Crowding Distance Calculation Research Articles

Learnable self-supervised support vector machine based individual selection strategy for multimodal multi-objective optimization

Multimodal multi-objective optimization problem (MMOP) is a frontier research problem, which can provide decision makers with more choices without making trade-offs. Many multimodal multi-objective evolutionary algorithms (MMOEAs) have been proposed to solve MMOP. However, most MMOEAs tend to prioritize the objective dominance of individuals in the process of individual selection, and only individuals with the same objective dominance will be considered the diversity, which leads to the loss of many promising solutions. To solve the above problem, this paper proposes a learnable self-supervised support vector machine (SVM) based multimodal multi-objective optimization algorithm (SVMEA). Support vector machine can learn the knowledge about distinguishing the advantages and disadvantages of individuals from the data in the existing training set and select individuals, in which the objective dominance of individuals is as important as diversity. Moreover, a crowding distance calculation method based on Manhattan distance is designed. Compared with the traditional method using Euclidean distance to calculate crowding distance, it can better evaluate the diversity of individuals in the decision space and assist the selection of elite solutions. Experimental results show that the proposed SVMEA is competitive with seven other advanced MMOEAs on 34 benchmark problems and a practical application problem.

A Hybrid Chaotic-Based Multiobjective Differential Evolution Technique for Economic Emission Dispatch Problem

The economic emission dispatch problem (EEDP) is a nonconvex and nonsmooth multiobjective optimization problem in the power system field. Generally, fuel cost and total emissions of harmful gases are the problem objective functions. The EEDP decision variables are output powers of thermal generating units (TGUs). To make the EEDP problem more practical, valve point loading effects (VPLEs), prohibited operation zones (POZs), and power balance constraints should be included in the problem constraints. In order to solve this complex and constrained EEDP, a new multiobjective optimization technique combining the differential evolution (DE) algorithm and chaos theory is proposed in this study. In this new multiobjective optimization technique, a nondomination sorting principle and a crowding distance calculation are employed to extract an accurate Pareto front. To avoid being trapped in local optima and enhance the conventional DE algorithm, two different chaotic maps are used in its initialization, crossover, and mutation phases instead of random numbers. To overcome difficulties caused by the equality constraint describing the power balance constraint, a slack TGU is defined to compensate for the gap between the total generation and the sum of the system load and total power losses. Then, the optimal power outputs of all thermal units except the slack unit are determined by the suggested optimization technique. To assess the effectiveness and applicability of the proposed method for solving the EEDP, the six-unit and ten-unit systems are used. Moreover, obtained results are compared with other new optimization techniques already developed and tested for the same purpose. The superior performance of the ChMODE is also evaluated by using various metrics such as inverted generational distance (IGD), hyper-volume (HV), spacing metric (SM), and the average satisfactory degree (ASD).

Energy-Saving Scheduling for Flexible Job Shop Problem with AGV Transportation Considering Emergencies

Emergencies such as machine breakdowns and rush orders greatly affect the production activities of manufacturing enterprises. How to deal with the rescheduling problem after emergencies have high practical value. Meanwhile, under the background of intelligent manufacturing, automatic guided vehicles are gradually emerging in enterprises. To deal with the disturbances in flexible job shop scheduling problem with automatic guided vehicle transportation, a mixed-integer linear programming model is established. According to the traits of this model, an improved NSGA-II is designed, aiming at minimizing makespan, energy consumption and machine workload deviation. To improve solution qualities, the local search operator based on a critical path is designed. In addition, an improved crowding distance calculation method is used to reduce the computation complexity of the algorithm. Finally, the validity of the improvement strategies is tested, and the robustness and superiority of the proposed algorithm are verified by comparing it with NSGA, NSGA-II and SPEA2.

Intelligent train operation based on deep learning from excellent driver manipulation patterns

In the application of deep learning to realize intelligent train operation, there are some problems, such as the single learning task. Especially when using the gradient descent approach to optimize the structure, weight and threshold of a deep network, it is easy in this task to fall into a local optimum. This leads to excessive reliance on manual tuning experience. Aiming at the above issues, this paper proposes a new approach of train manipulation and prediction based on a long short-term memory (LSTM) deep network. From the perspective of automatic hyper-parameter optimization, the gradient-free intelligent search method is principally chosen to optimize the architecture and parameters of a LSTM deep network, so as to improve the manipulation accuracy based on learning from excellent drivers. This method first selects excellent driver data through the Pareto dominance principle and crowding distance calculation; on this basis, a step-by-step method is used to optimize the structure, weight and threshold of the LSTM network. Particularly, in the first step, we adopt a genetic algorithm to search for the optimal deep network structure, which overcomes the problem that the structure is difficult to determine. In the second step, we optimize the parameters of the deep network, a process that is divided into two stages of ‘rough learning’ and ‘precise learning’. In the ‘rough learning’ stage, we use the multi-population chained multi-agent (MPCMA) algorithm to preliminarily optimize the LSTM network parameters. In the ‘precise learning’ stage, the Adam algorithm is applied to further finely optimize the network parameters. Finally, through simulation experiments, it is verified that the proposed method improves the accuracy of train manipulation and prediction, and shows strong robustness in situations of multiple manipulation sequences and different temporary speed limits.

