Schaffer, J. D., 1984, Some experiments in machine learning using vector evaluated genetic algorithms. PhD dissertation, Vanderbilt University, Nashville, USA.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....GA to converge on the smaller subset of acceptable solutions will then be introduced. In the light of this, six different ranking methods will be described: three commonly used methods ( sum of weighted objectives , non dominated sorting , and weighted maximum ranking based on Schaffer s VEGA [11]) and three new, or less commonly used methods ( weighted average ranking , sum of weighted ratios , and sum of weighted global ratios ) This paper will then describe the application of these six multiobjective techniques to four established test functions, and will examine the previously ....

....non Pareto approaches and the Pareto approaches. Many examples of aggregation approaches exist, from simple weighting and summing [7,15] to the multiple attribute utility analysis (MAUA) of Horn and Nafpliotis [9] Of the non Pareto approaches, perhaps the most well known is Schaffer s VEGA [11,12], who (as identified by Fonseca [3] does not directly make use of the actual definition of Pareto optimality. Many other non Pareto methods have been proposed (e.g. by Linkens [5] Ryan [10] and Sun [14] Finally the Pareto based methods, proposed first by Goldberg [7] have been explored by ....

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Schaffer, J. D., 1984, Some experiments in machine learning using vector evaluated genetic algorithms. PhD dissertation, Vanderbilt University, Nashville, USA.

....making. This information provides to the judgement of a human decision maker with the trade offs to establish interactions between different criteria, hence simplifying the decision process to choose an acceptable range of solutions for a multicriteria problem. Implemented first by Schaffer [36,37], Fourman [38] and then by Kursawe, 39,40] and others, cooperative population searches (CPS) with criterion selection [41] was used to build the Pareto front in selected multicriteria problems. The central idea in CPS is to make a parallel single criterion search, where all members of the ....

J.D. Schaffer, Some experiments in machine learning using vector evaluated genetic algorithms, Doctoral dissertation, Department of Electrical Engineering, Vanderbilt University, 1984.

....with a population of points, so we expect that they can find the Pareto optimal front easily. In this paper, we present the NSGA [2] and we analyze it regarding the solution of MOPs. Moreover, we compare its results with those obtained by the multiobjective evolutionary algorithms (MOEAs) VEGA [3], NPGA [4] and MOGA [5] and the classical method of objective weighting refereed as [1] We compare its performances in the solution of two analytical test problems. Finally, we apply the NSGA to solve the TEAM problem 22 without taking into account the quench Manuscript received July 05, 2001; ....

....by Schaffer opened a new avenue of research in this field. The algorithm, called vector evaluated genetic algorithm (VEGA) performs the selection operation based on the objective switching rule, i.e. selection is done for each objective separately, filling equally portions of mating pool [3]. Afterwards, the matting pool is shuffled, and crossover and mutation are performed as usual. Fonseca and Fleming [5] proposed a Pareto based ranking procedure (MOGA) where the rank of an individual is equal the number of solutions found in the population where its corresponding decision vector ....

J. D. Schaffer, "Some experiments in machine learning using Vector Evaluated Genetic Algorithms," Ph.D. dissertation, Vanderbilt Univ., Nashville, TN, 1984.

....process. Moreover, if available, a decision maker may be interested in knowing alternate solutions. Since genetic algorithms (GAs) work with a population of points, a number of Pareto optimal solutions may be captured using GAs. An early GA application on multiobjective optimization by Schaffer (1984) opened a new avenue of research in this field. Though his algorithm, VEGA, gave encouraging results, it suffered from biasness towards some Pareto optimal solutions. A new algorithm, Nondominated Sorting Genetic Algorithm (NSGA) is presented in this paper based on Goldberg s suggestion (Goldberg ....

