Evolutionary Multi-Criterion Optimization

One of the most important tasks in multi-objective optimization is "trade-off analysis" which aims to make the total balance among objective functions. The trade-off relation among alternatives can be shown as Pareto frontier. In cases with two or three objective functions, the set of Pareto optimal solutions in the objective function space (i.e., Pareto frontier) can be depicted relatively easily. Seeing Pareto frontiers, we can grasp the trade-off relation among objectives totally. Therefore, it would be the best way to depict Pareto frontiers in cases with two or three objectives. (It might be difficult to read the trade-off relation among objectives with three dimension, though). In cases with more than three objectives, however, it is impossible to depict Pareto frontier. There are some cases with a large number (e.g., a few hundreds) of objective functions in engineering applications such as erection management of cable stayed bridges and optical lens design. Under this circumstance, interactive methods can help decision makers (DMs) to make local trade-off analysis through interaction of DMs and computers by showing a Pareto solution nearest to their desire. Along this line, aspiration level methods were developed, and have been observed to be effective in many practical problems in various fields. Satisficing Trade-off Method proposed by the author is one of aspiration level methods, and has several devices for making trade-off analysis easily, i.e., automatic trade-off and exact trade-off. This paper discusses those methods for multi-objective optimization, in particular, from a viewpoint of engineering application.

Multi-objective evolutionary algorithms (MOEAs) are a class of stochastic optimization techniques that simulate biological evolution to solve problems with multiple objectives. Multi-objective (MO) optimization is a challenging research topic because it involves the simultaneous optimization of several (and normally conflicting) objectives in the Pareto optimal sense. It requires researchers to address many issues that are unique to MO problems, such as fitness assignment, diversity preservation, balance between exploration and exploitation, elitism and archiving. In this talk, a few advanced features for handling large and computationally intensive real-world MO optimization problems will be presented. These include a distributed cooperative coevolutionary approach to handle large-scale problems via a divide-and-conquer strategy by harnessing technological advancements in parallel and distributed systems and a hybridization scheme with local search heuristics for combinatorial optimization with domain knowledge. The talk will also discuss the application of these techniques to various engineering problems including scheduling and system design, which often involve different competing specifications in a large and highly constrained search space.

Current evolutionary multiobjective optimization (EMO) approaches tend to emphasize the approximation of the Pareto-optimal front as a whole, thereby dissociating the optimization process from the selection of the final compromise solution by a decision maker. This has the advantage of removing subjective preference information from the optimization problem formulation, but it also makes the resulting problem computationally more demanding. In order to concentrate the search effort on the regions of potential interest to the decision maker, techniques for the progressive articulation of preferences in EMO have been proposed, casting EMO as the interaction between an evolutionary search mechanism and a decision maker. It is worth noting that even the promotion of diversity across the Pareto-optimal front, which is generally regarded as an optimizer design issue, may be successfully addressed by the decision maker within this framework, as it has been proposed recently by others. Regarding the evolutionary search mechanism, the main question at each iteration consists of determining the next candidate solution(s) to be evaluated, given the information acquired since the beginning of the run. This may be seen as another decision-making problem, but one with (very) incomplete attribute information, since objective values are generally not known for most potential alternatives. Alternatively, it may be seen as a control problem, where actions (new solutions) are to be selected based on the feedback provided by the decision maker. Either way, some model, however weak, of the underlying optimization problem must be assumed. In this talk, both the evaluation of current solutions and the generation of new candidate solutions in EMO will be discussed from a decision making perspective. From the discussion, opportunities for incorporating more explicit decision making in EMO will be identified.

Advances in information and communications technology are changing network design techniques quantitatively and qualitatively. This technology is supporting the design of large scale network centric systems which are required in many contemporary real-world situations. These high-level robust centric systems by definition must provide improved information sharing and collaboration between network entities. Such systems enhance the quality of information awareness, improving sustainability, and mission effectiveness and efficiency. The hierarchical development of network centric systems includes all dynamic information elements and is applied so as to maximize the desired decision and action impact. Associated network information flow problems can have as objectives costs, delays, robustness, vulnerability, and reliability with related constraints of network flow capacities, rates, and quantities of information. The optimization of coupled complex capacitated network flow problems is therefore an integral and basic element of network centric systems design. Thus, the focus of the discussion is on the efficacy of multiobjective evolutionary algorithms (MOEAs) to solve effectively and efficiency variations of associated network flow problems, given sophisticated mathematical models. Also to be addressed are dynamic network environments where various information channels become non-available, change their characteristics, or information priorities are modified. Discrimination between possible MOEA operators (recombination, mutation, selection) and associated MOEA parameter values is discussed as related to solving effectively variations of multiobjective network centric information flow problems including real-time behavior. Example network flow applications provide insight to choosing appropriate MOEA characteristics. Included is a discussion of opportunities for future MOEA research in this arena.

This work proposes a method to control the dominance area of solutions in order to induce appropriate ranking of solutions for the problem at hand, enhance selection, and improve the performance of MOEAs on combinatorial optimization problems. The proposed method can control the degree of expansion or contraction of the dominance area of solutions using a user-defined parameter

S

. Modifying the dominance area of solutions changes their dominance relation inducing a ranking of solutions that is different to conventional dominance. In this work we use 0/1 multiobjective knapsack problems to analyze the effects on solutions ranking caused by contracting and expanding the dominance area of solutions and its impact on the search performance of a multi-objective optimizer when the number of objectives, the size of the search space, and the complexity of the problems vary. We show that either convergence or diversity can be emphasized by contracting or expanding the dominance area. Also, we show that the optimal value of the area of dominance depends strongly on all factors analyzed here: number of objectives, size of the search space, and complexity of the problems.

In this paper, we propose a new conceptual method for the design, investigation, and evaluation of multi-objective variation operators for evolutionary multi-objective algorithms. To this end, we apply a modified predator-prey model that allows an independent analysis of different operators. Using this model problem specific operators can be combined to more complex operators. Additionally, we review the simplex recombination, a new rotation-independent recombination scheme, and examine its impact concerning our design method. We show exemplarily as a first attempt the advantageous combination of several standard variation operators that lead to better results for selected test functions.

Recent works in evolutionary multiobjective optimization suggest to shift the focus from solely evaluating optimization success in the objective space to also taking the decision space into account. They indicate that this may be a) necessary to express the users requirements of obtaining distinct solutions (distinct Pareto set parts or subsets) of similar quality (comparable locations on the Pareto front) in real-world applications, and b) a demanding task for the currently most commonly used algorithms. We investigate if standard EMOA are able to detect and preserve equivalent Pareto subsets and develop an own special purpose EMOA that meets these requirements reliably.