An enhance multimodal multiobjective optimization genetic algorithm with special crowding distance for pulmonary hypertension feature selection

Multiobjective optimization assumes a one-to-one mapping between decisions and objective space, however, this is not always the case. When many variables have the same or equivalent objective value, a multimodal multiobjective issue develops in which more than one Pareto Set (PS) maps to the same Pareto Front (PF). Evolutionary computing research into multimodal multiobjective optimization issues has increased (MMOPs). This paper proposed an enhanced multimodal multiobjective genetic algorithm to crack MMOPs using a special crowding distance calculation (ESNSGA-II). This special crowding distance calculation can consider the diversity of the decision space while paying attention to the diversity of the object space. Then, a unique crossover mechanism is established by combining the simulated binary crossover (SBX) method with the capacity of Pareto solutions to generate offspring solutions. The balance between convergence and diversity in both decision space and object space can be guaranteed synchronously, and PS distribution and PF accuracy may both be enhanced. The proposed ESNSGA-II uses the CEC2020 benchmarks MMF1-MMF8 to assess its properties. Comparing the ESNSGA-II to other recently established multimodal multiobjective evolutionary techniques demonstrates that it is capable of efficiently searching numerous PSs of MMOPs. Finally, the suggested ESNSGA-II is used to address a real MMOP problem of pulmonary hypertension detection via arterial blood gas analysis. The statistical analysis reveals that the suggested ESNSGA-II algorithm outperforms other algorithms on this MMOP, and so may be considered a possible tool for pulmonary hypertension.

A multi-objective elitist feedback teaching–learning-based optimization algorithm and its application

Students generally attempt to improve themselves with different manners in their spare time. Inspired by this fact, this paper presents a multi-objective elitist feedback teaching–learning-based optimization (MEFTO) algorithm for multi-objective optimization problems. To promote the exploration capacity and convergence of the basic teaching–learning-based optimization (TLBO), a feedback phase that simulates the spare-time learning phenomenon is introduced at the end of the learner phase. Students should compare their score with the average class score to select an appropriate means for further improvement. Poorly performing students can learn from the teacher directly for rapid improvement, whereas high-performing students prefer to motivate themselves for reinforcement learning. Non-dominated sorting is incorporated to permit this heuristic to solve problems with several objectives. Crowding distance calculation is adopted to maintain the diversity of the obtained solution set in a single run. Elitism strategy is also employed to provide a great improvement for the algorithm. The performance of the MEFTO algorithm is compared with seven other well-known algorithms by using a set of unconstrained benchmark test problems and three constrained engineering optimization problems. The qualitative and quantitative results indicate that the proposed algorithm can provide considerably competitive results and outperforms the other algorithms.

Economic Emission Dispatch Considering Renewable Energy Resources—A Multi-Objective Cross Entropy Optimization Approach

The conventional electrical power system economic dispatch (ED) often only pursues immediate economic benefits but neglects the harmful environment impacts of gas emissions from thermal power plants. To address this shortfall, economic emission dispatch (EED) has drawn a lot of attention in recent years. With the increasing penetration of renewable generation, the intermittence and uncertainty of renewable energy such as solar power and wind power increase the difficulties of power system scheduling. To enhance the dispatch performance with significant penetration of renewable energy, a modified multi-objective cross entropy algorithm (MMOCE) is proposed in this paper. To solve multi-objective optimization problems, a crowding–distance calculation technique and a novel external archive mechanism are introduced into the conventional cross entropy method. Additionally, the population updating process is simplified by introducing a self-adaptive parameter operator that substitutes the smoothing parameters, while the solution diversity and the adaptability in large scale systems are improved by introducing the crossover operator. Finally, a two-stage evolutionary mechanism further enhances the diversity and the rate of convergence. To verify the efficacy of the proposed MMOCE, eight benchmark functions and three different test systems considering different mixes of renewable energy sources are employed. The dispatch results by the proposed MMOCE are compared with other multi-objective cross entropy algorithms and published heuristic methods, confirming the superiority of the proposed MMOCE over other methods in all test systems.

