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Computer and Information Science » Numerical Analysis and Scientific Computing » "Evolutionary Computation", book edited by Wellington Pinheiro dos Santos, ISBN 978-953-307-008-7, Published: October 1, 2009 under CC BY-NC-SA 3.0 license. © The Author(s).

Chapter 11

Applications of the Differential Ant-Stigmergy Algorithm on Real-World Continuous Optimization Problems

By Peter Korosec and Jurij Silc
DOI: 10.5772/9604

Article top

Overview

Stigmergy is an organizing principle in emergent systems in which the individual parts of the system communicate with one another indirectly by modifying their local environment. Ant colonies are a classic example. The ants communicate indirectly. Information is exchanged through modifications of the environment (local gradients of pheromone).
Figure 1. Stigmergy is an organizing principle in emergent systems in which the individual parts of the system communicate with one another indirectly by modifying their local environment. Ant colonies are a classic example. The ants communicate indirectly. Information is exchanged through modifications of the environment (local gradients of pheromone).
Search graph representation of a discrete multi-parameter optimization problem.
Figure 2. Search graph representation of a discrete multi-parameter optimization problem.
The Differential Ant-Stigmergy Algorithm (DASA).
Figure 3. The Differential Ant-Stigmergy Algorithm (DASA).
GUI used for setting the DASA options.
Figure 4. GUI used for setting the DASA options.
Turbo-compressor unit of a dry vacuum cleaner.
Figure 5. Turbo-compressor unit of a dry vacuum cleaner.
Parameters that define the rotor and stator geometries.
Figure 6. Parameters that define the rotor and stator geometries.
Laminations of the original engineering design with power losses of 177.9 W (left) and laminations of an optimized design acceptable from the production point of view with power losses of 129.1 W (right).
Figure 7. Laminations of the original engineering design with power losses of 177.9 W (left) and laminations of an optimized design acceptable from the production point of view with power losses of 129.1 W (right).
Parameters of a casing and ribs. Ribs A (top), B (center), and C (bottom); basic (left) and modify forms (center and right).
Figure 8. Parameters of a casing and ribs. Ribs A (top), B (center), and C (bottom); basic (left) and modify forms (center and right).
Existent (up) and optimized (bottom) casing and different loading cases.
Figure 9. Existent (up) and optimized (bottom) casing and different loading cases.
Parametric modeling: a) top view; b) 3D view, c) side view.
Figure 10. Parametric modeling: a) top view; b) 3D view, c) side view.
The existent impeller (left), the worst optimized impeller (middle), and the best optimized impeller (right).
Figure 11. The existent impeller (left), the worst optimized impeller (middle), and the best optimized impeller (right).
The Bonferroni-Dunn post-hoc statistical test.
Figure 12. The Bonferroni-Dunn post-hoc statistical test.

Applications of the Differential Ant-Stigmergy Algorithm on Real-World Continuous Optimization Problems

Peter Korošec1 and Jurij Šilc1

1. Introduction

Ants have always fascinated human beings. What particularly strikes the occasional observer as well as the scientist is the high degree of societal organization that these insects can achieve in spite of very limited individual capabilities (Dorigo et al., 2000). Ants have inspired also a number of optimization algorithms. These algorithms are increasingly successful among researches in computer science and operational research (Blum, 2005; Cordón et al., 2002; Dorigo & Stützle, 2004).

A particular successful metaheuristic—Ant Colony Optimization (ACO)—as a common framework for the existing applications and algorithmic variants of a variety of ant algorithms has been proposed in the early nineties by Marco Dorigo (Dorigo, 1992). ACO takes inspiration from the foraging behavior of some ant species. These ants deposit pheromone on the ground in order to mark some favorable path that should be followed by other members of the colony. ACO exploits a similar mechanism for solving combinatorial optimization problems.

In recent years ACO algorithms have been applied to more challenging and complex problem domains. One such domain is continuous optimization. However, a direct application of the ACO for solving continuous optimization problem is difficult.

The first algorithm designed for continuous function optimization was continuous ant colony optimization (Bilchev & Parmee, 1995) which comprises two levels: global and local; it uses the ant colony framework to perform local searches, whereas global search is handled by a genetic algorithm. Up to now, there are few other adaptations of ACO algorithm to continuous optimization problems: continuous interacting ant colony (Dréo & Siarry, 2002), ACO for continuous and mixed-variable (Socha, 2004), aggregation pheromone system (Tsutsui, 2006), and multilevel ant-stigmergy algorithm (Korošec & Šilc, 2008).