This paper proposes an idea of using evolutionary multiobjective optimization (EMO) to optimize scalarizing functions. We assume that a scalarizing function to be optimized has already been generated from an original multiobjective problem. Our task is to optimize the given scalarizing function. In order to efficiently search for its optimal solution without getting stuck in local optima, we generate a new multiobjective problem to which an EMO algorithm is applied. The point is to specify multiple objectives, which are similar to but different from the scalarizing function, so that the location of the optimal solution is near the center of the Pareto front of the generated multiobjective problem. The use of EMO algorithms helps escape from local optima. It also helps find a number of alternative solutions around the optimal solution. Difficulties of Pareto ranking-based EMO algorithms in the handling of many objectives are avoided by the use of similar objectives. In this paper, we first demonstrate that the performance of EMO algorithms as single-objective optimizers of scalarizing functions highly depends on the choice of multiple objectives. Based on this observation, we propose a specification method of multiple objectives for the optimization of a weighted sum fitness function. Experimental results show that our approach works very well in the search for not only a single optimal solution but also a number of good alternative solutions around the optimal solution. Next we evaluate the performance of our approach in comparison with a hybrid EMO algorithm where a single-objective fitness evaluation scheme is probabilistically used in an EMO algorithm. Then we show that our approach can be also used to optimize other scalarizing functions (e.g., those based on constraint conditions and reference solutions). Finally we show that our approach is applicable not only to scalarizing functions but also other single-objective optimization problems.

Uncertainties in design variables and problem parameters are inevitable and must be considered in an optimization task including multi-objective optimization, if reliable optimal solutions are to be found. Sampling techniques become computationally expensive if a large reliability is desired. In this paper, first we present a brief review of statistical reliability-based optimization procedures. Thereafter, for the first time, we extend and apply multi-objective evolutionary algorithms for solving two different reliability-based optimization problems for which evolutionary approaches have a clear niche in finding a set of reliable, instead of optimal, solutions. The use of an additional objective of maximizing the reliability index in a multi-objective evolutionary optimization procedure allows a number of trade-off solutions to be found, thereby allowing the designers to find solutions corresponding to different reliability requirements. Next, the concept of single-objective reliability-based optimization is extended to multi-objective optimization of finding a reliable frontier, instead of an optimal frontier. These optimization tasks are illustrated by solving test problems and a well-studied engineering design problem. The results should encourage the use of evolutionary optimization methods to more such reliability-based optimization problems.

Spatially structured populations have been used in evolutionary computation for many years. Somewhat surprisingly, in the multiobjective optimization domain, very few spatial models have been proposed. In this paper, we introduce a new multiobjective evolutionary algorithm on complex networks. Here, the individuals in the evolving population are mapped onto the nodes of alternative complex networks – regular, small-world, scale-free and random. A selection regime based on a non-dominance rating and a crowding mechanism guides the evolutionary trajectory. Our model can be seen as an extension of the standard cellular evolutionary algorithm. However, the dynamical behaviour of the evolving population is constrained by the particular network architecture. An important contribution of this paper is the detailed analysis of the impact that the structural properties of the network – node degree distribution, characteristic path length and clustering coefficient – have on the behaviour of the evolutionary algorithm using benchmark bi-objective problems.

Evolutionary algorithms have been adequately applied in solving single and multi-objective optimization problems. In the single-objective case various studies have shown the usefulness of combining gradient based classical methods with evolutionary algorithms. However there seems to be limited number of such studies for the multi-objective case. In this paper, we take two classical methods for unconstrained multi-optimization problems and discuss their use as a local search operator in a state-of-the-art multi-objective evolutionary algorithm. These operators require gradient information which is obtained using finite difference method and using a stochastic perturbation technique requiring only two function evaluations. Computational studies on a number of test problems of varying complexity demonstrate the efficiency of resulting hybrid algorithms in solving a large class of complex multi-objective optimization problems. We also discuss a new convergence metric which is useful as a stopping criteria for problems having an unknown Pareto-optimal front.

Archives are used in Multi-Objective Evolutionary Algorithms to establish elitism. Depending on the optimization problem, an unconstrained archive may grow to an immense size. With the growing number of nondominated solutions in the archive, testing a new solution for nondominance against this archive becomes the main bottleneck during optimization. As a remedy to this problem, we will propose a new data structure on the basis of Binary Decision Diagrams (BDDs) that permits a nondominance test with a runtime that is independent from the archive size. For this purpose, the region in the objective space weakly dominated by the solutions in the archive is represented by a BDD. We will present the algorithms for constructing the BDD as well as the nondominance test. Moreover, experimental results from using this symbolic data structure will show the efficiency of our approach in test cases where many candidates have to be tested but only few have to be added to the archive.

In this paper we study a number of issues related to the design of a cellular genetic algorithm (cGA) for multiobjective optimization. We take as an starting point an algorithm following the canonical cGA model, i.e., each individual interacts with those ones belonging to its neighborhood, so that a new individual is obtained using the typical selection, crossover, and mutation operators within this neighborhood. An external archive is used to store the non-dominated solutions found during the evolution process. With this basic model in mind, there are many different design issues that can be faced. Among them, we focus here on the synchronous/asynchronous feature of the cGA, the feedback of the search experience contained in the archive into the algorithm, and two different replacement strategies. We evaluate the resulting algorithms using a benchmark of problems and compare the best of them against two state-of-the-art genetic algorithms for multiobjective optimization. The obtained results indicate that the cGA model is a promising approach to solve this kind of problem.

We present a new multiobjective evolutionary algorithm (MOEA), called fast Pareto genetic algorithm (FastPGA). FastPGA uses a new fitness assignment and ranking strategy for the simultaneous optimization of multiple objectives where each solution evaluation is computationally- and/or financially-expensive. This is often the case when there are time or resource constraints involved in finding a solution. A population regulation operator is introduced to dynamically adapt the population size as needed up to a user-specified maximum population size. Computational results for a number of well-known test problems indicate that FastPGA is a promising approach. FastPGA outperforms the improved nondominated sorting genetic algorithm (NSGA-II) within a relatively small number of solution evaluations.

Among the multi-objective optimization methods proposed so far, Genetic Algorithms (GA) have been shown to be more effective in recent decades. Most of such methods were developed to solve primarily unconstrained problems. However, many real-world problems are constrained, which necessitates appropriate handling of constraints. Despite much effort devoted to the studies of constraint-handling methods, it has been reported that each of them has certain limitations. Hence, further studies for designing more effective constraint-handling methods are needed.

For this reason, we investigated the guidelines for a method to effectively handle constraints. Based on these guidelines, we designed a new constraint-handling method, Pareto Descent Repair operator (PDR), in which ideas derived from multi-objective local search and gradient projection method are incorporated. An experiment comparing GA that use PDR and some of the existing constraint-handling methods confirmed the effectiveness of PDR.

