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...one interested in this area who has a previous (at least basic) knowledge of genetic algorithms in general. Those who may need additional information about genetic algorithms should refer to Goldberg =-=[27]-=-, Holland [35], Michalewicz [54], and Mitchell [56] for more information. 2 Statement of the Problem Multiobjective optimization (also called multicriteria optimization, multiperformance or vector opt...

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...ompressor blade shapes. -- Rodr'iguez V'azquez et al. [78] extended MOGA to use it in genetic programming, introducing the so-called MOGP (Multiple Objective Genetic Programming). Genetic programming =-=[44]-=- replaces the traditional linear chromosomic representation by a hierarchical tree representation that, by definition, is more powerful, but also requires larger population sizes and specialized opera...

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...has a previous (at least basic) knowledge of genetic algorithms in general. Those who may need additional information about genetic algorithms should refer to Goldberg [27], Holland [35], Michalewicz =-=[54]-=-, and Mitchell [56] for more information. 2 Statement of the Problem Multiobjective optimization (also called multicriteria optimization, multiperformance or vector optimization) can be defined as the...

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...least basic) knowledge of genetic algorithms in general. Those who may need additional information about genetic algorithms should refer to Goldberg [27], Holland [35], Michalewicz [54], and Mitchell =-=[56]-=- for more information. 2 Statement of the Problem Multiobjective optimization (also called multicriteria optimization, multiperformance or vector optimization) can be defined as the problem of finding...

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..., known as a Nash equilibrium solution, represents a stable equilibrium condition in the sense that no player can deviate unilaterally from this point for further improvement of his/her own criterion =-=[57]-=-. This point has the characteristic that f 1 (x 1 ; x 2 )sf 1 (x 1 ; x 2 ) (25) and f 2 (x 1 ; x 2 )sf 2 (x 1 ; x 2 ) (26) where x 1 may be to the left or right of x 1 in (25) and x 2 may lie above or...

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... in this area who has a previous (at least basic) knowledge of genetic algorithms in general. Those who may need additional information about genetic algorithms should refer to Goldberg [27], Holland =-=[35]-=-, Michalewicz [54], and Mitchell [56] for more information. 2 Statement of the Problem Multiobjective optimization (also called multicriteria optimization, multiperformance or vector optimization) can...

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...the population is suitably ranked. Goldberg also suggested the use of some kind of niching technique to keep the GA from converging to a single point on the front. A niching mechanism such as sharing =-=[29]-=- would allow the GA to maintain individuals all along the non-dominated frontier. Applications -- Hilliard et al. [34] used a Pareto optimality ranking method to handle the objectives of minimizing co...

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...le to the shape or continuity of the Pareto front, whereas these two issues are a serious concern for mathematical programming techniques. 5.1 Multiple Objective Genetic Algorithm Fonseca and Fleming =-=[17]-=- have proposed a scheme in which the rank of a certain individual corresponds to the number of chromosomes in the current population by which it is dominated. Consider, for example, an individual x i ...

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...cessfully their approach with the two multiobjective optimization problems provided in the paper by Srinivas and Deb 5 P'eriaux et al. did not succeed at that in the example presented in their paper. =-=[86]-=-, but no further applications of this technique seem to be available at the moment. Strengths and Weaknesses The use of genders is really another way of defining separate subpopulations for each objec...

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...generation of non-inferior solutions for multiobjective optimization. This is an obvious consequence of the fact that it was implied by Kuhn and Tucker in their seminal work on numerical optimization =-=[45]-=-. The main strength of this method is its efficiency (computationally speaking), and can be applied to generate a strongly non-dominated solution that can be used as an initial solution for other tech...

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...etic operators Generation (t+1) Start all over again Fig. 2. Schematic of VEGA selection. It is assumed that the population size is N and that there are M objective functions. 4.1 VEGA David Schaffer =-=[83]-=- extended Grefenstette's GENESIS program [31] to include multiple objective functions. Schaffer's approach was to use an extension of the Simple Genetic Algorithm (SGA) that he called the Vector Evalu...

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...s is that it is more inefficient (both computationally and in terms of quality of the Pareto fronts produced) than MOGA, and more sentitive to the value of the sharing factor oe share . Other authors =-=[106, 105] report th-=-at the NSGA performed quite well in terms of "coverage" of the Pareto front (i.e., it spreads in a more uniform way the population over the Pareto front) when applied to the 0/1 knapsack pro...

