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
Figure 3.33. Generation wise convergence metric γ (average of 20 runs) for ZDT3 test function
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 3.34. Generation wise convergence metric γ (average of 20 runs) for ZDT4 test function
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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
Figure 3.35. Generation wise convergence metric γ (average of 20 runs) for ZDT6 test function
Table 3.10. 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
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
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3.8 Performance Evaluation Results and Discussion 99
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
Figure 3.36. Distribution and coverage of pareto front as spread metric ∆ (average of 20 runs) for ZDT1 testfunction
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).
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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
Figure 3.37. Distribution and coverage of pareto front as spread metric ∆ (average of 20 runs) for ZDT2 testfunction
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 3.38. Distribution and coverage of pareto front as spread metric ∆ (average of 20 runs) for ZDT3 testfunction
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3.8 Performance Evaluation Results and Discussion 101
Table 3.11. Statasical analysis of distribution and coverage as spread metric, ∆ for 20 simulation runs
Problem binGA realGA jgGA PUALGA
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
Figure 3.39. Pareto front for ZDT4 test function at the end of 200 generations
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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
Figure 3.40. Pareto front for ZDT6 test function at the end of 200 generations
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
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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.
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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
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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
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108 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
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4.2 Boundary Inspection Approach for Constraint Handling 109
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
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110 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
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4.2 Boundary Inspection Approach for Constraint Handling 111
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)
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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,
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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
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114 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
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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
Desai Rupande Nitinbhai Ph.D. thesis
116 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
Constraint on Decision Variable 0≤ x1,x2 ≤ 10
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
Constraint on Decision Variable 0≤ x1,x2 ≤ 10
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
Constraint on Decision Variable 0≤ x1,x2 ≤ 10
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
Ph.D. thesis Desai Rupande Nitinbhai
4.4 Sensitivity analysis of propose BI approach 117
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
Desai Rupande Nitinbhai Ph.D. thesis
118 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
medium feasible areaa
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-
Ph.D. thesis Desai Rupande Nitinbhai
4.4 Sensitivity analysis of propose BI approach 119
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.
Desai Rupande Nitinbhai Ph.D. thesis
120 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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.
The plot of performance probability for all the selected three cases of different feasible
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
Ph.D. thesis Desai Rupande Nitinbhai
4.4 Sensitivity analysis of propose BI approach 121
0 50 100 150 200
Generation No.
0
5
10
15
20
25
30
BI
tre
ate
d m
em
be
rssmall feasible area
medium feasible areaa
large feasible area
0 50 100 150 200
Generation No.
0
50
100
150
NC
Es
small feasible area
medium feasible areaa
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
Desai Rupande Nitinbhai Ph.D. thesis
122 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
The generation wise total count for BI treatment and NCEs are plotted in Fig. (4.12). The
small feasible area corresponds to 15%, medium feasible region to 35% and, large feasible
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
Ph.D. thesis Desai Rupande Nitinbhai
4.4 Sensitivity analysis of propose BI approach 123
0 50 100 150 200
Generation No.
0
10
20
30
40
50
BI
tre
ate
d m
em
be
rssmall feasible area
medium feasible areaa
large feasible area
0 50 100 150 200
Generation No.
0
50
100
150
200
NC
Es
small feasible area
medium feasible areaa
large feasible area
(a) Members requiring BI treatment (b) NCEs required for BI treatment
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
Desai Rupande Nitinbhai Ph.D. thesis
124 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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 4.13. Effect of feasible area on adoptive learning probability distribution of selecting a division rationvalue from an Ensemble pool
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
The generation wise total count for BI treatment and NCEs are plotted in Fig. (4.15). The
Ph.D. thesis Desai Rupande Nitinbhai
4.4 Sensitivity analysis of propose BI approach 125
small feasible area corresponds to 51%, medium feasible region to 75% and, large feasible
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
medium feasible areaa
large feasible area
0 50 100 150 200
Generation No.
0
50
100
150
NC
Es
small feasible area
medium feasible areaa
large feasible area
(a) Members requiring BI treatment (b) NCEs required for BI treatment
Figure 4.15. Generation wise infeasible members requiring BI treatment and NCEs required for FOsquare case
The plot of performance probability for all the selected three cases of different feasible
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
Desai Rupande Nitinbhai Ph.D. thesis
126 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
Figure 4.16. Effect of feasible area on adoptive learning probability distribution of selecting a division rationvalue from an Ensemble pool
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:
Ph.D. thesis Desai Rupande Nitinbhai
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)
Desai Rupande Nitinbhai Ph.D. thesis
128 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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:
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
130 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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:
Ph.D. thesis Desai Rupande Nitinbhai
4.6 Test Problems and Engineering Design Applications 131
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:
Desai Rupande Nitinbhai Ph.D. thesis
132 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
Ph.D. thesis Desai Rupande Nitinbhai
4.6 Test Problems and Engineering Design Applications 133
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:
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134 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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,
Ph.D. thesis Desai Rupande Nitinbhai
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
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136 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
Ignorance Infeasible
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
Ph.D. thesis Desai Rupande Nitinbhai
4.7 Results and Discussion 137
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
Ignorance Infeasible
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
Ignorance Infeasible
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
Ignorance Infeasible
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
Desai Rupande Nitinbhai Ph.D. thesis
138 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
Ignorance Infeasible
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
Ignorance Infeasible
Bounday Inspection
Figure 4.22. Average IGD values against NFEs for BNH test function
Ph.D. thesis Desai Rupande Nitinbhai
4.7 Results and Discussion 139
500 1000 1500 2000 2500
NCEs
0.4
0.5
0.6
0.7
0.8
me
an
IG
D
Augmented Penalty
Ignorance Infeasible
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
Ignorance Infeasible
Bounday Inspection
Figure 4.24. Pareto Front for BNH test function at 50 Generations
Desai Rupande Nitinbhai Ph.D. thesis
140 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
0 5 10 15 20
Run time (s)
0
5
10
15
20
25m
ea
n I
GD
Augmented Penalty
Ignorance Infeasible
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
Ignorance Infeasible
Bounday Inspection
Figure 4.26. Average IGD values against NFEs for OSY test function
Ph.D. thesis Desai Rupande Nitinbhai
4.7 Results and Discussion 141
0 0.5 1 1.5 2 2.5 3
NCEs 104
0
5
10
15
20
25
me
an
IG
D
Augmented Penalty
Ignorance Infeasible
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
Ignorance Infeasible
Bounday Inspection
-300 -250 -200 -150 -100 -50 0
f1
0
20
40
60
80
f 2
Augmented Penalty
Ignorance Infeasible
Bounday Inspection
(a) at 100 Generations (b) at 250 Generations
Figure 4.28. Pareto Front for OSY test function
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142 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
Ignorance Infeasible
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
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4.7 Results and Discussion 143
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
Ignorance Infeasible
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
Ignorance Infeasible
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
Ignorance Infeasible
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
Ignorance Infeasible
Bounday Inspection
(a) at 100 Generations (b) at 250 Generations
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
Desai Rupande Nitinbhai Ph.D. thesis
144 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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
Ignorance Infeasible
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.