The multi-objective covariance matrix adaptation evolution strategy (MO-CMA-ES) combines a mutation operator that adapts its search distribution to the underlying optimization problem with multi-criteria selection. Here, a generational and two steady-state selection schemes for the MO-CMA-ES are compared. Further, a recently proposed method for computationally efficient adaptation of the search distribution is evaluated in the context of the MO-CMA-ES.

Recent advances in quantum cascade lasers (QCLs) have enabled their use as (tunable) emission sources for chemical and biological spectroscopy, as well as allowed their demonstration in applications in medical diagnostics and potential homeland security systems. Finding the optimal design solution can be challenging, especially for lasers that operate in the terahertz region. The production process is prohibitive, so an optimization algorithm is needed to find high quality QCL designs. Past research attempts using multiobjective evolutionary algorithms (MOEAs) have found good solutions, but lacked a local search element that could enable them to find more effective solutions. This research looks at two memetic MOEAs that use a neighborhood search. Our baseline memetic MOEA used a simple neighborhood search, which is similar to other MOEA neighborhood searches found in the literature. Alternatively, our innovative multi-tiered memetic MOEA uses problem domain knowledge to change the temporal focus of the neighborhood search based on the generation. It is empirically shown that the multi-tiered memetic MOEA is able to find solutions that dominate the baseline memetic algorithm. Additional experiments suggest that using local search on only non-dominated individuals improves the effectiveness and efficiency of the algorithm versus applying the local search to dominated individuals as well. This research validates the importance of using multiobjective problem (MOP) domain knowledge in order to obtain the best results for a real world solution. It also introduces a new multi-tiered local search procedure that is able to focus the local search on specific critical elements of the problem at different stages in the optimization process.

In this paper, we propose a memetic EMO algorithm that enhances the similarity of two sets of non-dominated solutions. We employ our algorithm in vehicle routing problems (VRPs) where the demand of customers varies. We consider two periods of different demand in a problem that are Normal Demand Period (NDP) and High Demand Period (HDP). In each period, we can find a set of non-dominated solutions with respect to several objectives such as minimizing total cost for delivery, minimizing maximum cost, minimizing the number of vehicles, minimizing total delay to the date of delivery and so on. Although a set of non-dominated solutions can be searched independently in each period, drivers of vehicles prefer to have similar routes in NDP and HDP in order to reduce their fatigue to drive on a different route. In this paper, we propose a local search that enhance the similarity of routes in NDP and HDP. Simulation results show that the proposed memetic EMO algorithm can find a similar set of non-dominated solutions in HDP to the one in NDP.

When multi-objective genetic algorithms are applied to real-world problems for deriving Pareto-optimal solutions, the high calculation cost becomes a problem. One solution to this problem is to use a small population size. However, this often results in loss of diversity of the solutions, and therefore solutions with sufficient precision cannot be derived. To overcome this difficulty, the solutions should be replaced when they have converged on a certain point. To perform this replacement, inverse analysis is required to derive the design variables from objects as the solutions are located in the objective space. For this purpose, an Artificial Neural Network (ANN) is applied. Using ANN, the solutions concentrating on certain points are replaced and the diversity of the solutions is maintained. In this paper, a new mechanism using ANN to maintain the diversity of the solutions is proposed. The proposed mechanism was introduced into NSGA-II and applied to test functions. In some functions, the proposed mechanism was useful compared to the conventional method. In other numerical experiments, the results of the proposed algorithm with large populations are discussed and the effectiveness of the proposed mechanism is also described.

The Pareto-based Differential Evolution (PDE) algorithm is one of the current state-of-the-art Multi-objective Evolutionary Algorithms (MOEAs). This paper describes a series of experiments using PDE for evolving artificial neural networks (ANNs) that act as game-playing agents. Three systems are compared: (i) a canonical PDE system, (ii) a co-evolving PDE system (PCDE) with 3 different setups, and (iii) a co-evolving PDE system that uses an archive (PCDE-A) with 3 different setups. The aim of this study is to provide insights on the effects of including co-evolutionary techniques on a well-known MOEA by investigating and comparing these 3 different approaches in evolving intelligent agents as both first and second players in a deterministic zero-sum board game. The results indicate that the canonical PDE system outperformed both co-evolutionary PDE systems as it was able to evolve ANN agents with higher quality game-playing performance as both first and second game players. Hence, this study shows that a canonical MOEA without co-evolution is desirable for the synthesis of cognitive game AI agents.

The reliance of Tabu Search (TS) algorithms on a local search leads to a logical development of algorithms that use more than one search concurrently. In this paper we present a multi-threaded TS algorithm employing a number of threads that share information. We assess the performance of this algorithm compared to previous multi-objective TS algorithms, via the results obtained from applying the algorithms to a range of standard test functions. We also consider whether an optimal number of threads can be found, and what impact changing the number of threads used has on performance. We discover that, contrary to the popular belief that multi-threading is usually beneficial, performance only improves in a few special cases.

This paper presents a comprehensive comparison between the performance of state-of-the-art genetic algorithms NSGA-II, SPEA2 and IBEA and their differential evolution based variants DEMO

$^\text{NS-II}$

, DEMO

$^\text{SP2}$

and DEMO

$^\text{IB}$

. Experimental results on 16 numerical multiobjective test problems show that on the majority of problems, the algorithms based on differential evolution perform significantly better than the corresponding genetic algorithms with regard to applied quality indicators. This suggests that in numerical multiobjective optimization, differential evolution explores the decision space more efficiently than genetic algorithms.

This paper presents the Efficient Multi-Objective Particle Swarm Optimizer (EMOPSO), which is an improved version of a multi-objective evolutionary algorithm (MOEA) previously proposed by the authors. Throughout the paper, we provide several details of the design process that led us to EMOPSO. The main issues discussed are: the mechanism to maintain a set of well-distributed nondominated solutions, the turbulence operator that avoids premature convergence, the constraint-handling scheme, and the study of parameters that led us to propose a self-adaptation mechanism. The final algorithm is able to produce reasonably good approximations of the Pareto front of problems with up to 30 decision variables, while performing only 2,000 fitness function evaluations. As far as we know, this is the lowest number of evaluations reported so far for any multi-objective particle swarm optimizer. Our results are compared with respect to the NSGA-II in 12 test functions taken from the specialized literature.

To find solutions as close to the Pareto front as possible, and to make them as diverse as possible in the obtained non-dominated front is a challenging task for any multiobjective optimization algorithm.

