Teaching–Learning-Based Optimization (TLBO) seems to be a rising star from amongst a number of metaheuristics with relatively competitive performances. It is reported that it outperforms some of the well-known metaheuristics regarding constrained benchmark functions, constrained mechanical design, and continuous non-linear numerical optimization problems. Such a breakthrough has steered us towards investigating the secrets of TLBO’s dominance. This report’s findings on TLBO qualitatively and quantitatively through code-reviews and experiments, respectively.

1. TEACHING AND LEARNING
BASED OPTIMISATION

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CONTENTS
1. Optimization classification 4
2. Introduction to TLBO, Clustering with TLBO 6-7
3. Multi objective Optimisation with TLBO 8
4. Applications of TLBO
a. Multi objective optimization with TLBO Parameter
optimization of modern machining processes using
teaching– learning-based optimization algorithm
i. Ultrasonic machining 10
ii. Wire electrical discharge machining 11
b. Multi-objective optimization of heat exchangers using
a modified teaching-learning- based optimization
algorithm 12
c. Multi-objective optimization of two stage thermoelectric
cooler using a modified teaching–learning-based
optimization algorithm 13
d. Design of planar steel frames using Teaching–Learning
Based Optimization 14
e. A design of IIR based digital based aids using teaching
learning based optimization 15-19
f. Size and geometry optimization of trusses using teaching
and learning based optimization 20-24
5. Conclusion 25
References 26

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LIST OF TABLES
1 Table 1 Objective function on the basis of amplitude difference
2 Table 2
Results of sensitivity analysis of 18 bar truss for 30
independent runs
3 Table 3 Optimal results of TLBO with ps size 50 for 18-bar
truss
LIST OF DIAGRAMS-
Sr.
No
Fig. Page
no.
1 Fig 1 Plate-fin heat exchanger and rectangular offset
strip fin
13
2 Fig 2 Shell and tube heat exchanger geometry 13
3 Fig 3 Two stage TEC. (a) Electrically separated and
(b) electrically connected in series.
14
4 Fig 4 Effect of number of teachers on the
convergence rate of the modified TLBO
algorithm for multi-objective consideration
(electrically separated TEC).
14
5 Fig 5 Two-bay three-story frame design 15
6 Fig 6 Audiogram-1 19
7 Fig 7 The geometry of the 18-bar planar truss 22
8 Fig 8 Convergence of TLBO ps size 50 25
9 Fig 19 The optimized 18-bar truss 25

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ABSTRACT
Teaching–Learning-Based Optimization (TLBO) seems to be a rising
star from amongst a number of metaheuristics with relatively competitive
performances. It is reported that it outperforms some of the well-known
metaheuristics regarding constrained benchmark functions, constrained mechanical
design, and continuous non-linear numerical optimization problems. Such a
breakthrough has steered us towards investigating the secrets of TLBO’s
dominance. This report’s findings on TLBO qualitatively and quantitatively
through code-reviews and experiments, respectively. Findings have revealed three
important mistakes regarding TLBO:
(1) at least one unreported but important step;
(2) Incorrect formulae on a number of fitness function evaluations; and
(3) Misconceptions about parameter-less control. Additionally, unfair
experimental settings/conditions were used to conduct experimental
comparisons (e.g., different stopping criteria).
The ultimate goal of this paper is to provide reminders for
metaheuristics’ researchers and practitioners in order to avoid similar mistakes
regarding both the qualitative and quantitative aspects, and to allow fair
comparisons of the TLBO algorithm to be made with other metaheuristic
algorithms.

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1. OPTIMISATION CLASSIFICATION
Optimization is a mathematical discipline that concerns the finding of minima and
maxima of functions, subject to so-called constraints. Optimization originated in
the 1940s, when George Dantzig used mathematical techniques for generating
"programs" (training timetables and schedules) for military application. Since then,
his "linear programming" techniques and their descendents were applied to a wide
variety of problems, from the scheduling of production facilities, to yield
management in airlines. Today, optimization comprises a wide variety of
techniques from Operations Research, artificial intelligence and computer science,
and is used to improve business processes in practically all industries.
Discrete optimization problems arise, when the variables occurring in the
optimization function can take only a finite number of discrete values. Discrete
optimization aims at taking these decisions such that a given function is maximized
(for example revenue) or minimized (for example cost), subject to constraints,
which express regulations or rules,
Perhaps surprisingly, discrete optimization is more difficult than its "continuous"
counterpart, where variables are allowed to take fractional values or even "real
numbers". Linear programming has been applied to discrete optimization using so-
called "branch-and-bound" techniques, for example to solve facility location
problems.
1. Types of optimisation
Unconstrained Optimization: Optimizing Single-Variable Functions, conditions
for Local Minimum and Maximum, Optimizing Multi-Variable Functions.
Constrained Optimization: Optimizing Multivariable Functions with Equality
Constraint: Direct Search Method, Lagrange Multipliers Method, Constrained
Multivariable Optimization with inequality constrained: Kuhn-Tucker Necessary
conditions, Kuhn –Tucker Sufficient Condition.

