Page 58 - Fister jr., Iztok, and Andrej Brodnik (eds.). StuCoSReC. Proceedings of the 2016 3rd Student Computer Science Research Conference. Koper: University of Primorska Press, 2016
P. 58
Each cuckoo lays one egg, and dumps it in a randomly 4. CONSTRAINED DESIGN OPTIMIZATION
chosen nest, PROBLEMS
• Nests with high-quality egg, will be carried over to the The following design optimization problems have been used
next generations, in this study: welding beam, pressure vessel design, spring
design, and speed reducer design. The used problems are
• Any egg laid by a cuckoo, may be discovered by the thoroughly presented and formally defined in the remainder
host bird with a probability of pa ∈ (0, 1). When an of this section.
egg is discovered, the host bird may get rid of it or
simply abandon the nest and build a new one. 4.1 Welding beam
Each solution in the population of the cuckoo search algo- The goal of this problem is to design a welded beam subject
rithm corresponding to a cuckoo nest, represents the posi- to minimum cost, subject to some constraints. The prob-
tion of the egg in the search space. This position can be lem consists of four design variables, with the objective is
mathematically defined: to find the minimum fabrication cost, with constraints of
shear stress τ , bending stress σ, buckling load Pc, and end
xi = {xi,j}, f or i = 1, . . . , N p and j = 1, . . . , D, (1) deflection on the beam δ. The mathematical model can be
formulated as follows:
where N p represents the population size, and D the dimen-
sion of the problem to be solved.
Generating new solutions in the CS is done by executing a f (x) = 1.10471x12x2 + 0.04811 ∗ x3x4(14 + x2), (3)
random walk, with the use of the Levy flight distribution: subject to:
xi = xi + αL(s, λ). (2) g0 : τ − 13600 ≤ 0, g1 : σ − 30000 ≤ 0, g2 : x1 − x4 ≤ 0,
The term L(s, λ) determines the characteristic scale, and g3 : 0.10471x21+(0.04811x3x4(14+x2))−5 ≤ 0, g4 : 0.125−x1 ≤ 0,
α > 0 denotes the scaling factor of the step size s.
g5 : δ − 0.25 ≤ 0, g6 : 6000 − Pc ≤ 0, (4)
3.1 Hybrid self-adaptive Cuckoo Search where
According to [8] the CS was modified by adding the following τ= τ12 + 2τ1τ2 x2 + τ22, τ1 = √6000 , τ2 = MR,
mechanisms: balancing of the exploration strategies within 2R 2x1x2 J
the CS, self-adaptation of the parameters, and population
reduction. The used exploration employed by the HSA-CS M = 6000(14 + x2 ), R = x22 + ( x1 + x3 )2,
are: 2 42
J = √ x22 ) + ( x1 + x3 )2), σ= 504 ∗ 10e3
2( 2x1x2( 12 2 x4x23 ,
• random long distance exploration,
• stochastic short-distance exploration, and 65856 ∗ 10e3
• stochastic moderate-distance exploration. δ = 3 ∗ 10e6x4x33 ,
(12.039 ∗ 10e6 (x23x46)/36) (1 − x3 3∗10e6
196
Pc = 48∗10e6 ) (5)
28.0
The listed strategies have an impact on how the trial solution
will be generated. The random long distance exploration is The design variables are bounded as: 0.1 ≤ x2, x3 ≤ 10, and
implemented as the abandon operator. The second strategy 0.1 ≤ x1, x4 ≤ 2.
improves the current solution by using a local random walk,
with the help of Levy flights (Eq. 2). The last strategy 4.2 Pressure vessel design
is borrowed from the DE algorithm. Additionally the last
strategy adds a crossover operation to the CS algorithm. The idea of this problem is designing a compressed air stor-
These execution of these strategies is controlled by a single age design, with a working pressure of 1000 psi and and
parameter. minimum volume of 750 f t3. The problem is described us-
ing four variables, which represent shell thickness, spherical
As was stated all parameters are fully self-adaptive, except head thickness, radius and length of the shell. The objective
the starting population size, which must be experimentally of the problem is minimizing the manfuacturing cost of the
defined. Additionally the strategy balancing probability, the pressure vessel, and can be formulated as:
abandon rate, and the elitist parameter (controls whether a
random of best solution is taken as the basis trial vector cal- f (x) = 0.6224x1x3x4 +1.7781x2x32 +3.1661x21x4 +19.84x12x3,
culation) are determined by the user. Lastly the population (6)
reduction is implemented by using a simple linear reduction.
subject to
It was proven by the authors of the HSA-CS, that the biggest
impact on the results has the inclusion of multiple strategies, g0 : −x1 + 0.0193x3 ≤ 0, g1 : −x2 + 0.00954x3 ≤ 0,
than followed by self-adaptation. Population reduction did
not have a big impact on the results. For more information g2 : −πx23x4 − 4 πx33 + 1296000 ≤ 0, g3 : x4 − 240 ≤ 0.