ε

-dominance is a concept which can make genetic algorithm obtain a good distribution of Pareto-optimal solutions and has low computational time complexity,and the orthogonal design method can generate an initial population of points that are scattered uniformly over the feasible solution space.In this paper, combining

ε

-dominance and orthogonal design method, we propose a novel Differential Evolution (DE) algorithm for multiobjective optimization .Experiments on a number of two- and three-objective test problems of diverse complexities show that our approach is able to obtain a good distribution with a small computational time in all cases. Compared with several other state-of-the-art evolutionary algorithms, it achieves not only comparable results in terms of convergence and diversity metrics, but also a considerable reduction of the computational effort.

Molecular system possesses two main characteristics that seem to be applicable for the contrary goals of proximity and diversity in multiobjective optimization, namely the converging pressure in potential fields as dictated by the Maxwell-Boltzmann distribution and the inherent drift to a homogenous and uniform equilibrium with maximum entropy, even without any prior knowledge on the geometry and state of the enclosure. Inspired by this association, this paper explores the notion of exploiting molecular motion to solve multiobjective problems. By adapting the algorithmic structure of molecular dynamics, which essentially represents a technique for the computer simulation of molecular motion, a molecular system that is relevant for multiobjective optimization is proposed, known as molecular dynamics optimizer (MDO). The performance of MDO was subsequently compared with other conventional multiobjective optimizers, specifically EA and PSO, and the experimental results demonstrated that MDO is indeed a viable and practical approach for multiobjective optimization.

One of main issues in multi-objective optimization is to support for choosing a final solution from Pareto frontier which is the set of solution to problem. For generating a part of Pareto optimal solution closest to an aspiration level of decision maker, not the whole set of Pareto optimal solutions, we propose a method which is composed of two steps; i) approximate the form of each objective function by using support vector regression on the basis of some sample data, and ii) generate Pareto frontier to the approximated objective functions based on given the aspiration level. In addition, we suggest to select additional data for approximating sequentially the forms of objective functions by relearning step by step. Finally, the effectiveness of the proposed method will be shown through some numerical examples.

The design of a Hybrid Electric Vehicle (HEV) system is an energy management strategy problem between two sources of power. Traditionally, the drive train has been designed first, and then a driving strategy chosen and sometimes optimised. This paper considers the simultaneous optimisation of both drive train and driving strategy variables of the HEV system through use of a multi-objective evolutionary optimiser. The drive train is well understood. However, the optimal driving strategy to determine efficient and opportune use of each prime mover is subject to the driving cycle (the type of dynamic environment, e.g. urban, highway), and has been shown to depend on the correct selection of the drive train parameters (gear ratios) as well as driving strategy heuristic parameters. In this paper, it is proposed that the overall optimal design problem has to consider multiple objectives, such as fuel consumption, reduction in electrical energy stored, and the ‘driveability’ of the vehicle. Numerical results shows improvement when considering multiple objectives and simultaneous optimisation of both drive train and driving strategy.

This paper will investigate the application of multiobjective evolu-tionary neural networks in time series forecasting. The proposed algorithmic model considers training and validation accuracy as the objectives to be optimized simultaneously, so as to balance the accuracy and generalization of the evolved neural networks. To improve the overall generalization ability for the set of solutions attained by the multiobjective evolutionary optimizer, a simple algorithm to filter possible outliers, which tend to deteriorate the overall performance, is proposed also. Performance comparison with other existing evolutionary neural networks in several time series problems demonstrates the practicality and viability of the proposed time series forecasting model.

Understanding the results of a multi objective optimization process can be hard. Various visualization methods have been proposed previously, but the only consistently popular one is the 2D or 3D objective scatterplot, which cannot be extended to handle more than 3 objectives. Additionally, the visualization of high dimensional parameter spaces has traditionally been neglected. We propose a new method, based on heatmaps, for the simultaneous visualization of objective and parameter spaces. We demonstrate its application on a simple 3D test function and also apply heatmaps to the analysis of real-world optimization problems. Finally we use the technique to compare the performance of two different multi-objective algorithms.

Multiplex Polymerase Chain Reaction (PCR) assay is to amplify multiple target DNAs simultaneously using different primer pairs for each target DNA. Recently, it is widely used for various biology applications such as genotyping. For sucessful experiments, both the primer pairs for each target DNA and grouping of targets to be actually amplified in one tube should be optimized. This involves multiple conflicting objectives such as minimizing the interaction of primers in a group and minimizing the number of groups required for the assay. Therefore, a multiobjective evolutionary approach may be an appropriate approach. In this paper, a hybrid multiobjective evolutionary algorithm which combines

ε

-multiobjective evolutionary algorithm with local search is proposed for multiplex PCR assay design. The proposed approach was compared with another multiobjective method, called MuPlex, and showed comparative performance by covering all of the given target sequences.

This paper presents ParadisEO-MOEO, a white-box object-oriented generic framework dedicated to the flexible design of evolutionary multi-objective algorithms. This paradigm-free software embeds some features and techniques for Pareto-based resolution and aims to provide a set of classes allowing to ease and speed up the development of computationally efficient programs. It is based on a clear conceptual distinction between the solution methods and the multi-objective problems they are intended to solve. This separation confers a maximum design and code reuse. ParadisEO-MOEO provides a broad range of archive-related features (such as elitism or performance metrics) and the most common Pareto-based fitness assignment strategies (MOGA, NSGA, SPEA, IBEA and more). Furthermore, parallel and distributed models as well as hybridization mechanisms can be applied to an algorithm designed within ParadisEO-MOEO using the whole version of ParadisEO. In addition, GUIMOO, a platform-independant free software dedicated to results analysis for multi-objective problems, is briefly introduced.

The inadequacy of classical methods to handle

resource allocation problems

(RAPs) draw the attention of evolutionary algorithms (EAs) to these problems. The potentialities of EAs are exploited in the present work for handling two such RAPs of quite different natures, namely (1) university class timetabling problem and (2) land-use management problem. In many cases, these problems are over-simplified by ignoring many important aspects, such as different types of constraints and multiple objective functions. In the present work, two EA-based multi-objective optimizers are developed for handling these two problems by considering various aspects that are common to most of their variants. Finally, the similarities between the problems, and also between their solution techniques, are analyzed through the application of the developed optimizers on two real problems.

Multi-Objective Evolutionary Algorithms (MOEA) have been succesfully applied to solve control problems. However, many improvements are still to be accomplished. In this paper a new approach is proposed: the Multi-Objective Pole Placement with Evolutionary Algorithms (MOPPEA). The design method is based upon using complex-valued chromosomes that contain information about closed-loop poles, which are then placed through an output feedback controller. Specific cross-over and mutation operators were implemented in simple but efficient ways. The performance is tested on a mixed multi-objective

$\mathcal{H}_{2}$

/

$\mathcal{H}_{\infty }$

control problem.