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2. Types of Optimization Algorithms
o MMaatthheemmaattiiccaall AAllggoorriitthhmmss
 SSiimmpplleexx ((LLPP)),, BBFFGGSS ((NNLLPP)),, BB&&BB ((DDPP))
o MMeettaa--HHeeuurriissttiicc AAllggoorriitthhmmss
 GGAA,, SSAA,, TTSS,, AACCOO,, PPSSOO,, TTLLBBOO…………....
These algorithms are based on various nature inspired phenomenon as
follows-
Genetic algorithms - Survival of the genetically fittest
Particle swarm - Flock migration
Ant colony - Shortest path to food source
Shuffled frog leaping- Group search of frogs for food
TLBO -Influence of teacher on learners

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2. INTRODUCTION TO TLBO
Teaching-Learning based Optimization (TLBO) algorithm is a global
optimization method originally developed by Rao et al. (Rao et al. 2011a; Rao et
al. 2012; Rao & Savsani 2012a). It is a population- based iterative learning
algorithm that exhibits some common characteristics with other evolutionary
computation (EC) algorithms (Fogel 1995). However, TLBO searches for an
optimum through each learner trying to achieve the experience of the teacher,
which is treated as the most learned person in the society, thereby obtaining the
optimum results, rather than through learners undergoing genetic operations like
selection, crossover, and mutation (Shi & Eberhart 1998). Due to its simple
concept and high efficiency, TLBO has become a very attractive optimization
technique and has been successfully applied to many real world problems (Rao et
al. 2011a; Rao et al. 2012; Rao & Savsani 2012a), (Rao et al. 2011b; Rao & Patel
2012; Rao & Savsani 2012b; Vedat 2012; Rao & Kalyankar 2012; Suresh Chandra
& Anima 2011
The main motivation to develop a nature-based algorithm is its capacity
to solve different optimization problems effectively and efficiently. It is assumed
that the behavior of nature is always optimum in its performance. In this paper a
new optimization method, Teaching–Learning-Based Optimization (TLBO), is
proposed to obtain global solutions for continuous non-linear functions with less
computational effort and high consistency. The TLBO method is based on the
effect of the influence of a teacher on the output of learners in a class. Here, output
is considered in terms of results or grades.
A group of learners constitute the population in TLBO. In any
optimization algorithms there are numbers of different design variables. The
different design variables in TLBO are analogous to different subjects offered to
learners and the learners’ result is analogous to the ‘fitness’, as in other population-
based optimization techniques. As the teacher is considered the most learned
person in the society, the best solution so far is analogous to Teacher in TLBO.
The process of TLBO is divided into two parts. The first part consists of the
“Teacher phase” and the second part consists of the “Learner phase”. The “Teacher

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phase” means learning from the teacher and the “Learner phase” means learning
through the interaction between learners.
Initialization
Following are the notations used for describing the TLBO
N: number of learners in class i.e. “class size”
D: number of courses offered to the learners
MAXIT: maximum number of allowable iterations
2.1 CLUSTERING WITH TLBO
Satapathy and his collaborators in their works and demonstrated that TLBO
can be successfully applied to deal with the clustering. They investigated how to
use TLBO help k-means clustering and fuzzy c-means clustering to find the better
cluster-centers. The TLBO approach was compared against classical K-means
clustering and PSO clustering. From the simulation results it is observed that
TLBO may have a slow convergence but it has stable convergence trend much
earlier compared to other two algorithms and better clustering results. TLBO
algorithm was used to overcome cluster centers initialization problem in fuzzy c-
means clustering, which is very important in data clustering since the incorrect
initialization of cluster centers will lead to a faulty clustering process. The
experimental results reflected that TLBO algorithm can work globally and locally
in the search space to find the appropriate cluster-centers.

