Data-set at neural tree node input.
1. Introduction
Optimization is an important concept in science and engineering. Traditionally, methods are developed for unconstrained and constrained single objective optimization tasks. However, with the increasing complexity of optimization problems in the modern technological real world problems, multi-objective optimization algorithms are needed and being developed. With the advent of evolutionary algorithms in the last decades, multi-objective evolutionary algorithms (MOEAs) are extensively being investigated for solving associated optimization problems, e.g. (Deb et al., 2000, Zitzler & Thiele, 1999, Yao et al., 1999, Eiben et al., 1999). An updated survey of Ga-based MOEAs is given by (Coello, 1999). Evolutionary algorithms are particularly suitable for this, since they evolve simultaneously a population of potential solutions. These solutions are investigated in non-dominated solution space so that the optimized solutions in a multi-objective functions space form a front which is known as Pareto surface or front. Obtaining this simultaneous solution front in a single run is an appealing property that it is the incentive for a fast growing interest on MOEAs in the last decade. Although Pareto front is an important concept, its formation is not straightforward since the strict search of non-dominated regions in the multi-objective solution space prematurely excludes some of the potential solutions that results in an aggregated solutions in this very space. This means Pareto surface is not fully developed and the diversity of the solutions on the Pareto front is not fully exercised. Conventionally, non-dominated solutions with many objectives are usually low in number making the selection pressure toward the Pareto front also low, with aggregated solutions in the Pareto dominance-based MOEA algorithms (Sato, 2007). The purpose of this research is to investigate this issue and provide effective solutions with fast convergence together with diversity of solutions is maintained on the Pareto front. This goal has already attracted attention in the literature (Laumanns et al., 2002). This work addresses this issue with a novel concept of adaptive formation of Pareto front. This is demonstrated with an application from the domain of architectural design. The method is based on relaxed dominance domains, which basically refer to a degree of relaxation of the dominance in the terminology of MOEAs. In this book-chapter contribution, the relaxed dominance concept is explicitly described and applied. The organisation of this chapter is as follows. Section two describes the relaxed dominance concept. Section three describes the adaptive formation of Pareto front in a design application. This is followed by the conclusions in section four.
2. Design computation subject to multiobjective optimization
Multi-objective optimization deals with optimization where several objectives are involved. These objectives are conflicting or in competition among themselves. For a single objective case there are traditionally many algorithms in continuous search space, where gradient-based algorithms are most suitable in many instances. In discrete search spaces, in the last decade evolutionary algorithms are ubiquitously used for optimization, where genetic algorithms (GA) are predominantly applied. However, in many real engineering or design problems, more than two objectives need to be optimized simultaneously. To deal with multi-objectivity it is not difficult to realize that evolutionary algorithms are effective in defining the search direction. Basically, in a multi-objective case the search direction is not one but may be many, so that during the search a single preferred direction cannot be identified. In this case a population of candidate solutions can easily hint about the desired directions of the search and let the candidate solutions during the search process be more probable for the ultimate goal. Next to the principles of GA optimization, in MO algorithms, in many cases the use of Pareto ranking is a fundamental selection method. Its effectiveness is clearly demonstrated for a moderate number of objectives, which are subject to optimization simultaneously (Deb, 2001). Pareto ranking refers to a solution surface in a multidimensional solution space formed by multiple criteria representing the objectives. On this surface, the solutions are diverse but they are assumed to be equivalently valid. The eventual selection of one of the solutions among those many is based on some so-called higher order preferences, which require more insight into the problem at hand. This is necessary in order to make more refined decisions before selecting any solution represented along the Pareto surface.
In solving multi-objective optimization, the effectiveness of evolutionary algorithms has been well established. For this purpose there are quite a few algorithms which are running quite well especially with low dimensionality of the multidimensional space (Coello et al., 2003). However, with the increase of the number of objective functions, i.e. with high dimensionality, the effectiveness of the evolutionary algorithms is hampered. One measure of effectiveness is the expansion of Pareto front where the solution diversity is a desired property. For this purpose, the search space is exhaustively examined with some methods, e.g.
