Multiobjective Optimization with Fuzzy Classification-Assisted Environmental Selection

In recent years, multi-objective evolutionary algorithms (MOEAs) have been widely used to handle various water resources problems. One challenge within MOEAs’ applications is a lack of understanding on how various operators alter a MOEA’s behavior to achieve its final performance (i.e., MOEAs are black-boxes to practitioners), and hence it is difficult to select the most appropriate operators to ensure the MOEA’s best performance for a given real-world problem. To address this issue, this study proposes the use of the run-time measure metrics to reveal the underlying searching behavior of the MOEA’s operators. The proposed methodology is demonstrated by the non-dominated sorting genetic algorithm II (NSGA-II, a widely used MOEA in water resources) with five commonly used crossover operators applied to six water distribution system design problems. Results show that the simulated binary crossover (SBX) and the simplex crossover (SPX) operators possess great ability in extending the front and finding Pareto-front solutions, respectively, while the naive crossover (NVX) strategy exhibits the overall worst performance in identifying optimal fronts. The obtained understanding on the operators’ searching behavior not only offers guidance for selecting appropriate operators for real-world water resources problems, but also builds fundamental knowledge for developing more advanced MOEAs in future.

The potential and effectiveness of the newly developed Pareto-based multiobjective evolutionary algorithms (MOEA) for solving a real-world power system multiobjective nonlinear optimization problem are comprehensively discussed and evaluated in this paper. Specifically, nondominated sorting genetic algorithm, niched Pareto genetic algorithm, and strength Pareto evolutionary algorithm (SPEA) have been developed and successfully applied to an environmental/economic electric power dispatch problem. A new procedure for quality measure is proposed in this paper in order to evaluate different techniques. A feasibility check procedure has been developed and superimposed on MOEA to restrict the search to the feasible region of the problem space. A hierarchical clustering algorithm is also imposed to provide the power system operator with a representative and manageable Pareto-optimal set. Moreover, an approach based on fuzzy set theory is developed to extract one of the Pareto-optimal solutions as the best compromise one. These multiobjective evolutionary algorithms have been individually examined and applied to the standard IEEE 30-bus six-generator test system. Several optimization runs have been carried out on different cases of problem complexity. The results of MOEA have been compared to those reported in the literature. The results confirm the potential and effectiveness of MOEA compared to the traditional multiobjective optimization techniques. In addition, the results demonstrate the superiority of the SPEA as a promising multiobjective evolutionary algorithm to solve different power system multiobjective optimization problems.

As there is a growing interest in applications of multi-objective optimization methods to real-world problems, it is essential to develop efficient algorithms to achieve better performance in engineering design and resources optimization. An efficient algorithm for multi-objective optimization, based on swarm intelligence principles, is presented in this article. The proposed algorithm incorporates a Pareto dominance relation into particle swarm optimization (PSO). To create effective selection pressure among the non-dominated solutions, it uses a variable size external repository and crowding distance comparison operator. An efficient mutation strategy called elitist-mutation is also incorporated in the algorithm. This strategic mechanism effectively explores the feasible search space and speeds up the search for the true Pareto-optimal region. The proposed approach is tested on various benchmark problems taken from the literature and validated with standard performance measures by comparison with NSGA-II, one of the best multi-objective evolutionary algorithms available at present. It is then applied to three engineering design problems. The results obtained amply demonstrate that the proposed approach is efficient and is able to yield a wide spread of solutions with good coverage and convergence to true Pareto-optimal fronts.

In real-world multi-objective problems, each objective has a totally different scale. However, some frequently-used multi-objective evolutionary algorithms (MOEAs) have no objective space normalization mechanisms. The effect of objective space normalization on the performance of decomposition-based MOEAs (e.g., MOEA/D and NSGA-III) has already been examined for artificial test problems (e.g., DTLZ and WFG) in the literature. In this paper, we examine its practical usefulness for real-world multi-objective problems using various MOEAs. Our experimental results clearly show that objective space normalization is needed not only in decomposition-based MOEAs but also in hypervolume-based MOEAs. We also explain why objective space normalization is needed in these two types of MOEAs.