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3. MULTI-OBJECTIVE OPTIMIZATION WITH TLBO
Multi-objective optimization in automatic voltage regulator, power flow problem,
heat exchanger, and thermoelectric cooler [20] can also be solved with TLBO. The
authors gave the comprehensive and systematic discussions regarding how to use
TLBO to optimize the practical applications. Niknam et al. paper proposed a new
multi-objective optimization algorithm based on modified teaching-learning-based
optimization (MTLBO)algorithm in order to solve the optimal location of
automatic voltage regulators (AVRs) in distribution systems at presence of
distributed generators (DGs). Nayak et al. in[18] presented a non-domination based
sorting multi-objective teaching-learning-based optimization algorithm, for solving
the optimal power flow (OPF) problem which is a non linear constrained multi-
objective optimization problem where the fuel cost, Transmission losses and L-
index are to be minimized. Rao et al. used a modified version of the TLBO
algorithm to solve the multi-objective optimization of heat exchangers.
Maximization of heat exchanger effectiveness and minimization of total cost of the
exchanger are considered as the objective functions. Meanwhile, Rao et al. in [20]
also proposed a modified version of the TLBO algorithm which is introduced and
applied for the multi-objective optimization of a two stage thermoelectric cooler
(TEC). Maximization of cooling capacity and coefficient of performance of the
thermoelectric cooler are considered as the objective functions.

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4. APPLICATIONS OF TLBO
A. PARAMETER OPTIMISATION OF MODERN MACHINING
PROCESSES USING TEACHING LEARNING BASED
OPTIMISATION ALGORITHM
Modern machining processes are now-a-days widely used by
manufacturing industries in order to produce high quality precise and very
complex products. These modern machining processes involve large number of
input parameters which may affect the cost and quality of the products. Selection
of optimum machining parameters in such processes is very important to satisfy all
the conflicting objectives of the process. A newly developed advanced algorithm
named ‘teaching–learning-based optimization (TLBO) algorithm’ is applied for the
process parameter optimization of selected modern machining processes. The
important modern machining processes identified for the process parameters
optimization in this work are ultrasonic machining (USM), abrasive jet machining
(AJM), and wire electrical discharge machining (WEDM) process. The examples
considered for these processes were attempted previously by various researchers
using different optimization techniques such as genetic algorithm (GA), simulated
annealing (SA), artificial bee colony algorithm (ABC), particle swarm
optimization (PSO), harmony search (HS), shuffled frog leaping (SFL) etc.
However, comparison between the results obtained by the proposed algorithm and
those obtained by different optimization algorithms shows the better performance
of the proposed algorithm.

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a. Ultrasonic machining
Ultrasonic machining process (USM) is one of the widely used modern machining
processes in various industries. In USM process the material is removed due to the
action of abrasive grains. The abrasive particles are forced on the work surface by
a tool oscillating normal to the work surface at a high frequency. The tool is
shaped as the approximate mirror image of the configuration of cavity desired on
the workpiece. The various input parameters involved in USM process are
amplitude of tool oscillation, type of abrasive, grain size of the abrasives, feed
force, volume concentration of abrasive in water slurry, etc., which affect various
performance measures of the process such as material removal rate and surface
roughness.
Jain et al. (2007) used genetic algorithm to optimize the process
parameters of USM process. Singh and Khamba (2007) proposed an approach for
macro modelling of the material removal rate, tool wear rate, and surface
roughness during ultrasonic machining of titanium and its alloys and obtained the
relationship between these output parameters of USM with other controllable
machining parameters. Kumar and Khamba (2009) showed the effectiveness of the
ultrasonic machining of Stellite 6 in terms of tool wear rate and the material
removal rate of work piece and determined the optimum combination of various
input factors by applying the Taguchi's multi-objective optimization
technique. Rao et al. (2010a) attempted the parameter optimization of USM
process using ABC, HS and PSO algorithms and an example was presented. In
another work, Rao et al. (2010b) presented the application of simulated annealing
(SA) to the USM process.