The issue of solution diversity and effective solution for multi-objective optimization problem described above is especially the due to elimination of many acceptable solutions during the evolutionary computation process, in case orthogonal standard Pareto dominance is used. This is a kind of Greedy algorithm which considers the solutions at the search area delimited by orthogonal axes of the multidimensional space. To increase the pressure pushing the Pareto surface towards to the maximally attainable solution point is the main problem and relaxation of the orthogonality with a systematic approach is needed. By such a method next to non-dominated solutions also some dominated solutions are considered at each generation. Such dominated solutions can be potentially favourable solutions in the present generation, so that they can give birth to non-dominated solution in the following generation. Although, some relaxation of the dominance is addressed in literature (Branke et al., 2000, Deb et al., 2006), in a multidimensional space, to identify the size of relaxation corresponding to a volume is not explicitly determined. In such a volume next to non-dominated solutions, dominated but potentially favourable solutions, as described above, lie. To determine this volume optimally as to the circumstantial conditions of the search process is a major and a challenging task. The solution for this task is essentially due to the mathematical treatment of the problem where the volume in question is identified adaptively during the search that it yields a measured pressure to the Pareto front toward to the desired direction, at each generation. In the adaptive process reported in this work, the volume is determined by genetic search for each member of the population. The process is adaptive, because the Pareto front is converged progressively in the course of consecutive generations, where the rate of convergence is determined with volume size, which is subject to appropriate change at each generation. In non-adaptive case, the Pareto front is also converged progressively; however the rate of convergence, in contrast to the adaptive case, is monotonically exhausted. The adaptation is explained shortly afterwards below via contour lines in the objective-functions space. Here the volume with dominated solutions is termed as
Some important features of the latest generation MOEAs address the selection of the potential solutions during the optimization process, and diversity-preserving strategies in objective space. Next to the principles of GA optimization, in MO algorithms, in many cases the use of Pareto ranking is a fundamental selection method. Its effectiveness is demonstrated for a moderate number of objectives, which are subject to optimization simultaneously. With respect to the conflicting objectives in a MO optimization, one has to deal with the criteria as measures of the conflicts. The increased satisfaction of one criterion implies loss with respect to satisfaction of another criterion. Regarding to this, the formation of the Pareto front is based on some newly defined objective functions of the weighted N objectives
where
subject to identification as an ideal solution. The point P can be sought by means of appropriate methodologies one of which is the method of genetic algorithm applied in this approach. For the search process itself, different strategies can be followed. Below, two strategies are described for the sake of providing insight into the search process of the multi objective optimization approach applied in this work.
In one strategy the point P denotes the explicit ultimately attainable goal defined by the boundaries of the objective functions. The premise of the strategy is that beyond this point the solution is not acceptable or meaningless due to the limits of the objective functions.
The algorithm using the orthogonal lines is called
Figure 1b and its reference point serve as a conceptual explanation of the Pareto-optimality in order to point-out the ‘trade-off’ inherent to the relaxed dominance compared to the greedy dominance concept. Namely, with reference to the point P, by making the angle θ larger than 90 degrees the area of non-existent solutions is reduced compared to the greedy case. Therefore the Pareto front is allowed to establish closer to the reference point P, while at the same time the front is expected to be more diverse. In figure 1b it is seen how the greedy front comes closer to the point P through the widening of the angle θ. This is indicated by means of arrows. In the relaxed approach, the avoidance of aggregation to some extend is due to the distortion of the objective space, where the space becomes larger, and thus the density of solutions per unit length along the front is expected to become lower.
In a second strategy a hypothetical point designated as P’ denotes the explicit predefined sub-attainable goal. This goal is positioned somewhere in the convex hull defined via
For the
Based on the strategy depicted in figure 2b the search process can be relaxed in a controlled way concerning the trade-off between solution diversity and effective multi-objective optimisation. That is the mechanism pushing the Pareto front forward is well balanced. This is a novel approach and it is explained below.
