Swarm Optimization Algorithm Inspired in the Behavior of the Social-Spider for Microwave Filters
Optimization
Chahrazad Erredir, Mohamed Lahdi Riabi, Emir Bouarroudj, Halima Ammari Department of Electronic, University Constantine1
Laboratory of electromagnetic and telecommunication Constantine, Algeria
cerredir@yahoo.fr
Abstract— In this paper, a recent swarm optimization algorithm on the simulation of the cooperative behavior of social- spiders, called the Social Spider Optimization (SSO) is proposed for the optimization of microwave filters. In order to illustrate the proficiency and robustness of this optimization, it is compared to other literature of evolutionary algorithms, particle swarm optimization (PSO) and teaching–learning-based optimization (TLBO). The outcome shows a high performance of the proposed method.
Keywords—optimization ; microwave filter; social spider optimization
I. INTRODUCTION
Nature-Inspired algorithms of the swarm intelligence field perform powerfully and efficiently in solving global optimization problems. Recent research studies in optimization field [1] [2] have led to development of new approaches that exhibit certain advantages over traditional techniques in various aspects.
Recently, the social spider optimization (SSO), developed by Cuevas et al. [3] [4], was proposed. SSO is a swarm intelligence method based on the cooperative characteristics observed in social-spiders species. In SSO algorithm, individuals emulate a group of spiders which interact to each other based on the biological laws of the cooperative colony.
In this paper, we propose two waveguide microwave filters (rectangular waveguide H-plane three-cavity filter and Broad- band E-plan filters with Improved Stop band) optimization using SSO algorithm. The results of optimization obtained using SSO are validated by comparing them with those obtained using other optimization algorithms available in the literature Particle swarm optimization (PSO) [5] and Teaching-Learning Based Optimization (TLBO) [6] [7].
The remainder of this paper is organized as follows.
Sections II provide the description of the SSO. Section III presents the application examples and results. Finally, Section IV shows conclusions of this work.
II. SOCIALSPIDERSOPTIMIZATION
In 2013, Erik Cuevas et al [3] proposed a new swarm optimization algorithm called Social-Spiders Optimization (SSO). The SSO is based on the natural spider's colony behavior. Colonies are formed mainly by two elements:
spiders and communal web. Web is represented by the searching field domain while problem solutions are represented by the insects.
In nature, heavier individuals dominate the lighter ones.
This behavior is copied to the algorithm and the spider's weight is proportional to the solution evaluation.
Spiders are divided by gender and each one have a different behavior in the colony. This difference is implemented using unique evolutionary operators for males and females. Gender balance on the colony is normally around 70% of females.
The classical algorithm starts by populating randomly the first generation with uniform distribution in the search space.
A gender is addressed to each individual. The first spiders on the population matrix are addressed feminine and the rest as masculine. The cutting point is given by:
Nf=floor[(0.9 ‒rand* 0.25) *N] (1)
Nm=N‒Nf (2)
Where Nf and Nm are the number of females and males spiders,Nis the population size,floorrounds each element to the nearest integer, and rand is a random number in the unitary range [0, 1]. All elements are then evaluated on the objective function and the best solution (spider) and best objective function are recorded.
After the initialization process the algorithm starts the searching loop that only ends when the maximum number of function evaluations or the target function value is reached.
The first step in the searching loop is to calculate the spider's weight. This calculation is done according to:
Wi= (Worst‒f(xi)) / (Worst-Best) (3) WhereWiis the weight of theithspider,f(xi) is the fitness value of the spider xi. The values Worstand Bestare defined as follows (considering a minimization problem):
Best=mini=1, 2,…,N(f(xi)) andWorst=maxi=1,2,…,N(f(xi)) (4) In the colony, the spiders communicate with each other directly by mating or indirect by a small vibration to determine the potential direction of a food source, this vibration depend on the weight and distance of the spider which has generated them.
Vibi j=Wj*exp(‒ (di j)2) (5) Wherewjindicates the weight of thejthspider, anddi jis the Euclidean distance between ith and jth spiders. Every spider is able to consider three vibrations from other spiders as follows Fig.1.
Vibrations Vibciare perceived by the individual i (Si) as a result of the information transmitted by the member c (Sc) who is an individual that has two important characteristics: it is the nearest member to i and possesses a higher weight in comparison toi(Wc>
Wi).
The vibrationsVibbi perceived by the individualias a result of the information transmitted by the memberb (Sb), with b being the individual holding the best weight (best fitness value) of the entire population N, such thatWb= maxK=1, 2…..,N(Wk).
The vibrations Vibfiperceived by the individual i (Si) as a result of the information transmitted by the member f (Sf), with f being the nearest female individual toi.
Female spiders are moved either on a repulsion movement or on an attraction. This decision is made by attributing a random unitary number, to each female individual. If the result is greater than thePF, which is a constructive parameter, then the repulsion movement is selected and ruled by:
fi=fi+α Vibci(Sc‒fi) +β Vibbi(Sb‒fi) +δ(rand‒ 0.5) (6) Where fiis the ith female, Sc represents the closest heavier spider coordinates,Sbis the best spider's position, α, β, δ are unitary-ranged random numbers generated with uniform distribution. If the result is lower thanPFthen the attraction is selected and given by:
fi=fI‒α Vibci(Sc‒fi) ‒β Vibbi(Sb‒fi) +δ(rand‒ 0.5) (7) Male spiders are moved according to dominance. The dominance is given by the males median weight’s. Males lighter than the median are consider non-dominant and moved according to:
mi=mi+α((∑h=1..Nm mh*WNf+h)/(∑h=1..NmWNf+h)) ‒ mi) (8) Dominant males are those heavier than the median and are moved according to:
mi=mi+αVibfi(Sf‒mi) +δ (rand‒0.5) (9) Where the individual Sf represents the nearest female individual to the male memberi.