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b. Wire electrical discharge machining
The spark theory of a wire electrical discharge machining (WEDM)
process is basically the same as that of the vertical electrical discharge machining
(EDM) process. In WEDM, the conductive materials are machined with a series of
electrical discharges (sparks) that are produced between an accurately positioned
moving wire and the workpiece. High frequency pulses of alternating or direct
current is discharged from the wire to the workpiece with a very small spark gap
through an insulated dielectric fluid (water). The wire does not touch the
workpiece, so there is no physical pressure imparted on the workpiece compared to
grinding wheels or any other cutting tools used in other conventional machining
processes. The WEDM provides more flexibility in designing the dies and more
control of manufacturing as the process is completely automatic. The WEDM
process is controlled by large number of input parameters such as pulse-on time,
pulse-off time, table feed rate, flushing pressure, wire tension, wire speed, pulse
frequency, average gap voltage, discharge current, dielectric flow rate, etc.
It is observed that comparatively less work was carried out for the
parameter optimization of these modern machining processes. Few traditional
optimization techniques such as goal programming, feasible direction method, etc.,
had been reported to solve the problems of optimization of some of these
processes, but subsequently it was proved that the results obtained by these
traditional techniques are not the optimum and also these techniques are very
complex in nature and cannot handle multi-objective problems effectively. Hence,
recently developed new optimization technique named as teaching–learning-based
optimization (TLBO) proposed by Rao et al., 2011 and Rao et al., 2012 and Rao
and Patel (2012) is used here which does not require any algorithm-specific
parameter setting.
To check for any improvement in the results, this algorithm is considered here for
the parameters optimization of USM, AJM and WEDM processes.

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B.MULTI OBJECTIVE OPTIMISATION OF HEAT EXCHANGERS
USING MODIFIED TEACHING LEARNING BASED OPTIMISATION
ALGORITHM
In the present work, a modified version of the TLBO algorithm
is introduced and applied for the multi-objective optimization of heat exchangers.
Plate-fin heat exchanger and shell and tube heat exchanger are considered for the
optimization. Maximization of heat exchanger effectiveness and minimization of
total cost of the exchanger are considered as the objective functions. Two
examples are presented to demonstrate the effectiveness and accuracy of the
proposed algorithm. The results of optimization using the modified TLBO are
validated by comparing with those obtained by using the genetic algorithm (GA).
Figures -
Fig. 1. Plate-fin heat exchanger and rectangular offset strip fin
Fig. 2. Shell and tube heat exchanger geometry

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C. MULTI OBJECTIVE OPTIMISATION OF TWO STAGE
THERMOELECTRIC COOLER USING A MODIFIED TEACHING
LEARNING BASED OPTIMISATION ALGORITHM
In the present work, a modified version of the TLBO algorithm is
introduced and applied for the multi-objective optimization of a two stage
thermoelectric cooler (TEC). Two different arrangements of the thermoelectric
cooler are considered for the optimization. Maximization of cooling capacity and
coefficient of performance of the thermoelectric cooler are considered as the
objective functions. The results of optimization obtained by using the modified
TLBO are validated by comparing with those obtained by using the basic TLBO,
genetic algorithm (GA), particle swarm optimization (PSO) and artificial bee
colony (ABC) algorithms.
Fig. 3.. Two stage TEC. (a) Electrically separated and (b) electrically
connected in series.
Fig. 4. Effect of number of teachers on the convergence rate of the
modified TLBO algorithm for multi-objective consideration (electrically
separated TEC).

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D. DESIGN OF PLANER STEEL FRAMES USING TEACHING
LEARNING BASED OPTIMISATION ALGORITHM
This report presents a design procedure employing a
Teaching–Learning Based Optimization (TLBO) technique for discrete
optimization of planar steel frames.
The design algorithm aims to obtain minimum weight frames
subjected to strength and displacement requirements imposed by the American
Institute for Steel Construction (AISC) Load and Resistance Factor Design
(LRFD). Designs are obtained selecting appropriate W-shaped sections from a
standard set of steel sections specified by the AISC. Several frame examples from
the literature are examined to verify the suitability of the design procedure and to
demonstrate the effectiveness and robustness of the TLBO creating of an optimal
design for frame structures. The results of the TLBO are compared to those of the
genetic algorithm (GA), the ant colony optimization (ACO), the harmony search
(HS) and the improved ant colony optimization (IACO) and they shows that TLBO
is a powerful search and applicable optimization method for the problem of
engineering design applications.
Design examples-
Fig.5.Two-bay three-story frame design