2.1. Relaxed Pareto ranking
To avoid premature elimination of potential solutions, a relaxed dominance concept is implemented, where the angle
Let (1) be expressed by
In matrix equation form, (3) becomes
where
In a two-dimensional coordinate system the contour lines in figures 1a, 1b and 2b are orthogonal and non-orthogonal respectively. The search area in the latter case includes also the domains of relaxation. These are added to the search area of the orthogonal system, as illustrated in figure 2b.
In the non-orthogonal system the search area for the favourable solutions is wider. At the same time some of the solutions are not dominating the solution at the point P seen in figure 2b. However, as a trade-off it provides more diversity at the final Pareto front, while the front is not entirely non-dominated. The solutions at the front are more probably non-dominated in the middle part of the front. This is where
Interestingly, this situation is similar to the classical gradient-based optimization method where each iteration the step-length towards the global maxima or minima determined by a step-size parameter (Farhang-Boroujeny, 1998), also called convergence coefficient. The step-size parameter should be small enough to ensure the stability of the convergence (Bazaraa et al., 1993, Kuester, 1973). If the step-size parameter is zero, the approach to minima or maxima does not occur. If it is too big, convergence does not occur, due to instability. For similar reasons, in the evolutionary computation the angle
In (4) the small-enough designation of the parameters
which transforms the non-orthogonal system to the orthogonal system and vice versa via
where x’ denotes the non-orthogonal system
which corresponds to column vectors in (6), so that
In a two-dimensional case the directive cosines are shown in figure 3a and in relation to the set of transformation equations in (3), the directive cosine row vectors are given by
The coordinate transformation of point
It is interesting to note that since the cosine directive
and using the relationship
we obtain
which yields (5) in terms of the transformation angles in the form
In a general form, (17) is given by
The importance of the coordinate transformation becomes dramatic especially in higher dimensions. In such cases the spatial distribution of domains of relaxation becomes complex and thereby difficult to implement. Namely, in multidimensional space the volume of a relaxation domain is difficult to imagine. And more importantly it is difficult to identify the population in such domains. Therefore one needs a systematic approach for identification by computation and not by inspection or anything else. This systematic approach is based on the coordinate transformation as follows. Basically for each solution point, designated in general as P in figure 2b is temporarily considered to be a reference point as origin and all the other solution points in the orthogonal coordinate system are converted to the non-orthogonal system coordinate by (8). For instance for four objectives, we write
where the
so that, in view of (7) the cosine direction matrix and the corresponding weighted objectives
And in view of (8) the inverse of (21) becomes
After conversion, all points which have positive coordinates in the non-orthogonal system correspond to potential solutions contributing to the next generation in the evolutionary computation. If any point possesses a negative component in the new coordinate system, the respective solution does not dominate P and therefore is not counted. This is because otherwise such a solution may lead the search in a direction away from P. In general, the relaxation of the dominance in higher dimensions is extremely complex and therefore many different methods for effective Pareto front formation in the literature (Hughes, 2005, Jaszkiewicz, 2004) are reported. However (8) provides a decisive and easy technique revealed in this work for the same goal.
3. Adaptive formation of Pareto front in a design application
In this section adaptive formation of Pareto front with multi-objectivity is considered. The formation of Pareto front is explained by means of a design example where multi-objectivity is subjected to a Pareto front based solution. For the multi-objective optimization a genetic algorithm approach with a relaxed Pareto-ranking is used. The relaxation angle is computed adaptively for every chromosome, and at every generation. This is implemented by having the angle be a part of the chromosome of every solution. The fitness of a chromosome is obtained by considering two properties of the solution at the same time. One is the degree of dominance in terms of the amount of solutions dominating an individual, the second is the relaxation angle used to measure this amount. This is given by
In (23) and (24) the purpose is to reward a chromosome for affording a wide relaxation angle θ, relative to the average angle of the population
Since design is an intelligent activity it is composed of several considerations which are soft in nature. For this reason, to invoke methods of soft computing is most appropriate. In this respect fuzzy-neural tree is considered as a knowledge model in this work as it is explained below. The application concerns the design of a building. The building consists of a number of spatial units, referred to as design objects, where every unit is designated to a particular purpose in the building. The task is to locate the objects optimally with respect to three objectives. The objects are seen from figure 5.