In this chapter we will present so-called Differential Ant-Stigmergy Algorithm (DASA), a new approach to the continuous optimization problem (Korošec, 2006). We start with the DASA description followed by three case studies which show real-world application of the proposed optimization approach. Finally, we conclude with discussion of the obtained results.

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Figure 1.

Stigmergy is an organizing principle in emergent systems in which the individual parts of the system communicate with one another indirectly by modifying their local environment. Ant colonies are a classic example. The ants communicate indirectly. Information is exchanged through modifications of the environment (local gradients of pheromone).

2. A differential ant-stigmergy approach

2.1. Continuous optimization

The general continuous optimization problem is to find a set of parameter values, p=(p1p2pD) that minimizes a function, fcost(p) of D real variables, i.e.,

Find:

p*|fcost(p*)fcost(p)pD
(1)

To solve this problem, we created a fine-grained discrete form of continuous domain. With it we were able to represent this problem as a graph. This enabled us to use ant-based approach for solving numerical optimization problems.

2.2. The fine-grained discrete form of continuous domain

Let p'i be the current value of the i-th parameter. During the searching for the optimal parameter value, the new value, pi is assigned to the i-th parameter as follows:

Here, δi is the so-called parameter difference and is chosen from the set

Δi=Δi0Δi+
(3)

where

Δi={δik|δik=bk+Li1k=1,2,di}
(4)
and
Δi+={δik+|δik+=+bk+Li1k=1,2,di}
(5)

Here

Therefore, for each parameter pi the parameter difference, δi has a range from bLi to bUi where b is the so-called discrete base,

Li=lgb(εi)
(7)
and
Ui=lgb(max(pi)min(pi))
(8)

With the parameter εi the maximum precision of the parameter pi is set. The precision is limited by the computer's floating-point arithmetic. To enable a more flexible movement over the search space, the weight ω is added to Eq. 2:

where ω=RandomInteger(1,b1)

2.3. Graph representation

From all the sets Δi 1iD where D represents the number of parameters, a so-called search graph G=(VE) with a set of vertices, V, and a set of edges, E, between the vertices is constructed (see Fig. 2). Each set Δi is represented by the set of vertices,

Vi={vi,1vi,2vi,2di+1}
(10)
and

Then we have that

Δi={δidiδidij+1δi,1Δi,0,δi,1+δij+δidi+}Δi+
(12.1)

is equal to

Vi={vi,1vijvidi+10vidi+1+jvi,2di+1}
(12.2)

where j=1,2,di

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Figure 2.

Search graph representation of a discrete multi-parameter optimization problem.

Each vertex of the set Vi is connected to all the vertices that belong to the set Vi+1 Therefore, this is a directed graph, where each path ν from start vertex to any of the ending vertices is of equal length and can be defined with vi as:

ν=(v1v2vD)
(13)

where viVi 1iD

The optimization task is to find a path ν such that fcost(p)fcost(p') where p' is currently the best solution, and p=p'+Δ(ν) (using Eq. 9). Additionally, if the objective function fcost(p) is smaller than fcost(p') then the p' values are replaced with p values.

2.4. The differential ant stigmergy algorithm

The optimization consists of an iterative improvement of the currently best solution, p' by constructing an appropriate path ν that uses Eq. 9 and returns a new best solution. This is done as follows (see Fig. 3):

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Figure 3.

The Differential Ant-Stigmergy Algorithm (DASA).

  • Step 1. A solution p' is manually set or randomly chosen.

  • Step 2. A search graph is created and an initial amount of pheromone, τVi0 is deposited on all the vertices from the set ViV 1iD according to the Cauchy probability density function

    C(x)=1sπ+πs(xl)2
    (14)

    where l is the location offset and s=sglobalslocal is the scale factor. For an initial pheromone distribution the standard Cauchy distribution ( l=0, sglobal=1, and slocal=0 ) is used and each parameter vertices are equidistantly arranged between z=[4,4]

