Practical multi-criterion decision making not only involves the articulation of preferences in the objective space, but also a consideration of how the variables impact these preferences. Trend mining is a recently proposed visualization technique that offers the decision maker a quick overview of the variables' effect on the structure of the objective space and easily discover interesting variable trends. The original trend mining approach relies on a set of predefined reference directions along which an interestingness score is measured for each variable. In this paper, we relax this requirement by automating the approach to find optimal reference directions that maximize the interestingness for each variable. Additional extensions include the use of an Achievement Scalarizing Function (ASF) for ranking solutions along a given reference direction, and an updated interestingness score formulation for more appropriately handling discrete variables. We demonstrate the working of the extended approach on DTLZ2 and WFG2 benchmarks for up to five objectives and on a biobjective engineering design problem. The results show that the ability of the proposed approach to detect variable trends in high dimensional objective spaces is heavily dependent on the quality of the solutions used.

Metamodeling plays an important role in simulation-based optimization by providing computationally inexpensive approximations for the objective and constraint functions. Additionally metamodeling can also serve to filter noise, which is inherent in many simulation problems causing optimization algorithms to be mislead. In this paper, we conduct a thorough statistical comparison of four popular metamodeling methods with respect to their approximation accuracy at various levels of noise. We use six scalable benchmark problems from the optimization literature as our test suite. The problems have been chosen to represent different types of fitness landscapes, namely, bowl-shaped, valley-shaped, steep ridges and multi-modal, all of which can significantly influence the impact of noise. Each metamodeling technique is used in combination with four different noise handling techniques that are commonly employed by practitioners in the field of simulation-based optimization. The goal is to identify the metamodeling strategy, i.e. a combination of metamodeling and noise handling, that performs significantly better than others on the fitness landscapes under consideration. We also demonstrate how these results carry over to a simulation-based optimization problem concerning a scalable discrete event model of a simple but realistic production line.

The use of optimization to solve a simulation-based multi-objective problem produces a set of solutions that provide information about the trade-offs that have to be considered by the decision maker. An incomplete or sub-optimal set of solutions will negatively affect the quality of any subsequent decisions. The parameters that control the search behavior of an optimization algorithm can be used to minimize this risk. However, choosing good parameter settings for a given optimization algorithm and problem combination is difficult. The aim of this paper is to take a step towards optimal parameter settings for optimization of simulation-based problems. Two parameter tuning methods, Latin Hypercube Sampling and Genetic Algorithms, are used to maximize the performance of NSGA-II applied to a simulation-based problem with discrete variables. The strengths and weaknesses of both methods are analyzed. The effect of the number of decision variables and the function budget on the optimal parameter settings is also studied.

We consider a bilevel parameter tuning problem where the goal is to maximize the performance of a given multi-objective evolutionary optimizer on a given problem. The search for optimal algorithmic parameters requires the assessment of several sets of parameters, through multiple optimization runs, in order to mitigate the effect of noise that is inherent to evolutionary algorithms. This task is computationally expensive and therefore, in this paper, we propose to use sampling and metamodeling to approximate the performance of the optimizer as a function of its parameters. While such an approach is not unheard of, the choice of the metamodel to be used still remains unclear. The aim of this paper is to empirically compare 11 different metamodeling techniques with respect to their accuracy and training times in predicting two popular multi-objective performance metrics, namely, the hypervolume and the inverted generational distance. For the experiments in this pilot study, NSGA-II is used as the multi-objective optimizer for solving ZDT problems, 1 through 4.

In simulation-based Evolutionary Multi-objective Optimization the number of simulation runs is very limited, since the complex simulation models require long execution times. With the help of preference information, the optimization result can be improved by guiding the optimization towards relevant areas in the objective space, for example with the R-NSGA-II algorithm [9], which uses a reference point specified by the decision maker. When stochastic systems are simulated, the uncertainty of the objective values might degrade the optimization performance. By sampling the solutions multiple times this uncertainty can be reduced. However, resampling methods reduce the overall number of evaluated solutions which potentially worsens the optimization result. In this article, a Dynamic Resampling strategy is proposed which identifies the solutions closest to the reference point which guides the population of the Evolutionary Algorithm. We apply a single-objective Ranking and Selection resampling algorithm in the selection step of R-NSGA-II, which considers the stochastic reference point distance and its variance to identify the best solutions. We propose and evaluate different ways to integrate the sampling allocation method into the Evolutionary Algorithm. On the one hand, the Dynamic Resampling algorithm is made adaptive to support the EA selection step, and it is customized to be used in the time-constrained optimization scenario. Furthermore, it is controlled by other resampling criteria, in the same way as other hybrid DR algorithms. On the other hand, R-NSGA-II is modified to rely more on the scalar reference point distance as fitness function. The results are evaluated on a benchmark problem with variable noise landscape.

Evolutionary optimization algorithms typically use one or more parameters that control their behavior. These parameters, which are often kept constant, can be tuned to improve the performance of the algorithm on specific problems. However, past studies have indicated that the performance can be further improved by adapting the parameters during runtime. A limitation of these studies is that they only control, at most, a few parameters, thereby missing potentially beneficial interactions between them. Instead of finding a direct control mechanism, the novel approach in this paper is to use different parameter sets in different stages of an optimization. These multiple parameter sets, which remain static within each stage, are tuned through extensive bi-level optimization experiments that approximate the optimal adaptation of the parameters. The algorithmic performance obtained with tuned multiple parameter sets is compared against that obtained with a single parameter set. For the experiments in this paper, the parameters of NSGA-II are tuned when applied to the ZDT, DTLZ and WFG test problems. The results show that using multiple parameter sets can significantly increase the performance over a single parameter set.