Finally, after move all males and females on the web, the last operator is representing the mating behavior where only dominant males will participate. The code will check if there is any female closer than radius of mating to a dominant male.
The radius of mating is given by:
R= (∑d=1….D (pdh– pdl))/ 2D (10)
WhereRis the mating radius,phandplare respectively the upper and lower bound for a given dimension and D is the problem dimension.
Males and females which are under the mating radius generate new candidate spiders according to the roulette method. Each candidate spider is evaluated in the objective function and the result is tested against all the actual population members. If any member is worse than a new candidate, the new candidate will take the actual individual position assuming actual individual’s gender.
Fig. 1. Configuration of each special relation: (a) Vibci, (b) Vibbi (c) Vibfi.
The following outline summarizes how the SSO algorithm works [4]:
Step 1: Initialize the male and female spiders in the initial population using uniform distribution;
Step 2: Calculate the radius of mating;
Step 3: Calculate the fitness (objective function) of each spider (candidate solution);
Step 4: Calculate the weight of every spider in terms of its fitness;
Step 5: Move female spiders according to the female cooperative operator;
Step 6: Move male spiders according to the male cooperative operator;
Step 7: Perform the mating operation;
Step 8: If the stopping criteria is reached, the process is finished; otherwise, go back to Step 3.
III. APPLICATIONEXAMPLE ANDRESULTS
In this section, the application of a proposed algorithm is presented for the optimization of two microwave filters:
Rectangular waveguide H-plane three-cavity filter Fig. 2.
When the main guide is WR28, four parameters are to be optimized W1and W2(the opening of the iris), l1 and l2 (the distance between the iris). The thicknesses of the iris are fixed to t1=1.45mm, t2=1.1 mm . Table.1 contains the geometric variables of the structure and the corresponding ranges. As for the frequency range, it was chosen to bef∈(34, 35.5 GHz).
Fig. 2. Rectangular waveguide H-plane three-cavity filter
TABLE I. VARIABLES OF FIRST FILTER
Variables Min Max
W1(mm) 3.24 3.96
W2(mm) 2.16 2.64
l1(mm) 3.735 4.565
l2(mm) 4.23 5.17
Rectangular waveguide E-plane three-cavity filter Fig. 3.
When the main guide is WR28, four parameters are to be optimized W1 andW2 (the opening of the iris), l1 and l2 (the distance between the iris). The thicknesses of the iris are fixed tot1=t2=0.1 mm . Table.1 contains the geometric variables of the structure and the corresponding ranges. Table.2 contains the geometric variables of the structure and the corresponding ranges. As for the frequency range, it was chosen to bef∈(38, 39 GHz).
Fig. 3. Rectangular waveguide E-plane three-cavity filter
TABLE II. VARIABLES OF SECOND FILTER
Variables Min Max
W1(mm) 1.710 2.090
W2(mm) 2.475 3.025
l1(mm) 3.897 4.763
l2(mm) 4.050 4.950
The objective is to minimize the fitness function in the frequency range, where the fitness functions is the mean value of the coefficient of reflectionS11.
fitness= (∑f=f1…f2S11(f))/T (11) WithTis the number of points in the interval [f1,f2].
Figs. 4 and 5 show the convergence of the PSO, TLBO and SSO for the optimization of structures mentioned. Every algorithm is run 10 independent times with the population size of 30 and the maximum number of generations of 100. The other specific parameters of algorithms are given below:
TLBO Settings: for TLBO there is no such constant to set.
PSO Settings: c1and c2 are constant coefficientsc1=c2=2, the inertia weight decreased linearly from 0.9 to 0.2
SSO Settings:PF=0.7.
It is observed from Figs. 4 and 5 that the SSO algorithm performs better in terms of convergence.
Fig. 4. Convergence of algorithms for thefirst structure
Fig. 5. Convergence of algorithms for the second structure
The results of optimization for two structures are represents in Fig. 6, Table 3 and Fig.7, Table 4 respectively.
The optimization results are very effective.
Fig. 6. The reflection coefficient responses for the first structure
TABLE III. THE GEOMETRICAL PARAMETERS OPTIMIZED FOR THE FIRST STRUCTURE
Parameters Ref [8] SSO
w1 3.6 3.85
w2 2.4 2.47
l1 4.15 3.956
l2 4.7 4.2
Fig. 7. The reflection coefficient responses for the second structure
TABLE IV. THE GEOMETRICAL PARAMETERS OPTIMIZED FOR THE FIRST STRUCTURE
Parameters Ref [8] SSO
w1 1.90 1.932
w2 2.75 2.524
l1 4.33 4.340
l2 4.50 4.494
IV. CONCLUSION
In this paper, a swarm algorithm called the Social Spider Optimization (SSO) has been proposed for the optimization of two microwave filters (Rectangular waveguide H-plane three- cavity and Rectangular waveguide E-plane three-cavity).
Based on results obtained, SSO offers good performance when compared with the other tested optimization approaches.
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