  • Step 3. There are m ants in a colony, all of which begin simultaneously from the start vertex. Ants use a probability rule to determine which vertex will be chosen next. More specifically, ant α in step i moves from a vertex in set Vi to vertex vij{vi,1vi,2vi,2di+1 } with a probability, prob, given by:

    probj(αi)=τ(vij)1k2di+1τ(vik)
    (15)

    where τ(vik) is the amount of pheromone on vertex vik

    The ants repeat this action until they reach the ending vertex. For each ant, path ν is constructed. If for some predetermined number of tries we get ν=0 the search process is reset by randomly choosing new ptemporary_best and pheromone re-initialization. New solution p is constructed (see Eq. 9) and evaluated with a calculation of fcost(p)

    The current best solution, pcurrent_best out of m solutions is compared to the temporary best solution ptemporary_best If pcurrent_best is better than ptemporary_best then ptemporary_best values are replaced with pcurrent_best values. In this case sglobal is increased (in our case for 1 %) and pheromone amount is redistributed according to the associated path νbest Furthermore, if new ptemporary_best is better then pbest then pbest values are replaced with ptemporary_best values. So, global best solution is stored. If no better solution is found slocal is decreased (in our case for 3 %).

  • Step 4. Pheromone dispersion is defined by some predetermined percentage χ The probability density function C(x) is changed in the following way:

    l(1χ)l
    (16)
    and
    slocal(1χ)slocal
    (17)

    Pheromone dispersion has a similar effect as pheromone evaporation in classical ACO algorithm.

  • Step 5. The whole procedure is then repeated until some ending condition is met.

    It is a well known that ant-based algorithms have problems with convergence. This happens when on each step of the walk there is a large number of possible vertices from which ant can choose from. But this is not the case with the DASA where Cauchy distribution of pheromone over each parameter was used. Namely, such distribution reduces the width of the graph to only few dominant parameter values (i.e., vertices). On the other hand, with proper selection of the discrete base, b, we can also improve the algorithm's convergence (larger b reduces the search graph size).

2.5. Software implementation and parameter settings

The DASA was implemented in the Borland® Delphi™ programming language (see Fig. 4). The computer platform used to perform the optimizations was based on AMD Opteron™ 2.6-GHz processors, 2 GB of RAM, and the Microsoft® Windows® XP 32-bit operating system.

The DASA parameters were set as follows: the number of ants, m was set to 10, the pheromone evaporation factor, χ was set to 0.2, and the maximum parameter precision, ε dependent on the discrete step of each parameter.

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Figure 4.

GUI used for setting the DASA options.

3. Case studies

Case studies presented in this section are related to the development of a new dry vacuum cleaner (Korošec et al., 2007; Tušar et al., 2007). In particular, the efficiency of the turbo-compressor unit (see Fig. 5) was improved.

3.1. An electric motor power losses minimization

Home appliances, such as vacuum cleaners and mixers, are generally powered by a universal electric motor (UM). These appliances need as low as energy consumption, that is, input power, as possible, while still satisfying the needs of the user by providing sufficient output power. The optimization task is to find the geometrical parameter values that will generate the rotor and the stator geometries resulting in the minimum power losses.

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Figure 5.

Turbo-compressor unit of a dry vacuum cleaner.

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Figure 6.

Parameters that define the rotor and stator geometries.

There are several invariable and variable parameters that define the rotor and stator geometries. The invariable parameters are fixed; they cannot be altered, either for technical reasons (e.g., the air gap) or because of the physical constraints on the motor (e.g., the radius of the rotor's shaft). The variable parameters do not have predefined optimum values. Some variable parameters are mutually independent and without any constraints. Others are dependent, either on some invariable parameters or on mutually independent ones.

In our case, 11 mutually independent variable parameters defining the rotor and stator geometries are subject to optimization (see Fig. 6): rotor yoke thickness, p1 rotor external radius, p2 rotor pole width, p3 stator yoke horizontal thickness, p4 stator yoke vertical thickness, p5 stator middle-part length, p6 stator internal edge radius, p7 stator teeth radius, p8 stator slot radius, p9 stator width, p10 and stator height, p11 We optimized only 10 parameters (the ratio between p10 and p11 was set constant) of the UM rotor and stator geometries.

