6th International Science, Social Sciences, Engineering and Energy Conference
17-19 December, 2014, Prajaktra Design Hotel, Udon Thani, Thailand
I-SEEC 2014
http//iseec2014.udru.ac.th
A teaching-learning-based optimization algorithm for sizing
optimization of trusses
Anan Nimtawat
Faculty of Technology, Udon Thani Rajabhat University, Thailand
anannimt@gmail.com
Abstract
In this study, a modified teaching learning based optimization algorithm is applied to sizing design of
truss structures. Non-linear constrained engineering problems, such as truss design problems, are
popularly solved by using meta-heuristic search algorithms such as Genetic algorithms (GA), Particle
swarm optimization (PSO), and teaching learning based optimization (TLBO). It is because the metaheuristic search algorithms only need to iteratively evaluate problem functions without requiring gradient
information of the problem functions. The TLBO has been more widely used in that it can give good
solution with smaller numbers of predetermined parameters than other algorithms. Originally the TLBO
consists of two searching steps: teacher phase and learner phase. Firstly, in the teacher phase, all
searching points in the defined population, learners in the defined class, are encouraged to move toward
the promoted teacher that is the best searching point in the iteration. Secondly, all searching points are
motivated to move by randomly interacting with other points in the population. A modified TLBO
algorithm is proposed and validated by applying to truss design problems. The results show that the
modified TLBO can find good solutions of the design examples.
Keywords:Optimization ; teaching learning based optimization ; truss design
1. Introduction
The optimization of trusses is one of the most active researches in engineering [1-5]. The truss
optimization problems can be classified into three groups including sizing optimization, shaping
optimization and topology optimization. The sizing optimization is to find the optimal size of all truss
elements without changing its original shape and configuration. The shaping optimization is to optimize
the truss design by allowing to change the element size and the truss shape but no configuration changes
otherwise it is called the topology optimization. The sizing optimization of trusses is a kind of non-linear
2
optimization problem that may be hard to find the gradient of the objective function. Although there exists
a number of recognized gradient-based method to solve non-linear function such as the steepest descent
method and the conjugate gradient method, it is not easy to explore possible solutions in a quite large
search space. A number of the population-based evolutionary algorithms such as Genetic algorithms (GA)
and Particle swarm optimization (PSO) are recently employed to solve large scaled constrained non-linear
problems, for example, the optimization of trusses [4, 5]. However the performance of the mentioned
algorithms greatly depend on required arbitrary parameters used in the algorithms. For example, the
parameters required in the GA include the probabilistic of crossover and the probabilistic of mutation [5].
The parameters required in the PSO include inertia weight, and social and cognitive parameters [4].
Recently a novel algorithm called the teaching learning based optimization (TLBO) proposed by Rao
et al. [1] is employed to solve non-linear problems, and truss optimization problems [1-3]. The TLBO
algorithm mimics the teaching behaviors between a teacher and learners, and the learning behaviors
among learners in the classroom. The philosophy of the TLBO is that the teacher tries to share her
knowledge to her learners in the class. The teacher will try to improve the mean of grade of the class. In
additions, each learners in the class will also share their knowledge among other learners in the class.
The required input parameters of TLBO are only the population size and the number of generations
that are the common parameter of the population-based evolutionary algorithms such as GA and PSO.
Although there is a teaching factor that is arbitrary required in the TLBO algorithm, it can be randomly
generated in the algorithm. Therefore it may be said that the TLBO is a parameter-less algorithm [1].
In this study, the original TLBO is modified for finding the global optimum solution of sizing
optimization of trusses. The proposed modified TLBO will be described in the next section.
2. Teaching-learning based optimization
The basic concept of the TLBO algorithm is that it attempts to move the searching point toward the
best current point or called the teacher, and attempts to move the searching point by getting some
information from the other point in the same population [1]. There is only one parameter required in the
TLBO algorithm that is the teaching factor that will be explained later.