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...will satisfy the m inequality constraints: g i (��x)s0 i = 1; 2; : : : ; m (1) 2 Right after the submission of this paper, David A. Van Veldhuizen and Gary B. Lamont made available a technical rep=-=ort [99] tha-=-t contains another remarkable survey of the area that complements the material contained in this paper. the p equality constraints h i (��x) = 0 i = 1; 2; : : : ; p (2) and optimizes the vector fu...

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... to total weight, asymptotical stability and eigenvalues constraints. -- Yang and Gen [104] used a weighted sum approach to solve a bicriteria linear transportation problem. More recently, Gen et al. =-=[25, 26]-=- extended this approach to allow more than two objectives, and added fuzzy logic to handle the uncertainty involved in the decision making process. A weighted sum is still used in this approach, but i...

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... of oe share becomes another parameter with which the user has to experiment until a reasonable setting is found. Even when important work has been done in this area (see for example Deb and Goldberg =-=[15]-=- and Fonseca & Fleming [17]), most of that research is focused on single-objective optimization, or multimodal optimization. -- Some researchers have also found alternative applications of multiobject...

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... into an unconstrained multiobjective optimization problem, which is solved using Fonseca's MOGA [17]. This approach was used by Surry et al. to optimize gas supply networks [89]. Fonseca and Fleming =-=[19]-=- also proposed to handle constraints as objectives, and applied their approach to the design of a gas turbine [20]. Parmee and Purchase [67] implemented a version of VEGA [83] to handle constraints re...

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...s is that it is more inefficient (both computationally and in terms of quality of the Pareto fronts produced) than MOGA, and more sentitive to the value of the sharing factor oe share . Other authors =-=[106, 105] report th-=-at the NSGA performed quite well in terms of "coverage" of the Pareto front (i.e., it spreads in a more uniform way the population over the Pareto front) when applied to the 0/1 knapsack pro...

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...heuristic mutation that basically defined rules to exchange bit positions had to be used to avoid premature convergence of the population. Strenghts and Weaknesses It has been cited in the literature =-=[87, 14]-=- that the main weakness of MOGA is that it performs sharing on the objective value space, which implies that two different vectors with the same objective function values can not exist simultaneously ...

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...ated individuals to maintain diversity in the population. Strengths and Weaknesses Although Schaffer reported some success, and the main strength of this approach is its simplicity, Richardson et al. =-=[76]-=- noted that the shuffling and merging of all the sub-populations corresponds to averaging the fitness components associated with each of the objectives. Since Schaffer used proportional fitness assign...

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...nsformed into a single objective optimization problem by taking a linear combination of these 2 objectives. Applications -- Valenzuela-Rend'on and Uresti-Charre [96] obtained better results than NPGA =-=[36]-=- (see below) in 3 biobjective optimization problems, both in terms of the number of points in the Pareto front at the final iteration, and in terms of the total number of function evaluations. However...

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...n another version of this algorithm (that apparently worked quite well), an objective was randomly selected at each run. Fourman used this approach to design compact symbolic layouts [24]. -- Kursawe =-=[47]-=- formulated a multiobjective version of evolution strategies [84] (ESs) based on lexicographic ordering. Selection consisted of as many steps as objective functions had the problem. At each step, one ...

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... superior to a VEGA by Liepins et al. [48] when applied to a variety of set covering problems. -- Ritzel et al. [77] also used non-dominated ranking and selection combined with deterministic crowding =-=[53]-=- as the niching mechanism. They applied the GA to a groundwater pollution containment problem in which cost and reliability were the objectives. Though the actual Pareto front was unknown, Ritzel et a...

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...ome the difficulties involved in the aggregating approaches, much work has been devoted to the development of alternative techniques based on population policies or special handling of the objectives =-=[70]-=-. Some of the most popular approaches that fall into this category will be examined in this section. Individual 1 Individual 2 Individual N Individual 3 Sub-population 1 lation 2 Sub-popuSub -populati...

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...e used, unless there is some previous knowledge of the points which lie in the Pareto front (in which case there is obviously no need for a multiobjective optimization technique). Fonseca and Fleming =-=[23]-=- proposed the definition of certain (arbitrary) goals that we wish the GA to attain; then we can perform multiple runs and apply standard non-parametric statistical procedures to evaluate the quality ...