Power losses, Plosses are the main factor affecting the efficiency of a UM. The efficiency of a UM, η is defined as the ratio of the output power, Pout to the input power, Pinp:

η=PoutPinp=PoutPout+Plosses
(18)

and it is very dependent on various power losses (Sen, 1997):

Plosses=PCu+PFe+Pbrush+Pvent+Pfrict
(19)

The overall copper losses, PCu occurring in the rotor and the stator slots are as follows:

PCu=i(J2Aρlturn)i
(20)

where i stands for each slot, J is the current density, A is the slot area, ρ is the copper's specific resistance and lturn is the length of the winding turn. The calculation of the iron losses, PFe is less exact because iron has non-linear magnetic characteristics. The iron losses are therefore separated into two components: the hysteresis losses and the eddy-current losses. This means the motor's iron losses can be expressed with the formula

PFe=ceB2fr2mr+ceB2fs2ms+chB2fsms
(21)

where ce is the eddy-current material constant at 50Hz, ch is the hysteresis material constant at 50Hz, B is the maximum magnetic flux density, fr is the frequency of the magnetic field density in the rotor, fs is the frequency of the magnetic field density in the stator, mr is the mass of the rotor, and ms is the mass of the stator. Three additional types of losses occurring in the UM, i.e., the brush losses, Pbrush; the ventilation losses, Pvent; and the friction losses, Pfric; depend mainly on the speed of the motor. When optimizing the geometries of the rotor and the stator, the motor's speed is assumed to be fixed; hence Pbrush Pvent and Pfric have no impact on the motor's efficiency, and so these losses are not significantly affected by the geometries of the rotor and the stator. Therefore, in our case, the overall copper and iron losses can be used to estimate the cost function:

fc(p)=PCu+PFe
(22)

Our goal is to find such parameter-value settings, where fc(p) is minimized.

The DASA starts its search with the existent solution. The stopping criterion was set to 1,400 calculations. The calculation of a single solution via the ANSYS Multiphysics simulation takes approximately two minutes, which means that the execution of 1,400 calculations needs about two days. The optimization method was run 20 times. The obtained results in terms of the UM power losses are presented statistically in Table 1.

Existing solutionOptimized solutions
WorstMeanBest
177.9136.7129.5113.8

Table 1.

Optimized UM's power losses in Watts after 1,400 calculations.

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Figure 7.

Laminations of the original engineering design with power losses of 177.9 W (left) and laminations of an optimized design acceptable from the production point of view with power losses of 129.1 W (right).

The engineering rotor and stator design (the existent solution) results in power losses of 177.9 W, and can be seen in Fig. 7 (left). The figure shows the magnetic flux density in the laminations (higher magnetic flux density, B causes higher power losses). Figure 7 (right) present a typical example of a feasible rotor and stator geometry, with power losses of 129.1 W. This solution has very low iron losses in the rotor due to its small size and in spite of its high magnetic saturation. The small rotor and its saturation are compensated by large stator poles that ensure large enough a magnetic flux. This design is completely feasible from the technical and production points of view.

3.2. An electric motor casing stiffness maximization

The casing is a part of the dry vacuum cleaner motor which is built into vacuum cleaners. The casing is basically an axisymmetric shell structure which is built of steel suitable for forming. For this procedure which consists of eleven different phases it is important that the radii are growing or do not change while the height is growing. That is an important rule which must not be broken during the geometry optimization. The goal of optimization is to preserve the stiffness while using a thinner shell structure and consequently save material and reduce costs.

The computational model of the casing comprises 26 different parameters which generate the geometry variations. The classical parameters are: radius on the top, p1 radius on the side, p2 radius at the groove for brushes, p3 deviation of the hole for diffuser fixation, p4 radius of roundness at the beginning of the bearing groove, p5 radius on the top of the bearing groove, p6 radius on the top of air culverts, p7 radius at the bottom of air culverts, p8 roundness of air culverts, p9 angle of slope, p22 angle of culverts span, p23 slope of bearing groove, p24 height of bearing groove, p25 and slope of the brushes groove, p26

Besides the above listed 14 classical parameters there are 12 more (p 10 to p 21), with which the ribs on areas A, B and C are defined (Fig. 8). These are essential to improve the stiffness so we will search for those which maximize the cost function. With these parameters we can generate a lot of combinations of different rib shapes, depending on the number of given paths and accuracy of intervals for points deviations, positions and number of ribs. In our case there were several billion combinations for ribs. It is obvious that instead of a defined path at position j, we could define displacements for individual points but this would result in many more parameters and time consuming calculation procedure. The question still remains what would be the benefit of that. From previous designs there exist some reasonable shapes for ribs which have been defined intuitively and these then change their position, magnitude and number, which are probably sufficient for a good result, as there has been over ten different rib shapes defined earlier.

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Figure 8.