....one variable is considered to illustrate the concept of multiple Pareto optimality. This problem was used for the same purpose by Vincent and Grantham 5 Figure 1: Functions f 11 and f 12 are plotted versus x. Figure 2: The performance space of problem F1 is shown. 1981) and subsequently by Schaffer (1984). The problem has two objectives and is shown in figure 1 and figure 2: Minimize f 11 = x 2 , Minimize f 12 = x Gamma 2) 2 . 6) From the plot showing the performance space, it is clear that the Pareto optimal solutions constitute all x values varying from 0 to 2. The solution x = 0 is ....

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Schaffer, J. D. (1984). Some experiments in machine learning using vector evaluated genetic algorithms . Doctoral dissertation, Vanderbilt University, Electrical Engineering, Tennessee. (TCGA file No. 00314).

....genetic classification system. Sequential niching repeatedly runs a traditional GA, each time making sure that the population searches a new area of the space. Two previous learning systems that utilize implicit niching methods also deserve mention. In both systems (Greene Smith, 1993, 1994; Schaffer, 1984, 1985) the Financial Forecasting Using Genetic Algorithms 551 fitness function is decomposed into independent components, and different population elements are assigned to optimize each component. Fitness Function: Credit Assignment Most research on classification by GA has been for ....

Schaffer, J. D. 1984. Some experiments in machine learning using vector evaluated genetic algorithms. Doctoral dissertation. Vanderbilt University, Nashville, Tenn.

....methods applied to constrained optimization problems are equivalent to penalty approaches. It is also possible to experiment with evolutionary techniques for multi objective optimization, e.g. with Schaffer s VEGA (Vector Evaluated Genetic Algorithm) system for multi objective optimization (Schaffer 1984). The main idea behind the VEGA system was a division of the population into (equal sized) subpopulations; each subpopulation was responsible for a single objective. The selection procedure was performed independently for each objective, but crossover was performed across subpopulation ....

Schaffer, J.D. (1984). Some Experiments in Machine Learning Using Vector Evaluated Genetic Algorithms. Doctoral dissertation, Vanderbilt University.

....those of the goal programming approach, particularly for difficult problems with a strong focus on a narrow region. Test Problem 1 First we examine the algorithm s convergence properties using Schaffer s f2 Function. This is a very simple test function introduced by Schaffer in his dissertation [12]. Nevertheless nearly every multi objective EA is applied to it to test whether the population becomes and remains well distributed along the Pareto optimal front. In the tests reported here, no guiding was used (i.e. no transformation) 3 Any function could have served as boundary. We actually ....

J. D. Schaffer. Some experiments in machine learning using vector evaluated genetic algorithms. PhD thesis, Vanderbilt University, 1984.

....Pareto set: they are population based, require only objective function evaluations, use probabilistic transition rules which make then less prone to local optimum entrapment and allow for several types of parallel implementations. The first application of GAs to MOPs dates back to the mid eighties[1, 2] and numerous papers on the application of evolutionary techniques to MOPs have been published since then (see [3, 4, 5, 6] Although other techniques have been used for finding a suitable solution to MOPs we are interested here in those that try to approximate the Pareto set. The most popular ....

J.D. Schaffer. Some Experiments in Machine Learning using Vector Evaluated Genetic Algorithms. PhD thesis, Vanderbilt University, Nashville, TN, 1984.

....will enable researchers to test their algorithms for specific aspects of multi objective optimization. Keywords Genetic algorithms, multi objective optimization, niching, pareto optimality, problem difficulties, test problems. 1 Introduction After a decade since the pioneering work by Schaffer (1984), a number of studies on multiobjective genetic algorithms (GAs) have emerged. Most of these studies were motivated by a suggestion of a non dominated GA outlined in Goldberg (1989) The primary reason for these studies is a unique feature of GAs a population approach that is highly suitable ....

....champions, thereby causing an artificial bias towards some portion of the Pareto optimal region. In some multi objective optimization problems, the Pareto optimal front may not be continuous, instead it may be a collection of discretely spaced continuous sub regions (Poloni et al. in press; Schaffer, 1984). In such problems, although solutions within each sub region may be found, competition among these solutions may lead to extinction of some sub regions. It is also likely that the Pareto optimal front is not uniformly represented by feasible solutions. Some regions in the front may be ....