Like a Genetic algorithm (GA), the TLBO finds the best point or solution by evolving initial solutions
to better solutions. The TLBO also uses a population of learners or a population of individuals to search
the best solution by a number of iterations or generations. Each learner consists of a vector of design
variables representing marks of subjects. Unlike a GA, the TLBO has no crossover or mutation operators.
The TLBO has two operators called teacher phase and learner phase.
The TLBO processes can be summarized here. First, the number of starting points (learners) is
uniformly randomly initialized including setting up the termination criterion such as the number of
iterations. A searching point is a vector of design variables ( x ). Then, the mean of each component
(design variables) of x is calculated. Next, the best point, the point with minimum value of the objective
function at the current iteration, is identified and promoted as the teacher of the class (population). The
teacher will try to move the current mean to its own point, so the new mean will be the current teacher.
Then the difference value between the current mean of each direction and the corresponding current best
point are calculated. This different value is called the difference mean. After that all points in the
population will be updated in two step called the teacher phase and the learner phase.
Let N is the number of x in the population. At the current iteration t , the mean of x t is the calculated
by Eq. (1) designated as x mean ,t . As mentioned above, the new mean of x t is the current best point or the
teacher designated as xteacher ,t . Then, the difference mean Δt can be calculated using Eq. (2).
3
1 N
 x k ,t
N k 1
Δt  rt (xteacher ,t  TF x mean ,t )
xmean,t 
(1)
(2)
Here, Δt is the difference mean, rt is the random number in the range [0, 1], and TF is the teaching
factor that can be either 1 or 2.
In the teacher phase, all searching points (learners) are encouraged to move toward the best point
(teacher) of the current population. All points x old ,t are moved to the new positions x new,t using Eq. (3).
However, the point will move to the new position only when the new position is better than its current
position.
x new,t  xold ,t  Δt
(3)
In the learner phase, the point is moved by randomly interacting with others. For the minimization
problem, the point will be moved using Eq. (4). However, the point will be moved to the new position
only when the new position is better than its current position.
x A, new,t  x A,old ,t  rt (x A,old ,t  x B ,t )
if f (x A,t )  f (x B ,t )
x A, new,t  x A,old ,t  rt (x B ,t  x A,old ,t )
if f (x A,t )  f (x B ,t )
(4)
Here f ( x A,t ) and f ( x B ,t ) is the objective function value of the x A , t and x B , t respectively.
All points are iteratively moved until the termination criterion is reached, and the best point is
identified as the global optimum solution.
3. The formulation of the truss optimization problem
The truss optimization problem is to find the minimum weight of truss structures while the
configuration of the truss structure is fixed. The optimization problem can be written as:
find
x   A1 , A2 , A3 ,
minimize
W (x)    i Li Ai
, An 
m
(5)
i 1
subjected to
 L  i   U
 L   j  U
AL  Ai  AU
where W is the total weight of the design truss, m is the total number of element,  i is the material
unit weight of element i , Li is the length of element i , and Ai is the cross sectional area of element i .
The design truss must satisfy all three constraints: the element stress constraint  i , the nodal deflection
constraint  j , and the cross sectional element area constraint Ai . The cross sectional area constraints can
be satisfied just by limiting the search space of cross-sectional areas of elements.
4
The function for calculating the total weight of the truss design as shown in Eq. (5) is defined as the
objective function of the truss optimization problem. However, the truss design with very small weight
may violate any required constraint that is defined as an infeasible truss design. To handle the infeasible
design, a penalty function is multiplied to the objective function W to insure that the feasible design will
have the smaller weight than the infeasible design. Here, the penalized objective function W p is defined as
Eq. (6).
Wp  WE
(6)
Here, E is the total degree of constraint violation that is composed of the total degree of stress
constraint violation E , and the total degree of displacement constraint violation E of the truss design.
The total degree of constraint violation E is defined as Eq. (7).