The phylogeny reconstruction problem consists of determining the most accurate tree that represents evolutionary relationships among species. Different criteria have been employed to evaluate possible solutions in order to guide a search algorithm towards the best tree. However, these criteria may lead to distinct phylogenies, which are often conflicting among them. In this context, a multi-objective approach can be useful since it could produce a spectrum of equally optimal trees (Pareto front) according to all criteria. We propose a multi-objective evolutionary algorithm, named PhyloMOEA, which employs the maximum parsimony and likelihood criteria to evaluate solutions. PhyloMOEA was tested using four datasets of nucleotide sequences. This algorithm found, for all datasets, a Pareto front representing a trade-off between the criteria. Moreover, SH-test showed that most of solutions have scores similar to those obtained by phylogenetic programs using one criterion.

A perturbation method is proposed to detect convergence of the Pareto front for multi-objective algorithms and to investigate its effect on the rate of convergence of the optimization. Conventionally, evolutionary algorithms are allowed to run for a fixed number of trial solutions which can result in a premature convergence or in an unnecessary number of calls to a computationally intensive real world problem. Combination of evolutionary multi-objective algorithms with perturbation method will improve the rate of convergence of the optimization. This is a very important characteristic in reducing number of generations and therefore reducing the computational time which is important in real world problems where cost and time constraint prohibit repeated runs of the algorithm and the simulation. The performance of the method will be examined by its application to two water distribution networks from literature. The results will be compared with previously published results from literature and those generated by evolutionary multi-objective algorithm. It will be shown that the method is able to find the Pareto optimal front with less computational effort.

The importance of multi-objective optimization is globably established nowadays. Furthermore, a great part of real-world problems are subject to uncertainties due to,

e.g.

, noisy or approximated fitness function(s), varying parameters or dynamic environments. Moreover, although evolutionary algorithms are commonly used to solve multi-objective problems on the one hand and to solve stochastic problems on the other hand, very few approaches combine simultaneously these two aspects. Thus, flow-shop scheduling problems are generally studied in a single-objective deterministic way whereas they are, by nature, multi-objective and are subjected to a wide range of uncertainties. However, these two features have never been investigated at the same time.

In this paper, we present and adopt a proactive stochastic approach where processing times are represented by random variables. Then, we propose several multi-objective methods that are able to handle any type of probability distribution. Finally, we experiment these methods on a stochastic bi-objective flow-shop problem.

This paper presents the corrective process control system for achieving a target quality level in glass melting processes. Since automated data collection devices would monitor and log process attributes that are assumed to correlate to a quality level in the glass melting process, appropriate process control logics utilizing the collected data are definitely needed. In this paper, an evolutionary algorithm based search logic is newly proposed. The objective of the proposed logic is to find the best process condition composed of the process attributes which can generate the target quality level. The proposed logic tries to find the best process condition that needs to satisfy the following two criteria: 1) a process condition should require minimal changes from the current setting of the process attributes; and 2) a process condition can generate the exact or closest value against the target quality level. A case study and a developed process control system are presented.

This paper presents a multicriterion algorithm for dealing with joint facility location and network design problems, formulated as bi-objective problems. The algorithm is composed of two modules: a multiobjective quasi-Newton algorithm, that is used to find the location of the facilities; and a multiobjective genetic algorithm, which is responsible for finding the efficient topologies. These modules are executed in an iterative way, to make the estimation of whole Pareto set possible. The algorithm has been applied to the expansion of a real energy distribution system. The minimization of financial cost and the maximization of reliability have been considered as the design objectives in this case.

In this paper we present three path relinking approaches for solving a bi-objective permutation flowshop problem. The path relinking phase is initialized by optimizing the two objectives using Ant Colony System. The initiating and guiding solutions of path relinking are randomly selected and some of the solutions along the path are intensified using local search. The three approaches differ in their strategy of defining the heuristic bounds for the local search, i.e., each approach allows its solutions to undergo local search under different conditions. These conditions are based on local nadir points. Several test instances are used to investigate the performances of the different approaches. Computational results show that the decision which allows solutions to undergo local search has an influence in the performance of path relinking. We also demonstrate that path relinking generates competitive results compared to the best known solutions of the test instances.

Multi-objective metaheuristics have previously been applied to partial classification, where the objective is to produce simple, easy to understand rules that describe subsets of a class of interest. While this provides a useful aid in descriptive data mining, it is difficult to see how the rules produced can be combined usefully to make a predictive classifier. This paper describes how, by using a more complex representation of the rules, it is possible to produce effective classifiers for two class problems. Furthermore, through the use of multi-objective genetic programming, the user can be provided with a selection of classifiers providing different trade-offs between the misclassification costs and the overall model complexity.

In this paper, the design of systems using mechanical or electrical energy-transformation devices is treated as a knapsack problem. Due to the well-known NP-hard complexity of the knapsack problem, a combination of integer linear programming and evolutionary multi-criteria optimization is presented to solve this real problem with promising experimental results.

Genetic algorithms are one of the state-of-the-art metaheuristic techniques for optimal design of capital-intensive infrastructural water networks. They are capable of finding near optimal cost solutions to these problems given certain cost and hydraulic parameters. Recently, multi-objective genetic algorithms have become prevalent due to the conflicting nature of these hydraulic and cost objectives. The Pareto-front of solutions obtained enables water engineers to have more flexibility by providing a set of design alternatives. However, multi-objective genetic algorithms tend to require a large number of objective function evaluations to achieve an acceptable Pareto-front. This paper describes a novel hybrid cellular automaton and genetic algorithm approach, called CAMOGA for multi-objective design of urban water networks. The method is applied to four large real-world networks. The results show that CAMOGA can outperform the standard multi-objective genetic algorithm in terms of optimization efficiency and quality of the obtained Pareto fronts.

We introduce an important general Multiobjective Evolutionary Algorithm (MOEA) application – assessment of mechanistic simulation models in biology. These models are often developed to investigate the processes underlying biological phenomena. The proposed model structure must be assessed to reveal if it adequately describes the phenomenon. Objective functions are defined to measure how well the simulations reproduce specific phenomenon features. They may be continuous or binary-valued, e.g. constraints, depending on the quality and quantity of phenomenon data. Assessment requires estimating and exploring the model’s Pareto frontier. To illustrate the problem, we assess a model of shoot growth in pine trees using an elitist MOEA based on Nondominated Sorting in Genetic Algorithms. The algorithm uses the partition induced on the parameter space by the binary-valued objectives. Repeating the assessment with tighter constraints revealed model structure improvements required for a more accurate simulation of the biological phenomenon.