Ph.D. thesis Desai Rupande Nitinbhai
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
Ignorance Infeasible
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
Ignorance Infeasible
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
Ignorance Infeasible
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
Ignorance Infeasible
Bounday Inspection
(a) at 25 Generations (b) at 250 Generations
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
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146 4. Boundary Inspection Approach for Constrained handling in Evolutionary Optimization Algorithms
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.
Ph.D. thesis Desai Rupande Nitinbhai
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
150 5. Rubber Extruder Modelling and Simulation
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
Ph.D. thesis Desai Rupande Nitinbhai
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
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152 5. Rubber Extruder Modelling and Simulation
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
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5.2 Mathematical modelling of single screw extruder 153
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.
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154 5. Rubber Extruder Modelling and Simulation
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
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5.2 Mathematical modelling of single screw extruder 155
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)
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156 5. Rubber Extruder Modelling and Simulation
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
Ph.D. thesis Desai Rupande Nitinbhai
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)
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158 5. Rubber Extruder Modelling and Simulation
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
Ph.D. thesis Desai Rupande Nitinbhai
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
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160 5. Rubber Extruder Modelling and Simulation
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
Ph.D. thesis Desai Rupande Nitinbhai
5.4 Sensitivity of parameters influencing Extruder Throughput 161
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.
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162 5. Rubber Extruder Modelling and Simulation
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.
Ph.D. thesis Desai Rupande Nitinbhai
5.4 Sensitivity of parameters influencing Extruder Throughput 163
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
Desai Rupande Nitinbhai Ph.D. thesis
164 5. Rubber Extruder Modelling and Simulation
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.
Ph.D. thesis Desai Rupande Nitinbhai
5.4 Sensitivity of parameters influencing Extruder Throughput 165
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
Desai Rupande Nitinbhai Ph.D. thesis
166 5. Rubber Extruder Modelling and Simulation
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
Ph.D. thesis Desai Rupande Nitinbhai
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).
Desai Rupande Nitinbhai Ph.D. thesis
168 5. Rubber Extruder Modelling and Simulation
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
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
170 5. Rubber Extruder Modelling and Simulation
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.
Ph.D. thesis Desai Rupande Nitinbhai
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.
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
174 6. Multi-Objective Optimization: Application to Rubber Extruder Screw Design
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
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
176 6. Multi-Objective Optimization: Application to Rubber Extruder Screw Design
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)
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
178 6. Multi-Objective Optimization: Application to Rubber Extruder Screw Design
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
Ph.D. thesis Desai Rupande Nitinbhai
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)
Desai Rupande Nitinbhai Ph.D. thesis
180 6. Multi-Objective Optimization: Application to Rubber Extruder Screw Design
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
Ph.D. thesis Desai Rupande Nitinbhai
6.4 Results and Discussion 181
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
Desai Rupande Nitinbhai Ph.D. thesis
182 6. Multi-Objective Optimization: Application to Rubber Extruder Screw Design
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
Ph.D. thesis Desai Rupande Nitinbhai
6.4 Results and Discussion 183
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
Helix Angle (degree)
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
Desai Rupande Nitinbhai Ph.D. thesis
184 6. Multi-Objective Optimization: Application to Rubber Extruder Screw Design
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
Ph.D. thesis Desai Rupande Nitinbhai
6.4 Results and Discussion 185
-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
Desai Rupande Nitinbhai Ph.D. thesis
186 6. Multi-Objective Optimization: Application to Rubber Extruder Screw Design
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
188 6. Multi-Objective Optimization: Application to Rubber Extruder Screw Design
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
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
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
Ph.D. thesis Desai Rupande Nitinbhai
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
Desai Rupande Nitinbhai Ph.D. thesis
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.
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
197
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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)
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Figure A.4. Illustrative example of Pareto optimality in objective space (left) and the possible relations ofsolutions in objective space (right).
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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
Ph.D. thesis Desai Rupande Nitinbhai
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.
Desai Rupande Nitinbhai Ph.D. thesis
224 B. NSGA-II-JG
Figure B.1. Flowchart of NSGA-II-JG
Ph.D. thesis Desai Rupande Nitinbhai
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].