E  1  E  E
(7)
The total degree of stress constraint violation E of the truss design is the summation of the degree of
stress constraint violation, and can be written as Eq. (8).
m
E   e ,i
(8)
i 1
Here e ,i is the degree of element stress constraint violation of element i , and m is the number of
elements of the truss design.
The degree of stress constraint violation e ,i of each element is defined as Eq. (9).
if  L   i   U otherwise
e ,i  0
e ,i 
i 
a
(9)
a
Here  i is the element stress of element i , and  a is the lower bound  L or upper bound  U of the
allowable element stress of element i .
The total degree of displacement constraint violation E of the truss design is the summation of the
degree of displacement constraint violation, and can be written as Eq. (10).
n
E   e , j
j 1
(10)
Here, e , j is the degree of nodal displacement constraint violation in each direction of node j , and n is
the number of nodes of the truss design. If the planar truss design is considered, two directions x and y are
considered.
The degree of deflection constraint violation e , j of each node is defined as Eq. (11).
5
if  L   i   U otherwise
e , j  0
e , j 
(11)
 j  a
a
Here  i is the nodal displacement in each direction of node j , and  a is the lower bound  L or upper
bound  U of the allowable nodal displacement of node j .
4. A modified TLBO algorithm for sizing optimization of trusses
A modified TLBO algorithm is proposed in this study for sizing optimization of planar trusses. A
searching point or a design solution is a vector of cross-sectional area of a truss design. The best solution
is the vector of cross-sectional area that configures the optimal (minimum weight) truss design. The
configuration of the truss is not varied, but only the cross-sectional area of each element of the truss is set
as a design variable.
4.1. Initialization
Let a solution x is a vector of cross-sectional areas. The population of x is randomly generated
according to the number of population size N . All truss structures generated using cross-sectional areas
defined by x are analyzed by using the finite element method. The details of the finite element method for
truss structures can be found elsewhere [6]. Then the result of the penalized objective function W p are
calculated using Eq. (6).
4.2. Teacher phase
At the iteration t , the current best design xteacher ,t is identified, and the current mean x mean ,t is calculated
using Eq. (1), and the difference mean Δt is calculated using Eq. (2). The adaptive teaching factor TF as
shown in Eq. (12) is employed. It is noted TF is normally large in the early step of iterations and will
decreasingly approach to 1 in the later step of iterations.
TF ,t 
f (x worstfeas ,t )
f (xteacher ,t )
(12)
Here f (x worstfeas ,t ) and f (xteacher ,t ) is the result of penalized objective function W p according to
x worstfeas ,t and xteacher ,t respectively when x worstfeas ,t is the worst feasible design of the current population,
and xteacher ,t is the current best design in the tth iteration. If the f (xteacher ,t ) is zero, TF is set to 1.
The sign of the difference mean Δt is modified before calculating x new,t using Eq. (3). The concept is
the x new,t will always move toward the xteacher ,t . The x new,t will be accepted only when the x new,t is better
than the x old ,t . The rules for identifying that which x k ,t is better is proposed as followed:
 if x A , t is feasible and x B , t is infeasible, x A , t is better than x B , t
6
 if both are feasible, x A , t is better than x B , t when x A , t has smaller objective value than x B , t
 if both are infeasible, x A , t is better than x B , t when x A , t has smaller penalty value than x B , t .
It is noted that an infeasible x k ,t representing a truss design that has some constraint violations.
4.3. Learner phase
After the teacher phase, all solutions x k ,t in the population will be updated again using Eq. (4). At first,
a solution x A , t will be randomly mated to x B , t . Then x A, new,t is calculated using Eq. (4). The Eq. (4) is also
modified in such a way that the x A, mew,t will move toward the x B , t only when x B , t is better
than x A, old ,t otherwise x A, mew,t will move away x B , t . In addition, similar to the teacher phase, the new
solution x A, mew,t will be accepted only when it is better than the old one x A, old ,t by employing the proposed
comparison rules mentioned above.