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E. A DESIGN OF IIR BASD DIGITAL HEARING AIDS USING
TEACHING LEARNING BASED OPTIMISATION
This report describes a design of digital hearing aids. This report shows that
design of Infinite Impulse Response (IIR) digital filter with Teaching-learning-
based-optimization (TLBO) gives good result in digital hearing aids. TLBO
algorithm is used to optimize the filter coefficient of IIR filter. The error between
desired magnitude response and actual magnitude response will be minimized by
this algorithm. Three audiograms have been used to verify that high accuracy can
be achieved with direct IIR filter. The results of our study indicate that proposed
design is much simpler and able to maximize hearing.
Problem Formulation
It is always desirable to design a filter bank structure that is
simple, flexible and one which has least matching errors. Since the hearing loss is
compensated through the sub bands gains of the uniform or non uniform filter
banks and high accuracy depends on the fitting the increasing number of frequency
bands as per the hearing loss pattern [audiogram] which differs from person to
person. The main issue is built a filter that utilizes multiple frequency bands in
such a manner that are duced number filter sub bands or sub-filters do not degrade
the fitting flexibility and do not cause matching errors. The improper tuning of
filter specific parameters either increases computational effort or yields the local
optimal solution specific to a particular sub band. Therefore, the error between
desired magnitude response and actual magnitude response must be minimized by
some optimization method that requires few controlling parameters like population
size and number of generations, number of learners etc to yield optimal solution.
Error = max [|Hd(fi)-Ha(fi) |, fi ∈ F] (10)
where F= {F1,F2,F3,F4,F5,F6}, is an array offer quencies corresponding which a
hearing level is tested at which the subject is asked to hear a tone and points are
identified where the person can hear with loss or normally Ha(fi), Hd(fi) denotes
the actual magnitude response and desired magnitude response respectively. The
error term is minimized using Teaching-learning algorithm. In this study we

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assume that audiograms are measured at frequencies – 250Hz, 500Hz, 1kHz,
2kHz, 4kHz,8kHz.
IIR Digital Filter
A circuit which is designed to selectively filter one frequency or range of
frequencies out of a mix of different frequencies in a circuit is called a filter circuit
or simply a filter. Digital filters process digitized or sampled signals. Digital
hearing aids are preferred over analog hearing aid. In digital hearing aids almost
everything is digital from audio section to control circuitry. A digital filter
computes a quantized time-domain representation of the convolution of the
sampled input time function and a representation of the weighting function of the
filter. Poles and zeros are the roots of the denominator and numerator of the
transfer unction respectively .The transfer function of a digital filter H(z) is the
ratio of the z-transforms of the filter output and input given by:
H z = Yz ÷Xz
The digital filter transfer function is given by-
M N
H(z)= ( ∑ akz-k
) ÷ (1-∑ bkz-k
)
k=0 k=1
where k represents the order of filter.
The transfer function of the filter in Eq. is the ratio of two polynomials in the
variable z and maybe written in a cascade form as:
H( z) =H1 (z) H2 (z)
where H1(z) is the transfer function of a feed forward, all-zero, filter given by-
M
H1(z) = ∑ akz-k
k=0
and H2(z) is the transfer function of a feedback, all poles, recursive filter given
by:

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N
H2(z) = 1÷ (1-∑ bkz-k
)
k=1
Throughout this paper sixth order Yule walk recursive IIR filter is used as
prototype which is an all pole system and its transfer function is expressed as:
6
Hy(z) = 1÷ (1- ∑ bkz-k
)
k=1
That is numerator coefficient becomes 1.
Proposed Methodology
Step 1- Identify the extent and type of hearing loss levels on the basis of
audiogram.
Step 2- Define the limits to which amplification is to be carried out for the hearing
aid. This will depend upon the extent of hearing loss.
Step 3- Calculate the difference between actual value of amplitude and desired
value of amplitude.
Table 1: Objective function on the basis of amplitude difference
Audiogram Normal Hearing Level
(db)
Difference in Level
(db)
F1-Level at
250 Hz
-19 -19 = x-(c1)
F2-Level at 500 Hz -20
(F1>F2)
-20 = x-(c2)
F3-Level at 1000 Hz -21
(F2>F3)
-21=x-(c3)
F4-Level at 2000 Hz -22
(F4>F5)
-22=x-(c4)
F5-Level at 4000 Hz -24(
F5>F6)
-24=x-(c5)
F6-Level at 8000Hz -25 -25=x-(c6)