Many real-world optimization problems are characterized by multiple conflicting objectives, which are known as multi-objective optimization problems (MOPs). In the last two decades, evolutionary algorithms have shown promising performance in solving various MOPs, and a large number of multi-objective evolutionary algorithms (MOEAs) have been proposed. In order to determine the most suitable MOEA for a specific MOP, it is usually necessary to perform experiments to compare the performance of multiple candidate MOEAs. In 2017, an evolutionary multi-objective optimization platform was proposed by us, called PlatEMO, which provides the source codes of many state-of-the-art MOEAs and helps researchers perform batch experiments on these MOEAs. In this work, we illustrate the method of using the newest version of PlatEMO to solve MOPs in applications, by means of a case study on the feature selection problem, which is an important and difficult task in machine learning and data mining. This paper details the method of adding the feature selection problem to PlatEMO, and presents the experimental results of eight MOEAs on nine datasets in feature selection.

In data center companies, cloud computing can host multiple types of heterogeneous virtual machines (VMs) and provide many features, including flexibility, security, support, and even better maintenance than traditional centers. However, some issues need to be considered, such as the optimization of energy usage, utilization of resources, reduction of time consumption, and optimization of virtual machine placement. Therefore, this paper proposes an alternative multiobjective optimization (MOP) approach that combines the salp swarm and sine-cosine algorithms (MOSSASCA) to determine a suitable solution for virtual machine placement (VMP). The objectives of the proposed MOSSASCA are to maximize mean time before a host shutdown (MTBHS), to reduce power consumption, and to minimize service level agreement violations (SLAVs). The proposed method improves the salp swarm and the sine-cosine algorithms using an MOP technique. The SCA works by using a local search approach to improve the performance of traditional SSA by avoiding trapping in a local optimal solution and by increasing convergence speed. To evaluate the quality of MOSSASCA, we perform a series of experiments using different numbers of VMs and physical machines. The results of MOSSASCA are compared with well-known methods, including the nondominated sorting genetic algorithm (NSGA-II), multiobjective particle swarm optimization (MOPSO), a multiobjective evolutionary algorithm with decomposition (MOEAD), and a multiobjective sine-cosine algorithm (MOSCA). The results reveal that MOSSASCA outperforms the compared methods in terms of solving MOP problems and achieving the three objectives. Compared with the other methods, MOSSASCA exhibits a better ability to reduce power consumption and SLAVs while increasing MTBHS. The main differences in terms of power consumption between the MOSCA, MOPSO, MOEAD, and NSGA-II and the MOSSASCA are 0.53, 1.31, 1.36, and 1.44, respectively. Additionally, the MOSSASCA has higher MTBHS value than MOSCA, MOPSO, MOEAD, and NSGA-II by 362.49, 274.70, 585.73 and 672.94, respectively, and the proposed method has lower SLAV values than MOPSO, MOEAD, and NSGA-II by 0.41, 0.28, and 1.27, respectively.

Most of real-world optimization problems correspond to constrained multi-objective optimization problems (CMOPs). Multi-objective evolutionary algorithms (MOEAs) are useful to find out diverse Pareto-optimal solutions in CMOPs, and has been applied in various engineering fields. In order to expand the range of application of MOEAs further, we propose an improved evolutionary algorithm with parallel evaluation strategy (EAPES). In EAPES, feasible solutions and infeasible solution are separately stored in different populations, and infeasible solutions are evaluated in an unusual manner where not only feasible solutions but also useful infeasible solutions are used as parents to reproduce the populations for the next generation. Infeasible solutions are ranked based on a scalarizing function determined by objective function values and a total constraint violation value. This paper investigates the performance of the proposed EAPES to search for Pareto-optimal solutions compared to NSGA-II and the previous EAPES. The proposed EAPES with a well-tuned parameter is most capable to explore Pareto-optimal solutions with good diversity, spread, and convergence to the true Pareto front. The proposed EAPES assigns bad rank to the infeasible solutions that exist away from the true Pareto front, and does not store such solutions. Thus the proposed EAPES exhibits a higher searching capability than the previous EAPES by evaluating infeasible solutions in an appropriate balance between objective functions and total constraint violation. In addition, the results suggest that the proposed EAPES may exhibit high solution search performance even in more difficult CMOPs, which have more objective functions and/or more constraints.