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... which satisfies the equations of Steps 1 and 2 may be called the best compromise solution considering all the criteria simultaneously and on equal terms of importance. Applications -- Hajela and Lin =-=[33]-=- included the weights of each objective in the chromosome, and promoted their diversity in the population through fitness sharing. Their goal was to be able to simultaneously generate a family of Pare...

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...ormulation of the goal programming objective function is a weighted sum of the pth power of the deviation jf i (��x) \Gamma T i j [32]. Such a formulation has been called generalized goal programm=-=ing [37, 38]. This tec-=-hnique has also been called "target vector optimization" by other authors [12]. Applications -- Wienke et al. [102] used this approach in combination with a genetic algorithm to optimize sim...

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...it has the disadvantage of missing concave portions of the trade-off curve (in other words, the approach does not generate proper Pareto optimal solutions in the presence of non-convex search spaces) =-=[77]-=-, which is a serious drawback in most real-world applications. 3.2 Goal Programming Charnes and Cooper [5] and Ijiri [39] are credited with the development of the goal programming method for a linear ...

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...s0; j = 1; 2; : : : ; m (23) f l (��x) = f l ; l = 1; 2; : : : ; i \Gamma 1 (24) The solution obtained at the end, i.e., x k is taken as the desired solution x of the problem. Applications -- Four=-=man [24]-=- suggested a selection scheme based on lexicographic ordering. In a first version of his algorithm, objectives were assigned different priorities by the user and each pair of individuals were compared...

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...ions. The Pareto front is marked with a bold line. 3 Approaches That Use Aggregating Functions The notion of genetic search in a multicriteria problem dates back to the late 60s, in which Rosenberg's =-=[80]-=- study contained a suggestion that would have led to multicriteria optimization if he had carried it out as presented. His suggestion was to use multiple properties (nearness to some specified chemica...

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...eakness is its dependence on the value of oe share , but the idea of using a utility function that is dynamically modified, as in this case, has also been exploited more recently by other researchers =-=[96, 4, 30]-=-. 4.7 Use of the Contact Theorem to Detect Pareto Optimal Solutions Osyczka and Kundu [62] proposed the use of an algorithm based on the contact theorem (one of the main theorems in multiobjective opt...

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...is case, the vector function is normalized to the form �� f(��x) = [ �� f 1 (��x); �� f 2 (��x); : : : ; �� f k (��x)] T , where �� f i (��x) = f i (��x)=f =-=0 i . Applications -- Syswerda and Palmucci [90]-=- used weights in their fitness function to add or subtract values during the schedule evaluation of a resource scheduler, depending on the existence or absence of penalties (constraints violated). -- ...

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...opulation size, crossover and mutation rates, niche sizes, and elitism) and the way in which the selection technique adopted affects the performance of an algorithm. In this direction, Gunter Rudolph =-=[81]-=- has recently showed that theoretical results of convergence derived from single-objective evolutionary optimization cannot be used in the presence of multiple objectives. In his study, Rudolph propos...

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...considerable volume of research in evolutionary multiobjective optimization in the last 15 years, there have been only two surveys of this area published in the technical literature 2 : Tamaki et al. =-=[91]-=-, which is a very short and quick review of some of the main approaches, and Fonseca and Fleming [18, 21] which is a remarkable account of the issues that make this problem interesting from the evolut...

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...election ratio was defined as the ratio of the fraction of strings selected on the basis of the first objective (reliability) to the fraction selected via the second objective (cost). -- Surry et al. =-=[89]-=- proposed an interesting application of VEGA to model constraints in a single-objective optimization problem to avoid the need of a penalty function. Surry et al., however, modified the standard proce...

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...the deviation jf i (��x) \Gamma T i j [32]. Such a formulation has been called generalized goal programming [37, 38]. This technique has also been called "target vector optimization" by =-=other authors [12]-=-. Applications -- Wienke et al. [102] used this approach in combination with a genetic algorithm to optimize simultaneously the intensities of six atomic emission lines of trace elements in alumina po...