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Step 4- Run TLBO by taking two design variables(a, b) which is based on size of
population. Minimum and maximum value of the prototype filter coefficient
matrix.
Step 5- Stop optimization if stopping criteria met as optimal solution to the
problem is found i.e .optimized coefficients are found which gives the closest
matching response to normal hearing audiogram.
Step 6- On the basis of optimized solution obtained in step 5, error value is
calculated. As already stated error is the difference between actual amplification
attained through use of hearing aid and the desired amplification
Step 7- Plot the frequency matching result.
Step 8- Plot magnitude error.
Simulation results
In this section simulation of audiogram are investigated. Audiograms are
downloaded from the independent Hearing Aid Information, a public service by
Hearing Alliance of America . In this Yule walk filter is used as prototype.
The thresholds for the different types of hearing loss are as follows:
Normal hearing: 0-20dB
Mild hearing loss: 20-39dB
Moderate hearing loss: 40-59dB
Severe hearing loss: 60-89dB
Profound hearing loss: 90+ dB
Fig.7.shows Audiogram-1which has mild hearing loss at high frequency. The
maximum error is 0.5804 dB whereas that in [9] is 0.
Fig.7: Audiogram-1

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TLBO is used to optimize coefficient value of digital filter. It is apparent
from the above graphs that the TLBO bring the magnitude response close to the
desired hearing level and minimize the error. This is due to the fact that the
optimization algorithm is able to achieve the optimized results in terms of [a, b].
Results have shown that matching error between output values of hearing aid using
TLBO is smaller than method used in [9] that used Nelder-Mead algorithm.

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F.SIZE AND GEOMETRY OPTIMIZATION OF TRUSSES USING
TEACHING-LEARNING-BASED OPTIMIZATION
Structural optimal design has always been a concern for engineers in
practice. The focus is not only in construction cost, but also in geometry of
structures. It is responsible for engineers to design structures with high reliability
and low cost.
The suitability of TLBO for size and geometry optimization of structures
in structural optimal design was tested by truss examples. Meanwhile, these
examples were used as benchmark structures to explore the effectiveness and
robustness of TLBO. The results were compared with those of other algorithms. It
is found that TLBO has advantages over other optimal algorithms in convergence
rate and accuracy when the number of variables is the same. It is much desired for
TLBO to be applied to the tasks of optimal design of engineering structures.
In the problem of size and geometry optimization of truss structures, the
cross-sectional area and the geometry of primary structures both increase the
dimension of the design space. It has been proved that TLBO algorithm performs
well in problems with large dimensions .
Mathematical model for sizing and geometry optimisation of truss
Usually, there are two types of variables in the mathematical model for the
size and geometry optimization of the truss structures i.e. the cross-sectional area
variables and the node coordinate variables, which determine the geometry of the
structures. Compared with the truss size optimization problems which have been
extensively studied, the size and geometry optimization introduces node coordinate
variables. This not only makes the design space is of higher dimension, but also greatly
enhances the degree of nonlinearity, moreover, the optimization may lead to a local
optima. The mathematical model of size and geometry optimization problems can be
expressed as follows:

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n
Min. weight(Ai, Cj )= ∑ ρiAiLi
i=1
Li=Li(Cj)
s.t. gi
σ
=[σi] –σi>=0(i=1,2,….,k)
gu
jl= [ujl-ujl>=0(j=1,2,………M);(l=1,2……..N)
Ai є S (i=1,2,…..k)
where k is the total number of truss elements; M is the number of nodes; N is the
number of nodal freedoms; Ai , Li and ρirepresents the cross-sectional area, the
length and the density of the ith bar respectively; Cj represents the coordination of
jth node; gi
σ
and gjl
u
are the constraint violations for member stress (include
buckling stress) and joint displacements of the structure. σi is the stress of the ith
bar due to loading condition, [σi] is its allowable stress. Ujlis the nodal
displacement of the lth translational degree of the jth node, [ujl] is its allowable
joint displacements. S is a set of discrete cross-section of bars.
A 18-bar planar truss structure
The 18-bar planar truss is shown in Figure. The material density is 0.1 lb/in3,
and the modulus of elasticity is 10000 ksi. The stress limits of the members are subjected
to ± 20 ksi. Euler buckling stress constraints are , where buckling coefficient α=4. Node 1,
2, 4, 6 and 8 have -20 kips in y direction. Size variables are A1 = A4 = A8 = A12 = A16, A2 =
A6 = A10 = A14 = A18, A3 = A7 = A11 = A15, A5 = A9 = A13 = A17. The cross-sectional area
variables are set [2.00, 21.75] (in2) and the interval is 0.25 in2 .Side constraints for
geometry variables are -225 ≤ y3, y5, y7, y9 ≤ 245, 775 ≤ x3 ≤ 1225, 525 ≤ x5 ≤ 975, 275 ≤
x7 ≤ 725, 25 ≤ x9 ≤ 475 (in).