Parameters of a casing and ribs. Ribs A (top), B (center), and C (bottom); basic (left) and modify forms (center and right).

Besides the parameters for the creation of ribs there are other classical parameters which also have an influence on their shape. Namely, if the radii p1 or p2 are varying then this is reflected in the shape of ribs of model A or if the height of bearing groove, p25 and slope of groove, p24 are modified then this will be reflected in the ribs of model C, etc.

The cost function is relatively simply defined when loadings are deterministic. Usually we optimize such cases with the minimization of displacements or stresses, maximization of elasticity, etc. (Dražumerič & Kosel, 2005). It is difficult to determine the cost function for a dynamically loaded assembly where the loads are stochastic. So the question is how to define the stiffness of a shell structure if we do not know the lateral loads. Let us consider a certain arbitrarily shaped plane shell. We can write its potential energy, Wp as:

Wp=12σijεijdV
(23)

If we consider that if the Poissons shear modulus is neglected, the stress tensor, σij and strain tensor, εij are the following:

σij=z{2w(xy)x22w(xy)y22w(xy)xy}T
(24)
and
εij=zE{2w(xy)x22w(xy)y22w(xy)xy}T
(25)

Here w(xy) is displacement in z direction and E is elasticity modulus. The potential energy can then be written as:

Wp=12EV(σxx2+σyy2+2τxy2)dV
(26)

We can see that in this case the potential energy is an integral of squares of stresses over the volume. We define also the kinetic energy, Wk for plate vibrations:

Wk=ρω22Vzw2(xy)dV
(27)

where ρ represents the density of material. Here the second derivative of displacement is approximated with respect to time with the product of squared displacement and eigen frequency, ω

For conservative systems the maximal potential energy is equal to maximal kinetic energy and from this follows the expression for cost function which can be in our case the stiffness itself:

fc(p)=V(σxx2+σyy2+2τxy2)dVVzw2(xy)dV
(28)

The DASA was run 20 times and each run consisted of 2,000 calculations. The cost function was calculated by the ANSYS Multiphysics simulation tool. The obtained results are presented statistically in Table 2.

Existing solutionOptimized solutions
WorstMeanBest
5.87·10-36.13·10-36.81·10-37.34·10-3

Table 2.

Optimized casing's stiffness after 2,000 CFD calculations.

The results of optimization were quite surprising (Fig. 9 bottom row). It was expected that the ribs would form on all the given surface but they did not. The ribs were more distinct where the vertical part of the surface was turning into the horizontal one (ribs A). There were no ribs at the surface where the stator is in a tight fit, probably because of pre-stressing of the casing through the stator. Also in the groove for brushes (ribs B) there were no distinct ribs, probably because the groove itself is a kind of rib. Instead of this there is a given slope for a groove which was not there before. The radii of roundings were in most cases bigger than before as we can clearly see at the air culverts (Fig. 9 middle bottom).

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Figure 9.

Existent (up) and optimized (bottom) casing and different loading cases.

3.3. A turbo-compressor aerodynamic power maximization

Radial air impellers are the basic components of many turbo-machines. In the following we will concentrate on relatively small impellers and subsonic speeds used in a dry vacuum cleaner. Our main aim was to find an impeller shape that has a higher efficiency, i.e., greater aerodynamic power, than the one currently used in production.

An impeller is constructed from blades, an upper and a lower side. The sides enclose the blades and keep them together. The blades, which are all the same, were the main part of the optimization. The geometry of a blade is shown in Fig. 10, where the gray color represents the blade. The method of modeling is as follows: we construct the points at specific locations, draw the splines through them and spread the area on the splines. Once a blade is made an air channel must be constructed in a similar way.

In Fig. 10a the point 1 has two parameters: the radius p1 and the angle p2 Similarly, the points 2, 5 and 6 have parameter pairs p3 and p4 p5 and p6 p7 and p8 The points 3 and 4 are fixed on the x axis. This is because the impeller must have a constant outer radius p9 and the outer side of the blade must be parallel to the z axis. On the other hand, the outer angle of the blade p10 and the angle of the spline at points 3 and 4, can be varied. Analogously, the angles p11 and p12 are the inner-blade angles for the upper and lower edges of the blade at the input, respectively.

In Fig. 10c the points 1, 2, and 3 form the upper spline, and the points 4, 5, and 6, the lower spline. Between the points 1 and 6 is the point 7, which defines the spline of the input shape of the blade. In this figure, the points 1, 2, 5, and 6 have the parameters p13 p14 p15 and p16 respectively, describing their heights.