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...t is dynamically modified, as in this case, has also been exploited more recently by other researchers [96, 4, 30]. 4.7 Use of the Contact Theorem to Detect Pareto Optimal Solutions Osyczka and Kundu =-=[62]-=- proposed the use of an algorithm based on the contact theorem (one of the main theorems in multiobjective optimization [49]) to determine relative distances of a solution vector with respect to the P...

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...proach (i.e., the use of weights) and ranking techniques in which the level of preference may be defined. Greenwood et al. [30] used an approach called specified multi-attribute value theory (ISMAUT) =-=[101]-=- which, combined with a GA, allows the definition of preferences by the GA itself, rather than asking the intervention of the decision maker. However, the decision maker still gets to decide what part...

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...it against other similar techniques. However, these arbitrary goals are not easy to define either. Other (similar) metrics have been proposed in the literature. For example, Van Veldhuizen and Lamont =-=[98]-=- proposed the so-called generational distance, which is a measure of how close is our current Pareto front from the real Pareto front (assuming we know where it lies). Zitzler and Thiele [105] propose...

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...e proper Pareto optimal solutions in the presence of non-convex search spaces) [77], which is a serious drawback in most real-world applications. 3.2 Goal Programming Charnes and Cooper [5] and Ijiri =-=[39]-=- are credited with the development of the goal programming method for a linear model, and have played a key role in applying it to industrial problems. In this method, the decision maker has to assign...

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..., was taken from game theory, which deals with solving conflicting situations. The min-max approach to a linear model was proposed by Jutler [43] and Solich [85], and was further developed by Osyczka =-=[59, 60, 64]-=-, Rao [73] and Tseng and Lu [95]. The min-max optimum compares relative deviations from the separately attainable minima. Consider the ith objective function for which the relative deviation can be ca...

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...for more information. 2 Statement of the Problem Multiobjective optimization (also called multicriteria optimization, multiperformance or vector optimization) can be defined as the problem of finding =-=[65]-=-: a vector of decision variables which satisfies constraints and optimizes a vector function whose elements represent the objective functions. These functions form a mathematical description of perfor...

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.... to optimize gas supply networks [89]. Fonseca and Fleming [19] also proposed to handle constraints as objectives, and applied their approach to the design of a gas turbine [20]. Parmee and Purchase =-=[67]-=- implemented a version of VEGA [83] to handle constraints relating to a gas turbine design problem as objectives to allow the GA to locate a feasible region within the highly constrained search space ...

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...ould modify the way in which mating was performed. The idea was to model the sexual attraction that some individuals have over others in nature, which determines a not so random mating. Lis and Eiben =-=[50]-=- also incorporated gender in their GA, but in a more general sense. In this case, the number of genders (or sexes), was not limited to two, but it could be as many as objectives we had. Another distin...

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...cular set x 1 ; x 2 ; : : : ; x k which yields the optimum values of all the objective functions. 2.1 Pareto Optimum The concept of Pareto optimum was formulated by Vilfredo Pareto in the XIX century =-=[66], and constitute-=-s by itself the origin of research in multiobjective optimization. We say that a point �� x 2 F is Pareto optimal if for every �� x 2 F either, i 2 I (f i (��x) = f i (��x )) (4) or, t...

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...for the GA. Thus, through a process of running the GA numerous times with different values of the constrained objectives, a trade-off surface can be developed. Applications -- Quagliarella and Vicini =-=[71]-=- suggested the use of this technique coupled with a hybrid GA (a genetic algorithm that used gradient based optimization techniques to speed up the search in order to reduce the computational cost req...

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...eakness is its dependence on the value of oe share , but the idea of using a utility function that is dynamically modified, as in this case, has also been exploited more recently by other researchers =-=[96, 4, 30]-=-. 4.7 Use of the Contact Theorem to Detect Pareto Optimal Solutions Osyczka and Kundu [62] proposed the use of an algorithm based on the contact theorem (one of the main theorems in multiobjective opt...

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...s et al. [42] used weights for their genetic operators in order to reflect their effectiveness when a GA was applied to generate hyperstructures from a set of chemical structures. -- Wilson & Macleod =-=[103]-=- used this approach as one of the methods incorporated into a GA to design multiplierless IIR filters in which the two conflicting objectives were to minimize the response error and the implementation...

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... in their fitness function to add or subtract values during the schedule evaluation of a resource scheduler, depending on the existence or absence of penalties (constraints violated). -- Jakob et al. =-=[41]-=- used a weighted sum of the several objectives involved in a task planning problem : to move the tool center point of an industrial robot to a given location as precisely and quickly as possible, avoi...