Evolutionary optimization algorithms have parameters that are used to adapt the search strategy to suit different optimization problems. Selecting the optimal parameter values for a given problem is difficult without a-priori knowledge. Experimental studies can provide this knowledge by finding the best parameter values for a specific set of problems. This knowledge can also be constructed into heuristics (rule-of-thumbs) that can adapt the parameters for the problem. The aim of this paper is to assess the heuristics of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) optimization algorithm. This is accomplished by tuning CMA-ES parameters so as to maximize its performance on the CEC'15 problems, using a bilevel optimization approach that searches for the optimal parameter values. The optimized parameter values are compared against the parameter values suggested by the heuristics. The difference between specialized and generalized parameter values are also investigated.

Evolutionary algorithms are often applied to solve multi-objective optimization problems. Such algorithms effectively generate solutions of wide spread, and have good convergence properties. However, they do not provide any characteristics of the fou

Multi-objective evolutionary algorithms (MOEAs) are often criticized for their high-computational costs. This becomes especially relevant in simulation-based optimization where the objectives lack a closed form and are expensive to evaluate. Over the years, meta-modeling or surrogate modeling techniques have been used to build inexpensive approximations of the objective functions which reduce the overall number of function evaluations (simulations). Some recent studies however, have pointed out that accurate models of the objective functions may not be required at all since evolutionary algorithms only rely on the relative ranking of candidate solutions. Extending this notion to MOEAs, algorithms which can ‘learn’ Pareto-dominance relations can be used to compare candidate solutions under multiple objectives. With this goal in mind, in this paper, we study the performance of ten different off-the-shelf classification algorithms for learning Pareto-dominance relations in the ZDT test suite of benchmark problems. We consider prediction accuracy and training time as performance measures with respect to dimensionality and skewness of the training data. Being a preliminary study, this paper does not include results of integrating the classifiers into the search process of MOEAs.

In Evolutionary Multi-objective Optimization many solutions have to be evaluated to provide the decision maker with a diverse choice of solutions along the Pareto-front, in particular for high-dimensional optimization problems. In Simulation-based Optimization the modeled systems are complex and require long simulation times. In addition the evaluated systems are often stochastic and reliable quality assessment of system configurations by resampling requires many simulation runs. As a countermeasure for the required high number of simulation runs caused by multiple optimization objectives the optimization can be focused on interesting parts of the Pareto-front, as it is done by the Reference point-guided NSGA-II algorithm (R-NSGA-II) [9]. The number of evaluations needed for the resampling of solutions can be reduced by intelligent resampling algorithms that allocate just as much sampling budget needed in different situations during the optimization run. In this paper we propose and compare resampling algorithms that support the R-NSGA-II algorithm on optimization problems with stochastic evaluation functions.

Evolutionary Multi-objective Optimization aims at finding a diverse set of Pareto-optimal solutions whereof the decision maker can choose the solution that fits best to her or his preferences. In case of limited time (of function evaluations) for optimization this preference information may be used to speed up the search by making the algorithm focus directly on interesting areas of the objective space. The R-NSGA-II algorithm [1] uses reference points to which the search is guided specified according to the preferences of the user. In this paper, we propose an extension to R-NSGA-II that limits the Pareto-fitness to speed up the search for a limited number of function calls. It avoids to automatically select all solutions of the first front of the candidate set into the next population. In this way non-preferred Pareto-optimal solutions are not considered thereby accelerating the search process. With focusing comes the necessity to maintain diversity. In R-NSGA-II this is achieved with the help of a clustering algorithm which keeps the found solutions above a minimum distance ε. In this paper, we propose a self-adaptive ε approach that autonomously provides the decision maker with a more diverse solution set if the found Pareto-set is situated further away from a reference point. Similarly, the approach also varies the diversity inside of the Pareto-set. This helps the decision maker to get a better overview of the available solutions and supports decisions about how to adapt the reference points.

This paper presents a new efficient multi-objective evolutionary algorithm for solving computationally-intensive optimization problems. To support a high degree of parallelism, the algorithm is based on a steady-state design. For improved efficiency the algorithm utilizes a surrogate to identify promising candidate solutions and filter out poor ones. To handle the uncertainties associated with the approximative surrogate evaluations, a new method for multi-objective optimization is described which is generally applicable to all surrogate techniques. In this method, basically, surrogate objective values assigned to offspring are adjusted to consider the error of the surrogate. The algorithm is evaluated on the ZDT benchmark functions and on a real-world problem of manufacturing optimization. In assessing the performance of the algorithm, a new performance metric is suggested that combines convergence and diversity into one single measure. Results from both the benchmark experiments and the real-world test case indicate the potential of the proposed algorithm.

The development of control programs is a complex task that needs to be systematised and automated in order to reduce costs. It is demonstrated within the software engineering discipline that the use of analysis and modelling methodologies helps to reduce the software costs by systematising its development. Several methodologies have been designed to analyse and model batch and real-time systems, but not discrete event sequential systems (DESS). This paper demonstrates that the DESS control programs development process can benefit, as batch and real-time systems do, From the improvements in the software engineering, by designing a new object-oriented methodology adapted to the DESS peculiarities.