Considering evolutionary multiobjective algorithms for improving single objective optimization problems is focused in this work on introducing the concept of helper objectives in a computational mechanics problem: the constrained mass minimization in real discrete frame bar structures optimum design. The number of different cross-section types of the structure is proposed as a helper objective. It provides a discrete functional landscape where the non-dominated frontier is constituted of a low number of discrete isolated points. Therefore, the population diversity treatment becomes a key point in the multiobjective approach performance. Two different-sized test cases, four mutation rates and two codifications (binary and gray) are considered in the performance analysis of four algorithms: single-objective elitist evolutionary algorithm, NSGAII, SPEA2 and DENSEA. Results show how an appropriate multiobjective approach that makes use of the proposed helper objective outperforms the single objective optimization in terms of average final solutions and enhanced robustness related to mutation rate variations.

This work deals with the design of new shielding materials for the protection of electrical devices. Since there are many different requirements for modern materials, we have chosen a multi-objective approach to this problem. As material under consideration we chose conducting polymer composites due to their excellent electromagnetic properties in the microwave band and their high potential for the optimization process. In this paper, we start this process with the formulation of a novel model, deal further with the approximation of these solution sets, and finally consider the decision support related to this problem.

This paper presents results of extensive computational experiments in which evolutionary multiobjective algorithms were used to find Pareto-optimal solutions to a complex structural design problem. In particular, Strength-Pareto Evolutionary Algorithm 2 (SPEA2) was combined with a mathematical programming method to find optimal designs of steel structural systems in tall buildings with respect to two objectives (both minimized): the total weight and the maximum horizontal displacement of a tall building. SPEA2 was employed to determine Pareto-optimal topologies of structural members (topology optimization) whose cross-sections were subsequently optimized by the mathematical programming method (sizing optimization). The paper also presents the shape of the Pareto front in this two-dimensional objective space and discusses its dependence on the building’s aspect ratio. The results reported provide both qualitative and quantitative knowledge regarding the relationship between the two objectives. They also show the trade-offs involved in the process of conceptual and detailed design of complex structural systems in tall buildings.

Robust Design Optimization is the most appropriate approach to face problems characterized by uncertainties on operating conditions, which are peculiarity of aeronautical research activities. The Robust Design methodology illustrated in this paper is based on multi-objective approach. When a Pareto approach is used, a Multi Criteria Decision Method is needed for selecting the final optimal solution. This method is tested on an aeronautic case: the design of a transonic airfoil with uncertainties on free Mach number and angle of attack. The final solution is compared with a well known airfoil: the new design performs as the original one , especially concerning lift and drag stability.

In this paper a procedure for the multi-objective optimization of an automotive cooling duct is described. The two objectives considered are the minimization of the pressure drop between the inlet and the outlet of the duct and the maximization of the outlet flow velocity. Since there is no a single optimum to be found, the MOGA-II was used as multi-objective genetic algorithm. The optimization of the duct was obtained employing a parametric model, performing flow analysis with an open source suite and using a multi-objective optimization product. The distributed optimization search exploited the parallelization capabilities of the MOGA-II algorithm which allowed the evaluation of several designs configurations by running concurrent threads of the flow analysis solver. The results obtained are very satisfactory, and the procedure described can be applied to even more complex problems.

Since the pioneer work of Evolution Strategies, experiment-based optimization is one of the promising applications of evolutionary computation. Recent progress in automatic control and instrumentation provides a smart environment called Hardware In the Loop Simulation (HILS) for such application. However, since optimization through experiment has severe condition of limited evaluation time and fluctuation of observation, we have to develop methodologies that overcome these problems. This paper discusses application of Multi-Objective Evolutionary Algorithms (MOEAs) to experiment-based optimization of control parameters of dynamical systems. In such applications, we have to apply various parameter candidates spreading near the Pareto frontier to the system, and it causes fluctuation of the observed criteria due to the transient response by parameter switching. For reduction of loss time caused by such transient response in evaluation of criteria, we propose techniques called Evaluation Order Scheduling and Evaluation Time Scheduling. Numerical experiments using a formal test problem and experiment in a HILS environment for real internal-combustion engines have demonstrated the effectiveness of the proposed methods.

This paper presents a new approach for vehicle routing problems (VRPs), which are defined as problems of minimizing the total travel distance. The proposed approach treats VRPs as multi-objective problems using the concept of multiobjectivization. The multiobjectivization approach translates single-objective optimization problems into multi-objective optimization problems and then applies EMO to the translated problem. In the proposed approach, a newly defined objective related to assignment of customers is added, because the assignment has a more important influence on the search results than routing in VRPs. We investigated the characteristics and effectiveness of the proposed approaches by comparing the performance on conventional approaches and the proposed approaches.

Multi-Car Elevator (MCE) that has several elevator cars in a single shaft attracts attention for improvement of transportation in high-rise buildings. However, because of lack of experience of such novel systems, design of controller for MCE is very difficult engineering problem. One of the promising approaches is application of evolutionary optimization to from-scratch optimization of the controller through discrete event simulation of the MCE system. In the present paper, the authors propose application of evolutionary multi-objective optimization to design of traffic-sensitive MCE controller. The controller for MCE is optimized for different traffic conditions in multi-objective way. By combining the multi-objective optimization with the exemplar-based policy (EBP) representation that has adequate flexibility and generalization ability as a controller, we can successfully design a controller that performs well both in the different traffic conditions and works adequately by generalization in the conditions not used in the optimization process.

The article presents a framework for the resolution of rich vehicle routing problems which are difficult to address with standard optimization techniques. We use local search on the basis on variable neighborhood search for the construction of the solutions, but embed the techniques in a flexible framework that allows the consideration of complex side constraints of the problem such as time windows, multiple depots, heterogeneous fleets, and, in particular, multiple optimization criteria. In order to identify a compromise alternative that meets the requirements of the decision maker, an interactive procedure is integrated in the resolution of the problem, allowing the modification of the preference information articulated by the decision maker. The framework is implemented in a computer system. Results of test runs on multiple depot multi-objective vehicle routing problems with time windows are reported.

This paper introduces a real, unmodified

Many-Objective

optimisation problem for use in optimisation algorithm benchmarking. The radar waveform design problem has 9 objectives and an integer decision space that can be scaled from 4 to 12 decision variables. Proprietary radar waveform design software has been encapsulated in a fast and portable form to facilitate research groups in studying high-order optimisation of real engineering problems.

In most real world optimization problems several optimization goals have to be considered in parallel. For this reason, there has been a growing interest in Multi-Objective Optimization (MOO) in the past years. Several alternative approaches have been proposed to cope with the occurring problems, e.g. how to compare and rank the different elements. The available techniques produce very good results, but they have mainly been studied for problems of “low dimension”, i.e. with less than 10 optimization objectives.