Schaffer, J. D. (1984). Some experiments in machine learning using vector evaluated genetic algorithms. Doctoral Dissertation, Vanderbilt University, Nashville, Tennessee.

.... As early as in 1967, Rosenberg suggested, but did not simulate, a genetic search method for finding the chemistry of a population of single celled organisms with multiple properties or objectives [28] However, the first practical implementation was suggested by David Schaffer in the year 1984 [29]. Thereafter, no significant study was performed for almost a decade, except a revolutionary 10 line sketch of a new nondominated sorting procedure outlined in David Goldberg s book [15] The book came out in the year 1989. Getting a clue for an efficient multi objective optimization technique, ....

....written till to date. A year wise count of those papers is plotted in Figure 2, which shows an exponential growth in interest in the field over the past few years. We now present a brief summary of a few salient evolutionary multi objective optimization algorithms. 3. 1 Schaffer s VEGA Schaffer [29] modified the simple tripartite genetic algorithm by performing independent selection cycles according to each objective. He modified the public domain GENESIS software by creating a loop around the traditional selection procedure so that the selection method is repeated for each individual ....

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Schaffer, J. D. (1984). Some experiments in machine learning using vector evaluated genetic algorithms. (Doctoral Dissertation). Nashville, TN: Vanderbilt University.

....During the random generation of each new individual, the input parameters are allowed to take on any one of a pre determined set of values with equal probability. A full set of input parameters defines a variant, or individual, and the evaluation of a set of parameters determines 11 Schaffer [76, 77] was apparently the first to adapt a GA to search for Pareto optima in 1984. 90 establish initial population evaluation parent selection crossover mutation replacement satisfactory solution stop reproduction YES NO Figure 4 1: Flow diagram for a typical genetic algorithm. its ....

J. D. Schaffer. Some Experiments in Machine Learning Using Vector Evaluated Genetic Algorithms. PhD thesis, Vanderbilt University, December 1984.

....size equal to that used in the other EAs described below. The values of the various algorithm parameters are shown in Table 2. A mutation again corresponds to flipping a single bit. The first multi criteria EA alternative that we consider is the Vector Evaluated Genetic Algorithm (VEGA) (Schaffer, 1984; Schaffer, 1985) In VEGA, the population is divided into subpopulations, one associated with each of the criteria. In each subpopulation, the corresponding criterion is used as the fitness function. VEGA was used early on with some success in Pareto optimization and became well known as a ....

Schaffer, J. (1984). Some experiments in machine learning using vector evaluated genetic algorithms. PhD thesis, Vanderbilt University.

....A good parallel efficiency for this kind of an approach has been demonstrated [13] 15] For multiobjective optimization problems, it is necessary to make some modifications to the basic GA. There exist several variants of GAs for this kind of problem; see for example Vector Evaluated GA (VEGA) [18] and Nondominated Sorting GA (NSGA) 19] For further information on GAs for multiobjective optimization, see Reference [5] and references therein. In the following, we describe the basic ideas of NSGA. The fitness values are computed using the following procedure: ALGORITHM: Nondominated ....

J. D. Schaffer, Some experiments in machine learning using vector evaluated genetic algorithms, TCGA file no. 00314, PhD thesis, Vanderbilt University, Nashville, TN, 1984.

....by Ritzel et al. 1994) in which a genetic algorithm was used to solve for reliable and inexpensive solutions to a groundwater pollution containment problem. This paper compared two variations of a multiobjective GA: the vector evaluated GA (VEGA) and a Pareto GA. The VEGA approach, developed by Schaffer (1984), differs from a simple GA in that it changes the selection operation to ensure that solutions of each of the objectives are represented in the mating pool. This is accomplished by selecting members from the surviving population that have high fitness for each of the objectives. In essence, this ....

Schaffer, J.D. Some Experiments in Machine Learning Using Vector Evaluated Genetic Algorithms, Ph.D. Dissertation, Vanderbilt University, Nashville, Tenn., 1984.