The two primary goals of a multi-objective evolutionary algorithm (MOEA) for solving multi-objective optimization problems are to search as much non-dominated solutions as possible towards the true Pareto front and to maintain diversity of Pareto optimal solutions along tradeoff curves. This study presents a new hybrid MOEA, the niched Pareto tabu search combined with genetic algorithm (NPTSGA), to find Pareto-optimal solutions to multi-objective optimization problems. The NPTSGA is developed on the thoughts of integrating genetic algorithm (GA) with the improved tabu search (TS) based MOEA, niched Pareto tabu search (NPTS). The proposed NPTSGA is then tested through a simple test example and compared with other two techniques, NPTS and niched Pareto genetic algorithm (NPGA). Computational results indicate that the proposed NPTSGA is an efficient and effective method for solving multi-objective problems, while keeping the balance between the intensification of non-domination to the true Pareto-optimal solutions (TPS) and the diversification of the near Pareto-optimal solutions (NPS) along the tradeoff curves.

An Enhanced multi-objective evolutionary optimization algorithm based on decomposition (E-MOEA-D) proposed for optimal placement of Distributed Generation (DG) and Electric Vehicle (EV) Fast Charging Station (FCS) in distribution system. The diversity of the evolutionary algorithm improves the convergence and diverse solution in the process of evolutionary optimization. The proposed algorithm is improved using enhanced diversity algorithm, which yield diverse candidate solutions in population. The optimal placement of DGs and FCS are formulated using three objective functions as i) Active power loss ii) Voltage deviation iii) DG cost. The proposed algorithm is simulated on IEEE-33 bus distribution system. The proposed algorithm is compared with other competitive multi-objective evolutionary algorithms such as decomposition based multi-objective evolutionary algorithm (MOEA-D) and Non-dominated sorting multi-objective evolutionary algorithm (NSGA-II).

A computationally efficient procedure for multiobjective design optimization with variable-fidelity models and response surface surrogates is presented. The proposed approach uses the multiobjective evolutionary algorithm that works with a fast surrogate model, obtained with kriging interpolation of the low-fidelity model data enhanced by space-mapping correction exploiting a few high-fidelity training points. The initial Pareto front generated by multiobjective optimization of the surrogate using the multiobjective evolutionary algorithm can be iteratively refined by local enhancements of the surrogate model. The latter are realized with a space-mapping response correction based on a limited number of high-fidelity training points allocated along the initial Pareto front. The proposed method allows us to obtain, at a low computational cost, a set of designs representing tradeoffs between the conflicting objectives. The current approach is illustrated using examples of airfoil design: one in transonic flow, involving aerodynamics tradeoffs; and another one in low-speed flow, involving tradeoffs between the aerodynamic and the aeroacoustic performances.

A fast sampling based evolutionary algorithm for million-dimensional multiobjective optimization

A Simplex Crossover based evolutionary algorithm including the genetic diversity as objective

Broad learning approach to Surrogate-Assisted Multi-Objective evolutionary fuzzy clustering algorithm based on reference points for color image segmentation

Many real world problems are characterized by multiple goals, often conflicting in nature and compete with one another. Multi-objective optimization problems (MOOPs) require the simultaneous optimization of several non-commensurable and conflicting objectives. In the past, several studies have used conventional approaches to solve the MOOPs by adopting weighted approach or constrained approach, which may face difficulties while generating Pareto optimal solutions, if optimal solution lies on non-convex or disconnected regions of the objective function space. An effective algorithm should have an ability to learn from earlier performance to direct proper selection of weights for further evolutions. To achieve these goals, multi-objective evolutionary algorithms (MOEAs) have become effective means in recent past, which can generate a population of solutions in each iteration and offer a set of alternatives in a single run. This chapter presents an effective MOEA, namely multi-objective differential evolution (MODE) for problems of solving water, environmental systems.

Pareto compliance is a critical property of quality indicators (QIs) focused on measuring convergence to the Pareto front. This property allows a QI not to contradict the order imposed by the Pareto dominance relation. Hence, Pareto-compliant QIs are remarkable when comparing approximation sets of multi-objective evolutionary algorithms (MOEAs) since they do not produce misleading conclusions. However, the practical effect of Pareto-compliant QIs as the backbone of MOEAs' survival mechanisms is not entirely understood. In this paper, we study the practical effect of IGD++ (which is a combination of IGD+ and the hypervolume indicator), IGD+, and IGD, which are Pareto-compliant, weakly Pareto-compliant, and not Pareto-compliant QIs, respectively. To this aim, we analyze the convergence and diversity properties of steady-state MOEAs based on the previously mentioned QIs throughout the whole evolutionary process. Our experimental results showed that, in general, the practical effect of a Pareto-compliant QI in a selection mechanism is not very significant concerning weaker QIs, taking into account the whole evolutionary process.