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... and best-N selection (the best N individuals are selected for the next generation among N parents and N children) for the aerodynamic design of compressor blade shapes. -- Rodr'iguez V'azquez et al. =-=[78]-=- extended MOGA to use it in genetic programming, introducing the so-called MOGP (Multiple Objective Genetic Programming). Genetic programming [44] replaces the traditional linear chromosomic represent...

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...y in a scheduling problem. They tentatively concluded that the Pareto optimality ranking method outperformed the VEGA method. -- The Pareto method was found to be superior to a VEGA by Liepins et al. =-=[48]-=- when applied to a variety of set covering problems. -- Ritzel et al. [77] also used non-dominated ranking and selection combined with deterministic crowding [53] as the niching mechanism. They applie...

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...nd the decision variable domains, leading to what they called nested sharing. Applications -- Belegundu et al. [3] used the NPGA for the design of laminated ceramic composites. -- Poloni and Pediroda =-=[69]-=- used it for the design of a multipoint airfoil that has its minimum drag at two given lift values with a constraint in the maximum allowed pitching moment. -- A variation of the NPGA was proposed by ...

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... min-max tournament selection. Applications -- Coello and Christiansen applied these two approaches to the optimization of I-beams [8] and manufacturing problems [9], and to the design of a robot arm =-=[10]-=-. Strengths and Weaknesses The main strength of this technique is its efficiency and relative simplicity. The use of weights is its main weakness because it is not always easy to find an appropriate s...

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...ey had attempted before [22]. -- Aherne et al. [1] used MOGA to optimize the selection of parameters for an object recognition scheme called the Pairwise Geometric Histogram paradigm. -- Todd and Sen =-=[94]-=- used a variant of MOGA for the preplanning of containership layouts (a large scale combinatorial problem). In Todd and Sen's approach, a population of non-dominated individuals is kept and updated at...

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... the population, it may be desirable in some cases to devise ways of reducing the number of elements in such set, in order to facilitate the analysis for the decision maker. Kunha, Oliveira and Covas =-=[46]-=- proposed the incorporation of Roseman and Gero's algorithm [79] into the GA to cluster together points that are within a certain distance (defined by the user) of each other in the Pareto front. -- P...

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...native that the authors reported to work better (in terms of representation power) than the use of the conventional linear representation of MOGA that they had attempted before [22]. -- Aherne et al. =-=[1]-=- used MOGA to optimize the selection of parameters for an object recognition scheme called the Pairwise Geometric Histogram paradigm. -- Todd and Sen [94] used a variant of MOGA for the preplanning of...

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...hich the Pareto front can actually be found, although a cooperative game may be preferred in that case over a non-cooperative approach [75, 74]. 4.4 Using Gender to Identify Objectives Robin Allenson =-=[2]-=- used a population-based approach modelled after VEGA in which gender was used to distinguish between the two objective functions of a problem consisting of the planning of a route composed of a numbe...

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... in a real-world application) to solve multiobjective optimization problems. -- Ranjithan et al. [72] used this approach to solve groundwater pollution containment problems. -- Loughlin and Ranjithan =-=[52]-=- used a variation of this technique in which they incorporated target satisfaction levels (similar to those used in Goal Programming), and combined it with a neighboorhood selection procedure accordin...

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...ame theory, which deals with solving conflicting situations. The min-max approach to a linear model was proposed by Jutler [43] and Solich [85], and was further developed by Osyczka [59, 60, 64], Rao =-=[73]-=- and Tseng and Lu [95]. The min-max optimum compares relative deviations from the separately attainable minima. Consider the ith objective function for which the relative deviation can be calculated f...

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...tion pressure introduced by the use of min-max tournament selection. Applications -- Coello and Christiansen applied these two approaches to the optimization of I-beams [8] and manufacturing problems =-=[9]-=-, and to the design of a robot arm [10]. Strengths and Weaknesses The main strength of this technique is its efficiency and relative simplicity. The use of weights is its main weakness because it is n...

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...m as additional constraints. The objective function will then try to minimize the absolute deviations from the targets to the objectives. The simplest form of this method may be formulated as follows =-=[16]: min k X i=-=-1 jf i (��x) \Gamma T i j ; subject to �� x 2 F (9) where T i denotes the target or goal set by the decision maker for the ith objective function f i (��x), and F represents the feasible r...