....Because GAs operate on a whole population of points, they allow to nd a number of optimal solutions of the Pareto set. A rst application of GAs on multi objective optimization problems has been studied by Schaoeer in 1984 (see his experiments, called Vector Evaluated GA (VEGA) in his dissertation [15]) Schaoeer s objective was to minimize a cost and maximize a reliability. He distributed the population in two half populations and optimized the two objectives on each half. After a lot of generations, the population converged towards the optima of each sub region. This method gave a new avenue ....

J.D. SCHAFFER, "Some experiments in machine learning using Vector Evaluated Genetic Algorithms", Unpublished doctoral dissertation, Vanderbilt University, Nashville (TN), 1984.

....last. Distortion minimization must be balanced with size minimization to avoid overfitting; a minimal distortion is not always desirable, since it spells doom for generalization. Some types of vectorial fitness have already been described in the literature, mainly for multiobjective optimization [19, 20, 21]. In the case of Shaw and coauthors, vectorial fitness is also used to account for constraints (lateness of orders, for instance) in scheduling problems, and uses Pareto optimization to select the best candidates, which are then presented to a human operator. This approach is somewhat different to ....

J. D. Schaffer. Some experiments in machine learning using vector evaluated genetic algorithms. PhD thesis, Dept. of Electrical Engineering, Vanderbilt Univ, 1985.

....closely as does GA, it does offer the opportunity to directly evolve programs of unusual complexity, without having to define the structure or the size of the program or genetic material in advance. Others have described GA approaches that operate on variable length or program structured genomes [5,14]. These approaches typically require more constraints on the form of the final solution than does GP. The work described here uses GP as defined by Koza in [9, 11] as well as a further elaboration of GP developed by Craig Reynolds called Steady State GP (SSGP) 13] Reynolds developed SSGP through ....

J. D. Schaffer, Some Experiments in Machine Learning Using Vector Evaluated Genetic Algorithms. Doctoral Dissertation, Department of Electrical and Biomedical Engineering, Vanderbilt University, Nashville TN, 1984.

....material, and operate on this material with fitness proportionate selection, reproduction, crossover, and mutation [De Jong 1987, Goldberg 1989] A number of researchers have experimented with GA s applied to variable length genetic representations. Grefenstette [Grefenstette 1989] and Schaffer [Schaffer 1984] contain notable examples. Bickel has applied GA s to tree structured genetic material [Bickel 1987] John Koza has developed Genetic Programming (GP) using analogues of GA genetic operators to directly modify and evolve tree structured programs (typically LISP functions) Koza 1989, 1990, 1992] ....

Schaffer, J. D. (1984) Some Experiments in Machine Learning Using Vector Evaluated Genetic Algorithms.

....0.16 37.19 0.21 5769.41 15.22 Table 2.3: Pareto (PAR) random lexicographic (LEX) and weighted sum (WEI) optimisations, average over 50 runs. Here 0 e 0:005 and s(y) is the standard deviation of y. 2. 5 VEGA VEGA stands for Vector Evaluated Genetic Algorithm and it was developed by Schaffer [14, 3]. The basic algorithm is presented in Figure 2.4. ae oe ae oe ae oe ae oe ae oe ae oe ae oe ae oe POPULATION R XXXX Xz i i i i i1 Gamma Gamma Gamma Gamma Gamma . ....

J. David Schaffer. Some Experiments in Machine Learning using Vector Evaluated Genetic Algorithm. PhD thesis, Vanderbilt University, Nashville, 1984. TCGA file No. 00314.

....units. Since the rule ordering within a plan is irrelevant, the process of recombination can be viewed as simply selecting rules from each parent to create an offspring plan. Many genetic algorithms permit recombination within individual rules as a way of creating new rules (Smith, 1980; Schaffer, 1984; Holland, 1986) While such operators are easily defined for SAMUEL s rule language (Grefenstette, 1989) we prefer to use CROSSOVER solely to explore the space of rule combinations, and leave rule modification to other operators (i.e. SPECIALIZE, GENERALIZE,CREEP, MERGE and MUTATION) In ....