Object O1 is a hotel unit, O3 is a conference space, O4 and O5 are office wings, and O2 is a lobby with a restaurant. In order to let the computer generate a building from these parts, i.e. for the solution to be feasible, it is necessary to ensure that all solutions have some basic properties. These are that spaces should not overlap, and objects should be adjacent to the other objects around. This is realized in the present application by inserting the objects in a particular sequential manner into the site. This is illustrated in figure 5. One by one the objects are moved into the site starting from a point marked by a cross in the figures, then moving in west direction (lower part of the figures) until they reach an obstacle, that may be the site boundary or another object previously inserted. When they touch an object they change their movement direction from western to the northern direction, moving north until they again reach the site boundary or another object. As a third and final movement step the object will attempt to move once more in western direction, although often this is may not happen since often there is an adjacent object in the western direction blocking the way. An example of this third step is shown in figure 5c. The third step is to avoid that gaps between objects are reduced to some extend. This way of packing objects is known as
The multi objective optimization is accomplished using a multi-objective genetic algorithm with a relaxed Pareto ranking. It is used to determine the optimal sequence of insertion, so that three objectives are maximally fulfilled. Objects are numbered, and every chromosome contains the information for every object, at which rank in the insertion sequence it is to be inserted.
The objectives are functionality of the building, certain energy performance aspects, and some form related preferences. The design performance is obtained from these three objectives. Due to the linguistic nature of these objectives a special model is formed and used to assess satisfaction of the objectives. This model is a fuzzy neural tree model. In this model the ultimate goal is to have a good design, what we term as a design with a high design performance. That is, if all three objectives are highly fulfilled then the design has a high performance. The relation of the concept of design performance with the physical properties of possible solutions, which form the model inputs, is captured in the model through a hierarchical structure of logic operations. The method used is fuzzy neural tree.
3.1. Fuzzy-neural tree modeling domain knowledge
For human-like information processing the methods of soft computing are presumably the most convenient. The salient soft computing methods are in the paradigms of neural nets and fuzzy logic (Mitra et al., 2002). In this work a neural tree is considered to assess the suitability of a solution in a human-like manner. A neural tree is composed of terminal nodes, non-terminal nodes, and weights of connection links between two nodes. The non-terminal nodes represent neural units and the neuron type is an attribute introducing a non- linearity simulating a neuronal activity. In the present case, this attribute is established by means of a Gaussian function which has several desirable features for the intended goals; namely, it is a radial basis function ensuring a solution and the smoothness. At the same time it plays the role of a fuzzy membership function in the tree structure, which is considered to be a fuzzy logic system as its outcome is based on fuzzy logic operations and thereby associated reasoning. An instance of a neural tree is shown in figure 6.
Detailed structures of a neural tree are shown in figure 7. Figure 7a shows a terminal node connected to an inner node, and figure 8b and 8c show the connections among inner nodes. Each terminal node, also called
In the neural tree considered in this work the output of
where ϕ(.) is the Gaussian basis function,
where
The output of node
which reduces to
We can express (28) in the following form
This implies that the width of the Gaussian is scaled by the input weight
In figure 8 only two inputs are considered without loss of generality. The variables
Since
This is illustrated in figure 9b where the left part of the Gaussian is approximated by a straight line. In figure 9b, optimizing the
for the values
.1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |
.1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |
.1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |
.1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |
.1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |
.1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |
In general, the data sets given in Table 1 and Table 2 are named in this work as ‘
At this point a few observations are due, as follows. If a weight
The general properties of the present neural tree structure are as follows.
If an input of a node is small (i.e., close to zero) and the weight
If a weight
If all input values coming to a node are high (i.e., close to unity), the output of the node is also high complying with the AND operation.