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Figure 10.

Parametric modeling: a) top view; b) 3D view, c) side view.

Point 3 stays on the x axis and point 4 has a constant height p17 In other words, the designer of the impeller must know at least the outer diameter p9 and the height p17 The parameters p18 and p19 describe the input angles of the lower and upper parts of the blade with respect to the r z plane. Similarly, the parameters p20 and p21 describe the outer blade angle with respect to the same plane.

In Fig. 10b the meaning of point 7 is explained more precisely. The parameters p22 p23 and p24 define the radius, height, and angle, respectively. The radius and angle dictate where the point should appear with respect to the x y plane and the height with respect to the r z plane. Similarly, the angles p25 p26 p27 and p28 are needed to define the starting and ending angles of the spline constructed between the points 1, 7, and 6.

If we look closely at Fig. 10c then we can see the contour surrounding the blade. This is the air channel with the following parameters: the inner radius p29 (see Fig. 10a), which is needed for the hexahedral mesh (explained later), the air intake radius p30 the air outflow radius p31 the bolt radius p32 the bolt height p33 and the impeller height p34

In this way we have successfully modeled the impeller geometry with 34 parameters. For each parameter we have a predefined search interval with a given discrete step. Therefore, the size of the search space can be obtained as the product of the number of possible settings over all the parameters. It turns out that there are approximately 3 1034 possible solutions.

The mesh of the air channel between the two blades is constructed with more than 6,000 hexahedral elements. The boundary conditions are zero velocity at all the solid regions and symmetry boundary conditions at the fluid regions. At the influx and outflux the intake velocity, vin and reference pressure, pref are defined, respectively. The intake velocity is parabolically distributed, because we expect that the intake flow is laminar and so:

vin=v(Φ(t))6rrup(rrup1)
(29)

Here, v(Φ(t)) is a velocity dependent on the stream, which further depends on time, as we shall see later, rup is the upper radius, defined before, and r is the radius within the limits from rs to rup The reference pressure, pref is set to zero.

With respect to the maximum time, tmax , the flux is:

Φ(t)=vAinttmax
(30)

where Ain is the influx area and t is the current time.

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Figure 11.

The existent impeller (left), the worst optimized impeller (middle), and the best optimized impeller (right).

The distribution of the relative pressure can be used to estimate the cost function. The average pressure, pin is chosen from the air-intake area. Finally, the aerodynamic power, which represents the cost function, is as follows:

fc(p)=Pair=(poutpin)Φ(topt)
(31)

where pout is the output pressure at the radius rout and Φ(topt)=40 l/s is the flux near the desired optimum performance of the impeller. Our goal is to find such parameter-value settings, where Pair is maximized.

The DASA was run 10 times and each run consisted of 2,000 CFD calculations. For the CFD calculations we used the ANSYS Multiphysics package. A single CFD calculation takes approximately seven minutes. The obtained results, in terms of aerodynamic power, are presented statistically in Table 3.

Existing solutionOptimized solutions
WorstMeanBest
411.00432.00472.00483.00

Table 3.

Optimized impeller's aerodynamic power in watts after 2,000 CFD calculations.

Results show that we were able to increase aerodynamic power for approximately 20 %. Figure 11 shows a 3D view of the existent and two optimized impellers (best and worst of 10 runs).

4. Discussion

In this chapter the Differential Ant-Stigmergy Algorithm (DASA) was presented, where the main goal was an evaluation of the DASA on some real-world engineering applications. Case studies were selected from a R&D project where more efficient turbo-compressor for dry vacuum cleaner was developed. Here the DASA was used to improve the efficiency of an electric motor, increase casing stiffness, and increase impeller's aerodynamic power. In all these cases the improvement was evident.

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Figure 12.

The Bonferroni-Dunn post-hoc statistical test.

In (Korošec & Šilc, 2009a) we also compared the DASA to the eleven state-of-the-art stochastic algorithms for continuous optimization which were presented in the CEC'2005 Special Session on Real Parameter Optimization. The algorithms are:

Obtained results confirmed that the DASA is comparable to above algorithms and therefore generally applicable to global optimization problems.

The experimental results have shown that the DASA ranks among the best algorithms for real-life-like applications. Its main advantage is in consistent problem solving which is indicated by 19 rankings in top third, 4 ranking in middle third and only 2 in bottom third.