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...tion according to Pareto non-domination was superior to both VEGA and non-domination with deterministic crowding, at least for finding points near or on the front found by MICCP. -- Stanley and Mudge =-=[88]-=- implemented Goldberg's Pareto ranking technique to solve a microprocessor design problem in which the constraints were handled as additional objectives. Strengths and Weaknesses The main weakness of ...

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...] reported success in the use of MOGA for the multiobjective optimization of ULTIC controllers that satisfy a number of time domain and frequency domain specifications. Also, Chipperfield and Fleming =-=[7]-=- reported success in using MOGA for the design of a multivariable control system for a gas turbine engine. -- Obayashi [58] used Pareto ranking with phenotypic sharing and best-N selection (the best N...

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...eakness is its dependence on the value of oe share , but the idea of using a utility function that is dynamically modified, as in this case, has also been exploited more recently by other researchers =-=[96, 4, 30]-=-. 4.7 Use of the Contact Theorem to Detect Pareto Optimal Solutions Osyczka and Kundu [62] proposed the use of an algorithm based on the contact theorem (one of the main theorems in multiobjective opt...

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...s between target values and actually achieved values. A more general formulation of the goal programming objective function is a weighted sum of the pth power of the deviation jf i (��x) \Gamma T =-=i j [32]. Such a f-=-ormulation has been called generalized goal programming [37, 38]. This technique has also been called "target vector optimization" by other authors [12]. Applications -- Wienke et al. [102] ...

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...me domain and frequency domain specifications. Also, Chipperfield and Fleming [7] reported success in using MOGA for the design of a multivariable control system for a gas turbine engine. -- Obayashi =-=[58]-=- used Pareto ranking with phenotypic sharing and best-N selection (the best N individuals are selected for the next generation among N parents and N children) for the aerodynamic design of compressor ...

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...s w 1 ; w 2 ; : : : ; w k are normalized so that k X i=1 jw i j = 1 (12) If some w i = 0 (i = 1; 2; : : : ; k), it means that the maximum limit of objectivessf i (��x) is b i . It can be easily sh=-=own [6]-=- that the set of non-dominated solutions for a problem can be generated by varying the weights, with w is0 (i = 1; 2; : : : ; k) even for nonconvex problems. It should be pointed out that the optimum ...

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...ubject to further experimenting. Its main weakness is its simplicity and efficiency, because it does not require to check for non-dominance. 4.6 Two Variations of the Weighted Min-Max Strategy Coello =-=[12, 11]-=- proposed two variations of the weighted min-max strategy used by Hajela and Lin. In his first approach, the decision maker has to provide a predefined set of weights that will be used to spawn severa...

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...structures, as an alternative that the authors reported to work better (in terms of representation power) than the use of the conventional linear representation of MOGA that they had attempted before =-=[22]-=-. -- Aherne et al. [1] used MOGA to optimize the selection of parameters for an object recognition scheme called the Pairwise Geometric Histogram paradigm. -- Todd and Sen [94] used a variant of MOGA ...

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... center point of an industrial robot to a given location as precisely and quickly as possible, avoiding certain obstacles and aiming to produce a path as smooth and short as possible. -- Jones et al. =-=[42]-=- used weights for their genetic operators in order to reflect their effectiveness when a GA was applied to generate hyperstructures from a set of chemical structures. -- Wilson & Macleod [103] used th...

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... al. used fairly large population sizes (above 1000 individuals), the counter-effect of tournament selection may had been absorbed by the extra individuals in the population. -- Michielssen and Weile =-=[55]-=- used also the NSGA to design an electromagnetic system. Strengths and Weaknesses The main strengths of this technique is that can handle any number of objectives, and that does sharing iin the parame...

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..., was taken from game theory, which deals with solving conflicting situations. The min-max approach to a linear model was proposed by Jutler [43] and Solich [85], and was further developed by Osyczka =-=[59, 60, 64]-=-, Rao [73] and Tseng and Lu [95]. The min-max optimum compares relative deviations from the separately attainable minima. Consider the ith objective function for which the relative deviation can be ca...