In this paper we study MOO for high dimensional spaces. We first review existing techniques and discuss them in our context. The pros and cons are pointed out. A new relation called

ε-Preferred

is presented that extends existing approaches and clearly outperforms these for high dimensions. Experimental results are presented for a very complex industrial scheduling problem, i.e. a utilization planning problem for a hospital. This problem is also well known as

nurse rostering

, and in our application has more than 20 optimization targets. It is solved using an evolutionary approach. The new algorithms based on relation

ε-Preferred

do not only yield better results regarding quality, but also enhances the robustness significantly.

Many-objective optimization refers to optimization problems with a number of objectives considerably larger than two or three. In this paper, a study on the performance of the Fast Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) for handling such many-objective optimization problems is presented. In its basic form, the algorithm is not well suited for the handling of a larger number of objectives. The main reason for this is the decreasing probability of having Pareto-dominated solutions in the initial external population. To overcome this problem, substitute distance assignment schemes are proposed that can replace the crowding distance assignment, which is normally used in NSGA-II. These distances are based on measurement procedures for the highest degree, to which a solution is nearly Pareto-dominated by any other solution: like the number of smaller objectives, the magnitude of all smaller or larger objectives, or a multi-criterion derived from the former ones. For a number of many-objective test problems, all proposed substitute distance assignments resulted into a strongly improved performance of the NSGA-II.

Research within the area of Evolutionary Multi-objective Optimization (EMO) focused on two- and three-dimensional objective functions, so far. Most algorithms have been developed for and tested on this limited application area. To broaden the insight in the behavior of EMO algorithms (EMOA) in higher dimensional objective spaces, a comprehensive benchmarking is presented, featuring several state-of-the-art EMOA, as well as an aggregative approach and a restart strategy on established scalable test problems with three to six objectives. It is demonstrated why the performance of well-established EMOA (NSGA-II, SPEA2) rapidly degradates with increasing dimension. Newer EMOA like

ε

-MOEA, MSOPS, IBEA and SMS-EMOA cope very well with high-dimensional objective spaces. Their specific advantages and drawbacks are illustrated, thus giving valuable hints for practitioners which EMOA to choose depending on the optimization scenario. Additionally, a new method for the generation of weight vectors usable in aggregation methods is presented.

The scalability of EMO algorithms is an issue of significant concern for both algorithm developers and users. A key aspect of the issue is scalability to objective space dimension, other things being equal. Here, we make some observations about the efficiency of search in discrete spaces as a function of the number of objectives, considering both uncorrelated and correlated objective values. Efficiency is expressed in terms of a cardinality-based (scaling-independent) performance indicator. Considering random sampling of the search space, we measure, empirically, the fraction of the true PF covered after

p

iterations, as the number of objectives grows, and for different correlations. A general analytical expression for the expected performance of random search is derived, and is shown to agree with the empirical results. We postulate that for even moderately large numbers of objectives, random search will be competitive with an EMO algorithm and show that this is the case empirically: on a function where each objective is relatively easy for an EA to optimize (an NK-landscape with K=2), random search compares favourably to a well-known EMO algorithm when objective space dimension is ten, for a range of inter-objective correlation values. The analytical methods presented here may be useful for benchmarking of other EMO algorithms.

In our recent publication [1], we began with an understanding that many real-world applications of multi-objective optimization involve a large number (10 or more) of objectives but then, existing evolutionary multi-objective optimization (EMO) methods have primarily been applied to problems having smaller number of objectives (5 or less). After highlighting the major impediments in handling large number of objectives, we proposed a principal component analysis (PCA) based EMO procedure, for dimensionality reduction, whose efficacy was demonstrated by solving upto 50-objective optimization problems. Here, we are addressing the fact that, when the data points live on a non-linear manifold or that the data structure is non-gaussian, PCA which yields a smaller dimensional ’linear’ subspace may be ineffective in revealing the underlying dimensionality. To overcome this, we propose two new non-linear dimensionality reduction algorithms for evolutionary multi-objective optimization, namely C-PCA-NSGA-II and MVU-PCA-NSGA-II. While the former is based on the newly introduced correntropy PCA [2], the later implements maximum variance unfolding principle [3,4,5] in a novel way. We also establish the superiority of these new EMO procedures over the earlier PCA-based procedure, both in terms of accuracy and computational time, by solving upto 50-objective optimization problems.

With the popularity of efficient multi-objective evolutionary optimization (EMO) techniques and the need for such problem-solving activities in practice, EMO methodologies and EMO research and application have received a great deal of attention in the recent past. The first decade of research in EMO area has been spent on developing efficient algorithms for finding a well-converged and well-distributed set of Pareto-optimal solutions, although EMO researchers were always aware of the importance of procedures which would help choose one particular solution from the Pareto-optimal set for implementation. In this paper, we address this long-standing issue and suggest an interactive EMO procedure by collating most salient research in EMO and putting together a step-by-step EMO and decision-making procedure. The idea is implemented in a GUI-based, user-friendly software which allows a user to supply the problem mathematically or by using user-defined macros and enables the user to evaluate solutions directly or by calling an executable software, such as popularly-used MATLAB software for a local search or ANSYS software for finite element analysis, etc. Starting with standard EMO applications, continuing to finding robust, partial, and user-defined preferred frontiers through standard MCDM procedures, the well-coordinated software allows the user to first have an idea of the complete trade-off frontier, then systematically focus in preferred regions, and finally choose a single solution for implementation.

Most real-world optimization problems involve objectives, constraints, and parameters which constantly change with time. Treating such problems as a stationary optimization problem demand the knowledge of the pattern of change a priori and even then the procedure can be computationally expensive. Although dynamic consideration using evolutionary algorithms has been made for single-objective optimization problems, there has been a lukewarm interest in formulating and solving dynamic multi-objective optimization problems. In this paper, we modify the commonly-used NSGA-II procedure in tracking a new Pareto-optimal front, as soon as there is a change in the problem. Introduction of a few random solutions or a few mutated solutions are investigated in detail. The approaches are tested and compared on a test problem and a real-world optimization of a hydro-thermal power scheduling problem. This systematic study is able to find a minimum frequency of change allowed in a problem for two dynamic EMO procedures to adequately track Pareto-optimal frontiers on-line. Based on these results, this paper also suggests an automatic decision-making procedure for arriving at a dynamic single optimal solution on-line.