If a weight
It might be of value to point out that, the AND operation in a neural-tree node is executed in fuzzy logic terms and the associated connection weights play an important role on the effectiveness of this operation.
The neural tree employed in this work is shown in figure 10. The root node describes the ultimate goal subject to maximization, namely the design performance and the tree branches form the objectives constituting this goal. The connections among the nodes have a weight associated with them, as seen from the figure. The weight is given by a designer, as an expression of knowledge, and it specifies the relative significance a node has for the node one level closer to the root node. It is noted that in the multi-objective optimization case the weights connecting the nodes on the penultimate level of the weight tells how strongly the output of the lower node influences the output of the upper tree to the root node are not specified a-priori, but they are subject to identification after the optimization process is accomplished.
During evaluation of a design alternative the tree is provided with inputs at its leaf nodes and the fuzzification processes are carried out. The fuzzification yields the satisfaction of an elemental requirement at the terminal nodes of the neural tree. These requirements are some desirable features expressed by means of fuzzy membership functions at the terminal nodes of the tree. Three examples are shown in figure 11.
Figure 11a expresses the requirement x1 in figure 10. It demands that the conference space should be located close to the building services, which is an aspect of functionality, i.e. distance from service facilities should be low for convenient access of services. The requirement is fully satisfied, i.e. the membership degree
Figure 11b expresses the requirement x3. It demands that the conference space should be far from the hotel, to avoid acoustic disturbance and to keep people flows separate. From the figure we see that distances beyond 40m are considered to fully satisfy this demand. From figure 10 we note that, comparatively the requirement in figure 11a is considered 1.5 times more important than figure 11b regarding the functionality of the building.
Figure 11c expresses the requirement x19 in figure 10. It demands that the building should have an elongated shape. What is meant is that the shape of the floor plan should not be a square, but that the shape should clearly have a longer extend in north-south direction, termed length, than in east-west direction, termed width. This is an aspect of form preferences. To express this demand the input values to this membership function are the proportion length/width, being unit less. From the membership function we note that a square proportion yields a low membership degree
In the same way other properties of the design are measured and converted into satisfactions using specific membership functions at the terminals. The fuzzified information is then processed by the inner nodes of the tree. These nodes perform the AND operations using Gaussian membership functions as described above. Finally this sequence of logic operations starting from the model input yield, the performance at the penultimate node outputs of the model. This means the more satisfied the elemental requirements at the terminal level are, the higher the outputs will be at the nodes above, finally increasing the design performance at the root node of the tree. Next to the evaluation of the design performance score, due to the fuzzy logic operations at the inner nodes of the tree, the performance of any sub-aspect is obtained as well. This is a desirable feature in design, which is referred to as transparency.
Having established the performance evaluation model, it is used for the evolutionary search process aiming to identify designs with maximal design performance. In the present case we are interested in a variety of alternative solutions that are equivalent in Pareto sense. The design is therefore treated as a multi-objective optimization as opposed to a single-objective optimization. In single-objective case exclusively the design performance, i.e. the output at the root node of the neural tree, would be subject to maximization. In the latter case, the solution would be the outcome of a mere convergence and any cognition aspect would not be exercised. In the multi-objective implementation the outputs of the nodes
3.2. Design performance and the Pareto front
It is noted that generally multi-objective optimization involves no information on the relative importance among the objectives. Therefore, generally, Pareto optimal solutions cannot be distinguished without bringing into play other, i.e. higher-order criteria than the objectives used in the search. However, it is noted that the Pareto solutions may be distinguished as follows. From figure 10, at the root node, the performance score is computed by the defuzzification process given by
where
In this design exercise, the cognitive design viewpoint plays important role. This means it is initially uncertain what values
Normalising the components and equating them to the weights yields
In general, if there are
Above computation implies that, the performance
Therefore, (36) is computed for all the design solutions on the Pareto front. Then the
where
To this end, to make the analysis explicit we consider a two-dimensional objective space. In this case, (36) becomes
which can be put into the form
that defines a circle along which the performance is constant. To obtain the circle parameters in terms of performance, we write
From (40) we obtain the center coordinates
The performance circle with the presence of three progressive Pareto fronts are schematically shown in figure 12. From this figure, it is seen that the maximum performance is at the locations where the either objective is maximal at the Pareto front. If both objectives are equal, the maximal performance takes its lowest value and the degree of departing from the equality means a better performance in Pareto sense. This result is very significant since it reveals that, a design can have a better performance if some measured extremity in one way or other is exercised. It is meant that, if a better performance is obtained, then most presumably extremity will be observed in this design.