Statistical analysis following the Bonferroni-Dunn post-hoc test with α=0.10 showed that the DASA is significantly better than 8 out of 11 compared algorithms.

The algorithm was also applied to dynamic optimization problems with continuous variables proposed for CEC’2009 Special Session on Evolutionary Computation in Dynamic and Uncertain Environments (Korošec & Šilc, 2009b). If we compare the DASA to:

the results show that the DASA can find reasonable solutions for all of the problems. It can be seen that the DASA performs not many worse than a self-adaptive differential evolution and much better than the other three algorithms. One obvious advantage is that was no need any changes to the original algorithm. So, it can be used as such for both cases of numerical optimization, static and dynamic. Furthermore, the algorithm is unsusceptible to different types of changes and can be used with very limited knowledge about problem, only maximal dimension and input problem parameters.

References

1 - S. Alonso, J. Jimenez, H. Carmona, B. Galvan, G. Winter, 2005 Performance of a flexible evolutionary algorithm. www.ntu.edu.sg/home/EPNSugan.
2 - A. Auger, N. Hansen, 2005a A restart CMA evolution strategy with increasing population size, Proceedings of the IEEE Congress on Evolutionary Computation, 1769 1776 , 0-78039-363-5 UK, September 2005, IEEE.
3 - A. Auger, N. Hansen, 2005a Performance evaluation of an advanced local search evolutionary algorithm, Proceedings of the IEEE Congress on Evolutionary Computation, 1777 1784 , 0-78039-363-5 UK, September 2005, IEEE.
4 - P. J. Ballester, J. Stephenson, J. N. Carter, K. Gallagher, 2005 Real-parameter optimization performance study on the CEC-2005 benchmark with SPC-PNX, Proceedings of the IEEE Congress on Evolutionary Computation, 498 505, 0-78039-363-5 UK, September 2005, IEEE.
5 - G. Bilchev, I. C. Parmee, 1995 The ant colony metaphor for searching continuous design spaces, In: Evolutionary Computing, L. Fogarty (Ed.), 25-39, Springer, 0302-9743 Lecture Notes in Computer Science, 993
6 - C. Blum, 2005 Ant colony optimization: Introduction and recent trends, Physics of Life Reviews, 2 4 (December 2005) 35325-39373, 1571-0645
7 - J. Brest, A. Zamuda, B. Bosković, Maučec. M. Sepesy, V. Žumer, 2009 Dynamic optimization using self-adaptive differential evolution, Proceedings of the IEEE Congress on Evolutionary Computation, 415 422 , 978-1-42442-959-2 Trondheim, Norway, May 2009, IEEE.
8 - L. T. Bui, Y. Shan, F. Qi, H. A. Abbass, 2005 Comparing two versions of differential evolution in real parameter optimization, www.ntu.edu.sg/home/EPNSugan.
9 - O. Cordón, F. Herrera, T. Stützle, 2002 A review on the ant colony optimization metaheuristic: Basis, models and new trends, Mathware and Soft Computing, 9 3 (2002) 141-175, 1134-5632
10 - F. O. de França, Zuben. F. J. Von, 2009 A dynamic artificial immune algorithm applied to challenging benchmarking problems, Proceedings of the IEEE Congress on Evolutionary Computation, 423 430 , 978-1-42442-959-2 Trondheim, Norway, May 2009, IEEE.
11 - M. Dorigo, 1992 Optimization, learning and natural algorithms, Ph.D.Thesis, Politecnico di Milano, Italy.
12 - M. Dorigo, E. Bonabeaub, G. Theraulaz, 2000 Ant algorithms and stigmergy, Future Generation Computer Systems, 16 8 (June 2000) 851-871, 0016-7739X.
13 - M. Dorigo, T. Stützle, 2004 Ant Colony Optimization, The MIT Press, 978-0-26204-219-2 Cambridge, Massachusetts.
14 - R. Dražumerič, F. Kosel, 2005 Optimization of geometry for lateral buckling process of a cantilever beam in the elastic region, Thin-Walled Structures, 43 3 (March 2005) 515-529, 0263-8231
15 - J. Dréo, P. Siarry, 2002 A new ant colony algorithm using the heterarchical concept aimed at optimization of multiminima continuous functions, In: Ant Algorithms, M. Dorigo et al. (Eds.), 216-227, Springer, 0302-9743 Lecture Notes in Computer Science, 2463