Evolutionary Multi-objective Optimization (EMO) is ex-pected to be a powerful optimization framework for real world problems such as engineering design. Recent progress in automatic control and instrumentation provides a smart environment called Hardware In the Loop Simulation (HILS). It is available for our target application, that is, the experiment-based optimization. However, since Multi-objective Evolutionary Algorithms (MOEAs) require a large number of evaluations, it is difficult to apply it to real world problems of costly evaluation. To make experiment-based EMO using the HILS environment feasible, the most important pre-requisite is to reduce the number of necessary fitness evaluations. In the experiment-based EMO, the performance analysis of the evaluation reduction under the uncertainty such as observation noise is highly important, although the previous works assume noise-free environments. In this paper, we propose an evaluation reduction to overcome the above-mentioned problem by selecting the solution candidates by means of the estimated fitness before applying them to the real experiment in MOEAs. We call this technique Pre-selection. For the estimation of fitness, we adopt locally weighted regression. The effectiveness of the proposed method is examined by numerical experiments.

Optimization in changing environment is a challenging task, especially when multiple objectives are to be optimized simultaneously. The basic idea to address dynamic optimization problems is to utilize history information to guide future search. In this paper, two strategies for population re-initialization are introduced when a change in the environment is detected. The first strategy is to predict the new location of individuals from the location changes that have occurred in the history. The current population is then partially or completely replaced by the new individuals generated based on prediction. The second strategy is to perturb the current population with a Gaussian noise whose variance is estimated according to previous changes. The prediction based population re-initialization strategies, together with the random re-initialization method, are then compared on two bi-objective test problems. Conclusions on the different re-initialization strategies are drawn based on the preliminary empirical results.

In this paper, a new type of Multi-Objective Problems (MOPs) is introduced and formulated. The new type is an outcome of a motivation to find optimal solutions for different MOPs, which are coupled through communal components. Therefore, in such cases a multi-Multi-Objective Optimization Problem (m-MOOP) has to be considered. The solution to the m-MOOP is defined and an approach to search for it by applying an EMO algorithm sequentially is presented. This method, although not always resulting in the individual MOPs’ Pareto fronts, nevertheless gives solutions to the m-MOOP problem in hand. Several measures that allow the assessment of the introduced approach are offered. To demonstrate the approach and its applicability, academic examples as well as a "real-life," engineering example, are given.

The design of quality measures for approximations of the Pareto-optimal set is of high importance not only for the performance assessment, but also for the construction of multiobjective optimizers. Various measures have been proposed in the literature with the intention to capture different preferences of the decision maker. A quality measure that possesses a highly desirable feature is the hypervolume measure: whenever one approximation completely dominates another approximation, the hypervolume of the former will be greater than the hypervolume of the latter. Unfortunately, this measure—as any measure inducing a total order on the search space—is biased, in particular towards convex, inner portions of the objective space. Thus, an open question in this context is whether it can be modified such that other preferences such as a bias towards extreme solutions can be obtained. This paper proposes a methodology for quality measure design based on the hypervolume measure and demonstrates its usefulness for three types of preferences.

Considering external parameters during any evaluation leads to an optimization problem which has to handle several concurrent multi objective problems at once. This novel challenge, the Multiple Multi Objective Problem M-MOP, is defined and analyzed. Guidelines and metrics for the development of M-MOP optimizers are generated and exemplary demonstrated at an extended version of Deb’s NSGA-II algorithm. The relationship to the classical MOPs is highlighted and the usage of performance metrics for the M-MOP is discussed. Due to the increased number of dimensions the M-MOP represents a complex optimization task that should be settled in the optimization community.

Recent studies show that evolutionary optimizers are effective tools in solving real-world problem with complex and competing specifications. As more advanced multiobjective evolutionary optimizers (MOEO) are being designed and proposed, the issue of performance assessment has become increasingly important. While performance assessment could be done via theoretical and empirical approach, the latter is more practical and effective and has been adopted as the de facto approach in the evolutionary multiobjective optimization community. However, researches pertinent to empirical study have focused mainly on its individual components like test metrics and functions, there are limited discussions on the overall adequacy of empirical test in substantiating their statements made on the performance and behavior of the evaluated algorithm. As such, this paper aims to provide a holistic perspective towards the empirical investigation of MOEO and present a conceptual framework, which researchers could consider in the design and implementation of MOEO experimental study. This framework comprises of a structural algorithmic development plan and a general theory of adequacy in the context of evolutionary multiobjective optimization.

Multiobjective optimisation has traditionally focused on problems consisting of 2 or 3 objectives. Real-world problems often require the optimisation of a larger number of objectives. Research has shown that conclusions drawn from experimentations carried out on 2 or 3 objectives cannot be generalized for a higher number of objectives. The curse of dimensionality is a problem that faces decision makers when confronted with many objectives. Preference articulation techniques, and especially progressive preference articulation (PPA) techniques are effective methods for supporting the decision maker. In this paper, some of the most recent and most established PPA techniques are examined, and their utility for tackling

many

-objective optimisation problems is discussed and compared from the viewpoint of the decision maker.

Pareto optimization methods are usually expected to find well-distributed approximations of Pareto fronts with basic geometry, such as smooth, convex and concave surfaces. In this contribution, test-problems are proposed for which the Pareto front is the intersection of a Lamé supersphere with the positive ℝ

n

-orthant. Besides scalability in the number of objectives and decision variables, the proposed test problems are also scalable in a characteristic we introduce as

resolvability of conflict

, which is closely related to convexity/concavity, curvature and the position of knee-points of the Pareto fronts.

As a very basic bi-objective problem we propose a generalization of Schaffer’s problem. We derive closed-form expressions for the efficient sets and the Pareto fronts, which are arcs of Lamé supercircles. Adopting the bottom-up approach of test problem construction, as used for the DTLZ test-problem suite, we derive test problems of higher dimension that result in Pareto fronts of superspherical geomery.

Geometrical properties of these test-problems, such as concavity and convexity and the position of knee-points are studied. Our focus is on geometrical properties that are useful for performance assessment, such as the dominated hypervolume measure of the Pareto fronts. The use of these test problems is exemplified with a case-study using the SMS-EMOA, for which we study the distribution of solution points on different 3-D Pareto fronts.

Evolutionary algorithms have become a very popular approach for multiobjective optimization in many fields of engineering. Due to the outstanding performance of such techniques, new approaches are constantly been developed and tested to improve convergence, tackle new problems, and reduce computational cost. Recently, a new class of algorithms, based on ideas from the immune system, have begun to emerge as problem solvers in the evolutionary multiobjective optimization field. Although all these immune algorithms present unique, individual characteristics, there are some trends and common characteristics that, if explored, can lead to a better understanding of the mechanisms governing the behavior of these techniques. In this paper we propose a common framework for the description and analysis of multiobjective immune algorithms.