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Figure 8-. The geometry of the 18-bar planar truss
The optimal weight of 18-bar truss for different population under 30 independent
runs is shown in Table .
Table2: Results of sensitivity analysis of the 18-bar truss for 30 independent runs
PS No. of
structural
analyses
avg.d
No. of best
results for
structural
analyses
Best(lb) Mean(lb) Worst(lb) Std. dev
20 50021 4543.834 4672.787 5132.951 5135.951 116.962
30 50059 4532.538 4622.168 4815.306 4815.306 64.001
40 50029 4535.251 4590.072 4590.072 4750.639 53.406
50 50003 4526.708 4597.752 4597.752 4727.466 54.070
It is observed from table that strategy with population size of 50 and
number of iterations of 500 produced the best result than other strategies. The
standard deviation (std. dev) is relatively large as well, this indicates that
computation is easy to trap in local optimum. Similarly, the increase of the
population has little impact on the results when the number of structural analyses is
about the same. Good global search ability and weak local search ability are also
expressed. The best results of TLBO for population size50 were selected to
contrast with those obtained from other algorithms and were shown in table.
Figure 9 and Figure 10 are the convergence curves of TLBO and the optimized 18-
bar structure respectively.

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Table3: Optimal results of TLBO with ps size 50 for 18-bar truss
Variables Rajeev
[23]
Hasanqehi
[24]
Kaveh
[25]
GSO [22] TLBO
A1 12.5 12.5 13 12.25 12.5
A2 16.25 18.25 18.25 18.25 18
A3 8 5.5 5.5 4.75 5.25
A4 4 3.75 3 4.25 3.75
X3 891.9 933 913 916.9 914.524
Y3 145.3 188 182 191.971 188.793
X5 610.6 658 648 654.224 647.351
Y5 118.2 148 152 156.1 149.683
X7 385.4 422 417 423.5 416.831
Y7 72.5 100 103 102.571 101.332
X9 184.4 205 204 207.519 204.165
Y9 23.4 32 39 28.579 31.662
Weight
(lb)
4616.800 4574.280 4566.210 4538.768 4526.708

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Figure9 : Convergence Of TLBO with ps size 50
Figure10 : The optimized18-bar truss
TLBO has almost the same number of structural analyses with GSO. It is obvious from
table that the result of TLBO is the best. It is obvious that the TLBO requires less
computation effort to reach convergence and its convergence rate is faster than that of
GSO.

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6. Conclusions
The performance comparisons are done with TLBO and other evolutionary
computation techniques like particle swarm optimization (PSO), Differential evolution (DE),
artificial bee colony (ABC) and several of variants of these algorithms suggested by other
researchers. From the results analysis it is evident that TLBO outperforms all other approaches.
The efficiency of the proposed approach is compared with other algorithms in terms of number
of function evaluations (FEs). We can conclude by saying that TLBO is a very powerful
approach of optimizing different types of problems which are separable, non-separable,
unimodal and multimodal in providing quality optimum results in faster convergence time
compared to very popular evolutionary techniques like PSO, DE, ABC and its variants. This may
be used to multi-objective optimization problems and also some engineering applications from
mechanical, chemical or data mining may be investigated.
It does not require any algorithm-specific parameters, Only the common control
parameters are needed. Within the examples considered, the results of TLBO obtained are as
good as or better than that of other algorithms in terms of both convergence rate and convergence
accuracy. Thus TLBO proves to be a rising star from amongst a number of metaheuristics.

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