3.3. Application results
The results from the design with multi-objective optimization are presented in figures 13, and 14-17. Figure 13 shows the maximum design performance
To exemplify the solutions on the Pareto front, four resulting Pareto-optimal designs D1-D4 are shown in figures 14-17, respectively. The left part of the figure shows the instantiation of the solutions in decision space. The right part of the figure shows the same solution in objective space together with the other Pareto optimal solutions obtained. It is noted that in objective space a solution is represented by a sphere. The size of the sphere indicates the maximal performance value of the corresponding solution. That is, a large sphere indicates a high maximal design performance, and conversely a small sphere indicates a low performance.
Design D1 is the design among the Pareto solutions having the highest maximal design performance, as obtained by (37), namely
Design D2 has a high energy performance (.83), while form and functionality are moderate (.54 and.41). Its maximal design performance is
Design D3 has a high form related performance (.81), while functionality is low (.26) and energy performance moderate (.68). Its maximal design performance is
Design D4 has a high
From the results we note that all four Pareto solutions investigated, although being located at different extremities of the front, have shared properties. Namely the hotel object O1 is always located at the street corner. This means that this feature is desirable for any optimal solution, revealing a principle applicable to this design task. This discovery is referred to as
From the results we note that design D1 has a maximal performance that is higher than the other Pareto optimal designs described by factor 1.15. That is, D1 clearly outperforms the others regarding their respective maximal performance. This means that when there is no a- priori bias for any of the three objectives, it is more proficient to be less concerned with energy, but to aim for maximal functionality and form qualities instead in the particular design task at hand. That is, in absence of second-order preferences, design D1 should be built, rather than the other designs.
4. Conclusions
A novel adaptive approach for formation of the Pareto front in multi-objective optimization is presented. The approach is an adaptive stochastic search, where a relaxed dominance concept is introduced, and the relaxation angle is adapted during the search. This approach is a dual counterpart of gradient-based stochastic adaptive algorithms. In this duality the fitnessfunction is the dual of stochastic gradient, and the degree of relaxation is the dual of the step-size parameter. The adaptation is found to be significantly favorable for convergence and diversity in the multi-objective optimization. This is demonstrated in an application from architectural design, where a building consisting of several volumes is obtained. In this design the location of the volumes relative to each other is subject to optimization. This is accomplished by identifying an optimal sequence of arranging the volumes, so that three objectives pertaining to the building are satisfied. These objectives are functionality, energy aspects, and form related aspects. The linguistic nature of these requirements is treated by using a fuzzy neural tree approach, that is able to handle the imprecision and complexity inherent to the concepts, forming a model. This model plays the role of fitness function in the adaptive MOEA, so that the search is endowed with some human like reasoning during the search. The designs obtained are diverse, and have a high design performance. This performance is the maximum suitability one can attribute to a solution without preference for any objectives. The location where this maximal performance is constant in objective space constitutes a circle for two objectives, and a sphere for three or more objectives etc. Using this location information and associating it with the Pareto front, it is shown that a solution satisfying each objective approximately equally yields a minimal performance among the Pareto solutions. A Pareto solution deviating from this equality is preferred. This situation in two dimensional cases can metaphorically be referred to the preference for the
This result implies that Q is an orthogonal matrix.
Appendix - Coordinate Transform
Let us consider the elements of an n-dimensional numerical vector
which lie along the axes of reference rectangular coordinates x1, x2, …., xn in n-dimensional space. The numerical vector
Let a new coordinate system in n-dimensional space is choisen whose geometrical unit vectors
which can be expressed as a matrix equation
From (1.3), one can write the set of equation in the following form.