Schaffer, J. D. (1984). Some experiments in machine learning using vector evaluated genetic algorithms, Doctoral dissertation, Department of Electrical and Biomedical Engineering, Vanderbilt University, Nashville.

.... As early as in 1967, Rosenberg suggested, but did not simulate, a genetic search method for finding the chemistry of a population of single celled organisms with multiple properties or objectives [35] However, the first practical implementation was suggested by David Schaffer in the year 1984 [36]. Thereafter, no significant study was performed for almost a decade, except a revolutionary 10 line sketch of a new non dominated sorting procedure outlined in David Goldberg s book [20] The book came out in the year 1989. Getting a clue for an efficient multi objective optimization technique, ....

....studies in the field from different perspectives [18, 25, 42, 3] These reviews list many different techniques of multi criterion optimization that exist to date. We now present a brief summary of a few salient evolutionary multicriterion optimization algorithms. 1.4. 1 Schaffer s VEGA Schaffer [36] modified the simple tripartite genetic algorithm by performing independent selection cycles according to each objective. He modified the public domain GENESIS software by creating a loop around the traditional selection procedure so that the selection method is repeated for each individual ....

[Article contains additional citation context not shown here]

Schaffer, J. D. (1984). Some experiments in machine learning using vector evaluated genetic algorithms. (Doctoral Dissertation). Nashville, TN: Vanderbilt University.

....having other difficult and interesting problem features. More importantly, these difficult test problems will enable researchers to test their algorithms for specific aspects of multiobjective optimization in the coming years. 1 Introduction After about a decade since the pioneering work by Schaffer (1984; 1985) a number of studies on multiobjective genetic algorithms (GAs) have been pursued since the year 1994, although most of these studies took a hint from Goldberg (1989) The primary reason for these studies is a unique feature of GAs population approach that make them highly suitable to ....

....with individual champions, thereby causing an artificial bias towards some portion of the Paretooptimal region. In some multi objective optimization problems, the Pareto optimal front may not be continuous, instead it is a set of discretely spaced continuous sub regions (Poloni et al. in press; Schaffer, 1984). In such problems, although solutions within each sub region may be found, competition among these solutions may lead to extinction of some sub regions. It is also likely that the Pareto optimal front is not uniformly represented by feasible solutions. Some regions in the front may be represented ....

Schaffer, J. D. (1984). Some experiments in machine learning using vector evaluated genetic algorithms.

....frontier 2 . The goal of a Pareto GA is to find a representative sampling of solutions all along the Pareto front. II. Previous Work We assume the reader is familiar with the simple GA [3] Here we review previous approaches to multiobjective optimization with GAs. In his 1984 dissertation [10], and later in [11] Schaffer proposed his Vector Evaluated GA (VEGA) for finding multiple solutions to multiobjective (vector valued) problems. He created VEGA to find and maintain multiple classification rules in a set covering problem. VEGA tried to achieve this goal by selecting a fraction of ....

....on the front, and only one or two at each end point of the front, it should be harder to maintain equal size subpopulations at the extreme points. B. Problem 2: Schaffer s F2 Next we compare our algorithm to Schaffer s VEGA by running it on one of the test functions from Schaffer s dissertation [10]. This is the simple function F2, with a single decision variable, the real valued x, and two attributes, f21 andf22 to be minimized: f21(x) x 2 f22(x) x Gamma 2) 2 The decision variable is mapped to a 14 bit string as a binary coded integer. Thus 00000000000000 = xmin = Gamma6:00 ....

Schaffer, J. D., (1984). Some experiments in machine learning using vector evaluated genetic algorithms, Unpublished doctoral dissertation, Vanderbilt University.

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Schaffer, J. 1984, Some experiments in machine learning using vector evaluated genetic algorithms, PhD Thesis, Vanderbilt University, Nashville.

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Schaffer, J. D. (1984). "Some Experiments in Machine Learning Using Vector Evaluated Genetic Algorithms".

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