TEACHING AND LEARNING BASED OPTIMISATION

TEACHING AND LEARNING

BASED OPTIMISATION

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).

http://www.sciencedirect.com/science/article/pii/S095219761200053X#gr1

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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

http://www.sciencedirect.com/science/article/pii/S0141029611003634


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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)


(F2>F3)

-21=x-(c3)


(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. giσ=[σi] –σi>=0(i=1,2,….,k)

gujl= [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; giσand gjl

uare 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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Engineering

TEACHING AND LEARNING BASED OPTIMISATION