about HSA-CS readers are referred to [8]. 3
(7)
StuCoSReC Proceedings of the 2016 3rd Student Computer Science Research Conference 58
Ljubljana, Slovenia, 12 October
chosen nest, PROBLEMS
• Nests with high-quality egg, will be carried over to the The following design optimization problems have been used
next generations, in this study: welding beam, pressure vessel design, spring
design, and speed reducer design. The used problems are
• Any egg laid by a cuckoo, may be discovered by the thoroughly presented and formally defined in the remainder
host bird with a probability of pa ∈ (0, 1). When an of this section.
egg is discovered, the host bird may get rid of it or
simply abandon the nest and build a new one. 4.1 Welding beam
Each solution in the population of the cuckoo search algo- The goal of this problem is to design a welded beam subject
rithm corresponding to a cuckoo nest, represents the posi- to minimum cost, subject to some constraints. The prob-
tion of the egg in the search space. This position can be lem consists of four design variables, with the objective is
mathematically defined: to find the minimum fabrication cost, with constraints of
shear stress τ , bending stress σ, buckling load Pc, and end
xi = {xi,j}, f or i = 1, . . . , N p and j = 1, . . . , D, (1) deflection on the beam δ. The mathematical model can be
formulated as follows:
where N p represents the population size, and D the dimen-
sion of the problem to be solved.
Generating new solutions in the CS is done by executing a f (x) = 1.10471x12x2 + 0.04811 ∗ x3x4(14 + x2), (3)
random walk, with the use of the Levy flight distribution: subject to:
xi = xi + αL(s, λ). (2) g0 : τ − 13600 ≤ 0, g1 : σ − 30000 ≤ 0, g2 : x1 − x4 ≤ 0,
The term L(s, λ) determines the characteristic scale, and g3 : 0.10471x21+(0.04811x3x4(14+x2))−5 ≤ 0, g4 : 0.125−x1 ≤ 0,
α > 0 denotes the scaling factor of the step size s.
g5 : δ − 0.25 ≤ 0, g6 : 6000 − Pc ≤ 0, (4)
3.1 Hybrid self-adaptive Cuckoo Search where
According to [8] the CS was modified by adding the following τ= τ12 + 2τ1τ2 x2 + τ22, τ1 = √6000 , τ2 = MR,
mechanisms: balancing of the exploration strategies within 2R 2x1x2 J
the CS, self-adaptation of the parameters, and population
reduction. The used exploration employed by the HSA-CS M = 6000(14 + x2 ), R = x22 + ( x1 + x3 )2,
are: 2 42
J = √ x22 ) + ( x1 + x3 )2), σ= 504 ∗ 10e3
2( 2x1x2( 12 2 x4x23 ,
• random long distance exploration,
• stochastic short-distance exploration, and 65856 ∗ 10e3
• stochastic moderate-distance exploration. δ = 3 ∗ 10e6x4x33 ,
(12.039 ∗ 10e6 (x23x46)/36) (1 − x3 3∗10e6
196
Pc = 48∗10e6 ) (5)
28.0
The listed strategies have an impact on how the trial solution
will be generated. The random long distance exploration is The design variables are bounded as: 0.1 ≤ x2, x3 ≤ 10, and
implemented as the abandon operator. The second strategy 0.1 ≤ x1, x4 ≤ 2.
improves the current solution by using a local random walk,
with the help of Levy flights (Eq. 2). The last strategy 4.2 Pressure vessel design
is borrowed from the DE algorithm. Additionally the last
strategy adds a crossover operation to the CS algorithm. The idea of this problem is designing a compressed air stor-
These execution of these strategies is controlled by a single age design, with a working pressure of 1000 psi and and
parameter. minimum volume of 750 f t3. The problem is described us-
ing four variables, which represent shell thickness, spherical
As was stated all parameters are fully self-adaptive, except head thickness, radius and length of the shell. The objective
the starting population size, which must be experimentally of the problem is minimizing the manfuacturing cost of the
defined. Additionally the strategy balancing probability, the pressure vessel, and can be formulated as:
abandon rate, and the elitist parameter (controls whether a
random of best solution is taken as the basis trial vector cal- f (x) = 0.6224x1x3x4 +1.7781x2x32 +3.1661x21x4 +19.84x12x3,
culation) are determined by the user. Lastly the population (6)
reduction is implemented by using a simple linear reduction.
subject to
It was proven by the authors of the HSA-CS, that the biggest
impact on the results has the inclusion of multiple strategies, g0 : −x1 + 0.0193x3 ≤ 0, g1 : −x2 + 0.00954x3 ≤ 0,
than followed by self-adaptation. Population reduction did
not have a big impact on the results. For more information g2 : −πx23x4 − 4 πx33 + 1296000 ≤ 0, g3 : x4 − 240 ≤ 0.
about HSA-CS readers are referred to [8]. 3
(7)
StuCoSReC Proceedings of the 2016 3rd Student Computer Science Research Conference 58
Ljubljana, Slovenia, 12 October