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... a genetic algorithm to optimize simultaneously the intensities of six atomic emission lines of trace elements in alumina powder as a function of spectroscopic excitation conditions. -- Eric Sandgren =-=[82]-=- also used goal programming coupled with a genetic algorithm to optimize plane trusses and the design of a planar mechanism. Strengths and Weaknesses This technique will yield a dominated solution if ...

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... to total weight, asymptotical stability and eigenvalues constraints. -- Yang and Gen [104] used a weighted sum approach to solve a bicriteria linear transportation problem. More recently, Gen et al. =-=[25, 26]-=- extended this approach to allow more than two objectives, and added fuzzy logic to handle the uncertainty involved in the decision making process. A weighted sum is still used in this approach, but i...

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...of the Contact Theorem to Detect Pareto Optimal Solutions Osyczka and Kundu [62] proposed the use of an algorithm based on the contact theorem (one of the main theorems in multiobjective optimization =-=[49]-=-) to determine relative distances of a solution vector with respect to the Pareto set. In this paper [62], the contact theorem was used to determine the fitness of each individual in the population. T...

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...ethods incorporated into a GA to design multiplierless IIR filters in which the two conflicting objectives were to minimize the response error and the implementation cost of the filter. -- Liu et al. =-=[51]-=- used this technique to optimize the layout and actuator placement of a 45-bar plane truss in which the objectives were to minimize the linear regulator quadratic control cost, the robustness and the ...

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...andled. Applications -- P'eriaux et al. [68] used the NSGA to find an optimal distribution of active control elements which minimizes the backscattering of aerodynamic reflectors. -- Vedarajan et al. =-=[97]-=- used the NSGA for investment portfolio optimization, but interestingly they used binary tournament selection instead of stochastic remainder selection as suggested by Srinivas and Deb [86]. The autho...

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...in the objective domain, and suggested the use of a metric combining both the objective and the decision variable domains, leading to what they called nested sharing. Applications -- Belegundu et al. =-=[3]-=- used the NPGA for the design of laminated ceramic composites. -- Poloni and Pediroda [69] used it for the design of a multipoint airfoil that has its minimum drag at two given lift values with a cons...

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...x 2 )sf 1 (x 1 ; x 2 ) (25) and f 2 (x 1 ; x 2 )sf 2 (x 1 ; x 2 ) (26) where x 1 may be to the left or right of x 1 in (25) and x 2 may lie above or below x 2 in (26). Applications -- P'eriaux et al. =-=[68]-=- proposed a GA-based approach that uses the concept of Nash equilibrium to solve a biobjective optimization problem (the optimal distribution of active control elements which minimizes the backscateri...

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...f a problem), and to have several Nash equilibrium points, with which the Pareto front can actually be found, although a cooperative game may be preferred in that case over a non-cooperative approach =-=[75, 74]-=-. 4.4 Using Gender to Identify Objectives Robin Allenson [2] used a population-based approach modelled after VEGA in which gender was used to distinguish between the two objective functions of a probl...

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...s of reducing the number of elements in such set, in order to facilitate the analysis for the decision maker. Kunha, Oliveira and Covas [46] proposed the incorporation of Roseman and Gero's algorithm =-=[79]-=- into the GA to cluster together points that are within a certain distance (defined by the user) of each other in the Pareto front. -- Probably one of the most difficult problems in multiobjective opt...

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...s with solving conflicting situations. The min-max approach to a linear model was proposed by Jutler [43] and Solich [85], and was further developed by Osyczka [59, 60, 64], Rao [73] and Tseng and Lu =-=[95]. Th-=-e min-max optimum compares relative deviations from the separately attainable minima. Consider the ith objective function for which the relative deviation can be calculated from z 0 i (��x) = jf i...

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...j [32]. Such a formulation has been called generalized goal programming [37, 38]. This technique has also been called "target vector optimization" by other authors [12]. Applications -- Wien=-=ke et al. [102]-=- used this approach in combination with a genetic algorithm to optimize simultaneously the intensities of six atomic emission lines of trace elements in alumina powder as a function of spectroscopic e...

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...ar regulator quadratic control cost, the robustness and the modal controllability of the controlled system subject to total weight, asymptotical stability and eigenvalues constraints. -- Yang and Gen =-=[104]-=- used a weighted sum approach to solve a bicriteria linear transportation problem. More recently, Gen et al. [25, 26] extended this approach to allow more than two objectives, and added fuzzy logic to...