16 - C. García-Martínez, M. Lozano, 2005 Hybrid real-coded genetic algorithm with female and male differentiation, Proceedings of the IEEE Congress on Evolutionary Computation, 896 903 , 0-78039-363-5 UK, September 2005, IEEE.
17 - P. Korošec, 2006 Stigmergy as an approach to metaheuristic optimization, Ph.D.Thesis, Jožef Stefan International Postgraduate School, Ljubljana, Slovenia.
18 - P. Korošec, K. Oblak, F. Kosel, J. Šilc, 2007 Multi-parameter numerical optimization of selected thin-walled machine elements using a stigmergic optimization algorithm, Thin-Walled Structures, 45 12 (December 2007) 991-1001, 0263-8231
19 - P. Korošec, J. Šilc, 2008 Using stigmergy to solve numerical optimization problems, Computing and Informatics, 27 3 (2008) 377-402, 1335-9150
20 - P. Korošec, J. Šilc, 2009a A stigmergy-based algorithm for continuous optimization tested on real-life-like environment, In: EvoWorkshops 2009, M. Giacobini et. al (Eds.), 675-684, Springer, 0302-9743 Lecture Notes in Computer Science, 5484
21 - P. Korošec, J. Šilc, 2009b The differential ant-stigmergy algorithm applied to dynamic optimization problems, Proceedings of the IEEE Congress on Evolutionary Computation, 407 414 , 978-1-42442-959-2 Trondheim, Norway, May 2009, IEEE.
22 - C. Li, S. Yang, 2009 A clustering particle swarm optimizer for dynamic optimization, Proceedings of the IEEE Congress on Evolutionary Computation, 439 446 , 978-1-42442-959-2 Trondheim, Norway, May 2009, IEEE.
23 - D. Molina, F. Herrera, M. Lozano, 2005 Adaptive local search parameters for real-coded memetic algorithms, Proceedings of the IEEE Congress on Evolutionary Computation, 888 895 , 0-78039-363-5 UK, September 2005, IEEE.
24 - P. Pošík, 2005 Real-parameter optimization using the mutation step co-evolution, Proceedings of the IEEE Congress on Evolutionary Computation, 872 879 , 0-78039-363-5 UK, September 2005, IEEE.
25 - J. Rönkkönen, S. Kukkonen, K. V. Price, 2005 Real-parameter optimization with differential evolution, Proceedings of the IEEE Congress on Evolutionary Computation, 506 513 , 0-78039-363-5 UK, September 2005, IEEE.
26 - P. C. Sen, 1997 Principles of Electric Machines and Power Electronics, 2nd edition, John Wiley & Sons, 978-0-47102-295-4 New York.
27 - A. Sinha, S. Tiwari, K. Deb, 2005 A population-based, steady-state procedure for real-parameter optimization, Proceedings of the IEEE Congress on Evolutionary Computation, 514 521 , 0-78039-363-5 UK, September 2005, IEEE.
28 - K. Socha, 2004 ACO for continuous and mixed-variable optimization, In: Ant Colony Optimization and Swarm Intelligence, M. Dorigo et al. (Eds.), 25-36, Springer, 0302-9743 Lecture Notes in Computer Science, 3172
29 - S. Tsutsui, 2006 An enhanced aggregation pheromone system for real-parameter optimization in the ACO metaphor, In: Ant Colony Optimization and Swarm Intelligence, M. Dorigo et al. (Eds.), 60-71, Springer, 0302-9743 Lecture Notes in Computer Science, 4150
30 - T. Tušar, P. Korošec, P. , G. Papa, B. Filipič, J. Šilc, 2007 A comparative study of stochastic optimization methods in electric motor design, Applied Intelligence, 27 2 (2007) 101-111, 0092-4669X.
31 - E. L. Yu, P. Suganthan, 2009 Evolutionary programming with ensemble of external memories for dynamic optimization, Proceedings of the IEEE Congress on Evolutionary Computation, 431 438 , 978-1-42442-959-2 Trondheim, Norway, May 2009, IEEE.
32 - B. Yuan, M. Gallagher, 2005 Experimental results for the special session on real-parameter optimization at CEC 2005: A simple, continuous EDA, Proceedings of the IEEE Congress on Evolutionary Computation, 1792 1799 , 0-78039-363-5 UK, September 2005, IEEE.