The geometrical vector determined by the components of the numerical vector x can be expressed by the components of the numerical vector x’ as follows
Substitution of (1.4) into (1.5) yields
Comparison of (1.2) and (1.7) yields
The matrix equation form of (1.8) is given by
where Q is the transformation matrix which is
One should note that, Q is the transpose of the coefficient matrix of the equations in (1.4). In order that the new unit vectors be linearly independent, and hence span n-dimensional space, the matrix Q must be non-singular. In this case from (1.9) we can write
It may be interesting to note that, if the new unit vectors are mutually orthogonal, we obtain
References
- 1.
Bazaraa M. S. Sherali H. D. Shetty C. M. 1993 , John Wiley & Sons, Inc,0-47155-793-5 York - 2.
Branke J. Kaussler T. Schmeck H. 2000 Guiding multi-objective evolutionary algorithms towards interesting regions, Technical Report399 Institute AIFB University of Karlsruhe, Germany - 3.
Branke J. Kaussler T. Schmeck H. 2001 Guidance in evolutionary multi-objective optimization, ,32 499 507 - 4.
Ciftcioglu Ö. Bittermann M. S. 2008 Solution diversity in multi-objective optimization: A study in virtual reality, , Hong Kong, 2008, - 5.
Coello C. A. C. 1999 An updated survey of GA-based multi-objective optimization techniques, ,32 2 109 143 - 6.
Coello C. A. C. Veldhuizen D. A. Lamont G. B. 2003 , Kluwer Academic Publishers, Boston - 7.
Deb K. Pratap A. Agarwal S. Meyarivan T. 2000 A fast and elitist multi-objective genetic algorithm: NSGA-II,6 182 197 - 8.
Deb K. 2001 , John Wiley & Sons - 9.
Deb K. Zope P. Jain A. 2003 Distributed computing of Pareto-optimal solutions with evolutionary algorithms, ,532 549 , 978-1-59593-697-4 London, England, ACM New York - 10.
Deb K. Srinivasan A. 2006 Innovization: Innovating design principles through optimization, ,1629 1636 , Seattle, Washington, July 8-12, 2006 - 11.
Deb K. Sundar J. Bhaskara U. Chaudhuri S. 2006 Reference point based multi-objective optimization using evolutionary algorithm, ,2 3 273 286 - 12.
Eiben A. E. Hinterding R. Michaelwicz Z. 1999 Parameter control in evolutionary algorithms, ,3 2 124 141 - 13.
Farhang-Boroujeny B. 1998 , John Wiley & Sons, New York - 14.
Horn J. Nafploitis N. Goldberg D. E. 1994 A niched pareto genetic algorithm for multiobjective optimization, ,82 87 , 1994, IEEE Press - 15.
Hughes E. J. 2005 Evolutionary many-objective optimisation: Many once or one many? ,222 227 , Edinburgh, Scotland, 2005, IEEE Service Center - 16.
Jaszkiewicz A. 2004 On the computational efficiency of multiple objective metaheuristics: The knapsack problem case study, ,158 2 418 433 - 17.
Kuester J. 1973 , McGraw-Hill College, 0070356068 - 18.
Laumanns M. Thiele L. Deb K. Zitzler E. 2002 Combining convergence and diversity in evolutionary multiobjective optimization, ,10 3 262 282 - 19.
Mitra S. Pal S. K. Mitra P. 2002 Data mining in soft computing framework: A survey, ,13 1 3 14 - 20.
Sato H. Aguirre H. E. Tanaka K. 2007 Controlling dominance area of solutions and its impact on the performance of MOEAs, ,0302-9743 Springer Berlin/Heidelberg,5 20 , May - 21.
Yao X. Liu Y. Lin G. M. 1999 Evolutionary programming made faster, ,3 82 102 - 22.
Zitzler E. Thiele L. 1999 Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, ,3 4 257 271