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...om converging to a single point on the front. A niching mechanism such as sharing [29] would allow the GA to maintain individuals all along the non-dominated frontier. Applications -- Hilliard et al. =-=[34]-=- used a Pareto optimality ranking method to handle the objectives of minimizing cost and minimizing delay in a scheduling problem. They tentatively concluded that the Pareto optimality ranking method ...

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...ormulation of the goal programming objective function is a weighted sum of the pth power of the deviation jf i (��x) \Gamma T i j [32]. Such a formulation has been called generalized goal programm=-=ing [37, 38]. This tec-=-hnique has also been called "target vector optimization" by other authors [12]. Applications -- Wienke et al. [102] used this approach in combination with a genetic algorithm to optimize sim...

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...the population over the Pareto-optimal region, but instead of performing sharing on the parameter values, they have used sharing on the objective function values [87]. Applications -- Chen Tan and Li =-=[93]-=- reported success in the use of MOGA for the multiobjective optimization of ULTIC controllers that satisfy a number of time domain and frequency domain specifications. Also, Chipperfield and Fleming [...

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... metrics that allows us to obtain a single value (or utility function) that will guide the search to the particular Pareto region that is of interest to the decision maker. Finally, Voget and Kolonko =-=[100]-=- proposed the use of a fuzzy controller that regulates the selection pressure automatically by using a set of predefined goals that define the `desirable' behavior of the population. An interesting as...

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...rate a certain amount of individuals from one sub-population to another. They used these and other traditional multiobjective optimization approaches for preliminary airframe design. -- Tamaki et al. =-=[92, 91]-=- developed a technique in which at each generation, non-dominated individuals in the current population are kept for the following generation. This approach is really a mixture of Pareto selection (se...

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...used to overcome the high selection pressure introduced by the use of min-max tournament selection. Applications -- Coello and Christiansen applied these two approaches to the optimization of I-beams =-=[8]-=- and manufacturing problems [9], and to the design of a robot arm [10]. Strengths and Weaknesses The main strength of this technique is its efficiency and relative simplicity. The use of weights is it...

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..., was taken from game theory, which deals with solving conflicting situations. The min-max approach to a linear model was proposed by Jutler [43] and Solich [85], and was further developed by Osyczka =-=[59, 60, 64]-=-, Rao [73] and Tseng and Lu [95]. The min-max optimum compares relative deviations from the separately attainable minima. Consider the ith objective function for which the relative deviation can be ca...

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...based optimization techniques to speed up the search in order to reduce the computational cost required in a real-world application) to solve multiobjective optimization problems. -- Ranjithan et al. =-=[72]-=- used this approach to solve groundwater pollution containment problems. -- Loughlin and Ranjithan [52] used a variation of this technique in which they incorporated target satisfaction levels (simila...

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...mum and applying it to multiobjective optimization problems, was taken from game theory, which deals with solving conflicting situations. The min-max approach to a linear model was proposed by Jutler =-=[43]-=- and Solich [85], and was further developed by Osyczka [59, 60, 64], Rao [73] and Tseng and Lu [95]. The min-max optimum compares relative deviations from the separately attainable minima. Consider th...

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...they used only two simple biobjective optimization functions in their paper), Osyczka's algorithm for detecting Pareto optimality has been applied before to several problems, mainly in machine design =-=[63, 59, 60, 61, 64]-=-. Strengths and Weaknesses The main strengths of this approach are its efficiency and relative simplicity. Additionally, it does not require an explicit sharing function. However, its main weakness is...

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...they used only two simple biobjective optimization functions in their paper), Osyczka's algorithm for detecting Pareto optimality has been applied before to several problems, mainly in machine design =-=[63, 59, 60, 61, 64]-=-. Strengths and Weaknesses The main strengths of this approach are its efficiency and relative simplicity. Additionally, it does not require an explicit sharing function. However, its main weakness is...

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... it to multiobjective optimization problems, was taken from game theory, which deals with solving conflicting situations. The min-max approach to a linear model was proposed by Jutler [43] and Solich =-=[85]-=-, and was further developed by Osyczka [59, 60, 64], Rao [73] and Tseng and Lu [95]. The min-max optimum compares relative deviations from the separately attainable minima. Consider the ith objective ...