4.4. Termination
After the process of the learner phase is applied, the algorithm will be getting in the teacher phase
again, and the learner phase will be applied again. The algorithm will be operated iteratively until the
termination criterion is accomplished. In this study, the termination criterion is simply set as the required
number of iterations that is given as the required input of the algorithm.
5. Design problem
The proposed modified TLBO algorithm is implemented and is verified by testing a problem of sizing
optimization of the planar ten-bar cantilever truss. The test results are compared with results in literatures
that also used the TLBO as an optimizer.
The truss configured as shown in Fig. (1) is one of the benchmark problem in sizing optimization of
trusses [2, 3]. The range of the cross-sectional area used is 0.1 in 2 to 35.0 in 2 . For the two types of
constraint, the stress for each truss element is limited to 25 ksi , and the displacement in horizontal and
vertical directions of all nodes of the truss design is limited to 2 in. The unit weight of the material is set
as 0.1 lb/in3 , and the modulus of elasticity is set as 107 psi.
(5)
[1]
[8]
360 in
(3)
[7]
[10]
[2]
(1)
[9]
[5]
[6]
360 in
360 in
y
x [3]
(6)
[4]
(4)
100 kips
(2)
100 kips
Fig. 1. The configuration of the planar ten-bar truss.
7
The number of populations used is 20, and the required number of iterations is 2,000. The number of
runs is 30. The results are shown in Table 1. The minimum weight obtained is 5061.1295 lb without any
constraint violation. However, it is noted that the results from Degertekin SO and Hayalioglu MS [2] has
some degree of displacement constraint violation 0.000051, and the result from Camp CV [3] has some
degree of constraint violation 0.000035.
Table 1. Results after 30 runs of the example
Elements No.
This study
Degertekin and
Hayalioglu [2]
Camp [3]
1
30.53329
30.4286
30.6684
2
0.10000
0.1000
0.1000
3
23.15025
23.2436
23.1584
4
15.27638
15.3677
15.2226
5
0.10002
0.1000
0.1000
6
0.55563
0.5751
0.5421
7
21.04019
20.9665
21.0255
8
7.46202
7.4404
7.4654
9
0.10001
0.1000
0.1000
10
21.51140
21.5330
21.4660
Minimum weight (lb)
5061.1295
5060.96
5060.973
Average weight (lb)
5061.3946
5062.08
5064.808
Standard Deviation (lb)
0.2236
0.79
6.3707
Average number of structure analyses
37,600
16,872
13,767
6. Conclusions and future researches
The teaching-learning-based optimization (TLBO) mimics the behavior of learners in a classroom. The
learners or the searching points move toward the best learner of the population, and move toward the
other better learner. This study found that the adaptive teaching factor is recommended. The modified
TLBO moves searching points quite fast in the early iterations and then gradually move searching points
finely to explore the better point in a quite small step. A topology optimization of trusses using TLBO is
one of the interesting future researches.
References
[1] Rao RV, Savasani VJ, Vakharia DP. Teaching-learning-based optimization: a novel for comstrained mechanical design
optimization problems. Computer-Aided Design 2011;43:303-315.
[2] Degertekin SO, Hayalioglu MS. Sizing truss structures using teaching-learning-based optimization. Computers and
Structures 2013;119:177-88.
[3] Camp CV, Farshchin M. Design of space trusses using modified teaching-learning based optimization. Engineering
Structures 2014;62-63:87-97.
[4] Schutte JF, Groenwold AA. Sizing design of truss structures using particle swarms. Struct Multidisc Optim 2003;25:261-9.
[5] Camp CV, Pezeshk S, Cao G. Optimized design of two-dimentional structures using a genetic algorithm. J Struct Eng
1998;124(5):551-9.
[6] Chandrupatla TR, Belegundu AD. Introduction to Finite Elements in Engineering. 3rd ed. New Jersey: Prentice Hall; 2002.