Differential evolution using improved crowding distance for multimodal multiobjective optimization

In multiobjective optimization, the relationship between decision space and objective space is generally assumed to be a one-to-one mapping, but it is not always the case. In some problems, different variables have the same or similar objective value, which means a many-to-one mapping. In this situation, there is more than one Pareto Set (PS) mapping to the same Pareto Front (PF) and these problems are called multimodal multiobjective problems. This paper proposes a multimodal multiobjective differential evolution algorithm to solve these problems. In the proposed method, the difference vector is generated taking the diversity in both decision and objective space into account. The way to calculate crowding distance is quite different from the others. In the crowding distance calculation process, all the selected individuals are taken into account instead of considering each Pareto rank separately. The crowding distance in decision space is replaced with the weighted sum of Euclidean distances to its neighbors. In the environmental selection process, not all the individuals in top ranks are selected, because some of them may be very crowded. Instead, the potential solutions in the bottom rank are given a chance to evolve. With these operations, the proposed algorithm can maintain multiple PSs of multimodal multiobjective optimization problems and improve the diversity in both decision and objective space. Experimental results show that the proposed method can achieve high comprehensive performance.

Optimization of Wheel Reprofiling Based on the Improved NSGA-II

Wheels are the key components of a train, and the shape of the wheel flange should be maintained to ensure the security of train operations. As a method to maintain the shape at the cost of the diameter size, reprofiling has significant impacts on the lifecycle of a train. A wheel model is built in this paper based on the analysis of the wheel wear features and datasets from Taiyuan locomotives. With the decision variables T i , T i ′ , which describe the reprofiling strategy, we formulate a multiobjective optimization problem simultaneously minimizing the reprofiling numbers and maximizing the serving years. To find the solutions of the multiobjective model, the NSGA-II (nondominated sorting genetic algorithm II) is extended with an alteration of the crowding distance calculation and genetic operators. The improved NSGA-II performs better than other approaches (e.g., fixed reprofiling strategy, changeable reprofiling strategy, and NSGA-II). Meanwhile, outstanding solutions with longer servicing years and less reprofiling are listed in this paper. Our study reveals the relationship between the diameter, flange thickness, and their individual attrition rates and proposes a wear model, multiobjective model, and improved NSGA-II. Compared with existing reprofiling strategies, the strategy recommended in our work can significantly increase the lifecycle of the wheel coupled with a low repair frequency.

Multiobjective Drilling Trajectory Optimization Considering Parameter Uncertainties

This article is concerned with the trajectory optimization problem of a drilling process, which is of significance to drilling efficiency and safety. Due to the difference between an actual trajectory and a planned trajectory, this problem is a multiobjective optimization problem (MOP) with parameter uncertainties. To solve the problem, this study devises a new approach named nondominated sorting genetic Algorithm II using outlier removal (OR-NSGA-II). First, the optimization problem with parameter uncertainties is formulated, including two objective functions: 1) a trajectory length and 2) an expected value of drill-string torque. Then, an outlier-removal mechanism is devised in the sorting process of NSGA-II to reduce the negative effects of parameter uncertainties. Next, the crowding distance calculation in NSGA-II is improved to ensure population diversity. Comparison of simulation results show that our method is effective on the MOP with parameter uncertainties.

An Elitist Multi-Objective Particle Swarm Optimization Algorithm for Sustainable Dynamic Economic Emission Dispatch Integrating Wind Farms

In recent years, wind energy has been widely used as an alternative energy source as it is a clean energy with a low running cost. However, the high penetration of wind power (WP) in power networks has created major challenges due to their intermittency. In this study, an elitist multi-objective evolutionary algorithm called non-dominated sorting particle swarm optimization (NSPSO) is proposed to solve the dynamic economic emission dispatch (DEED) problem with WP. The proposed optimization technique referred to as NSPSO uses the non-dominated sorting principle to rank the non-dominated solutions. A crowding distance calculation is added at the end of all iterations of the algorithm. In this study, WP is represented by a chance-constraint which describes the probability that the power balance cannot be met. The uncertainty of WP is described by the Weibull distribution function. In this study, the chance constraint DEED problem is converted into a deterministic problem. Then, the NSPSO is applied to simultaneously minimize the total generation cost and emission of harmful gases. To proof the performance of the proposed method, the ten-unit and forty-unit systems—including wind farms—are used. Simulation results obtained by the NSPSO method are compared with other optimization techniques that were presented recently in the literature. Moreover, the impact of the penetration ratio of WP is investigated.

A self-organizing multimodal multi-objective pigeon-inspired optimization algorithm

Multi-objective optimization algorithms have recently attracted much attention as they can solve problems involving two or more conflicting objectives effectively and efficiently. However, most existing studies focus on improving the performance of the solutions in the objective spaces. This paper proposes a novel multimodal multi-objective pigeon-inspired optimization (MMOPIO) algorithm where some mechanisms are designed for the distribution of the solutions in the decision spaces. First, MMOPIO employs an improved pigeon-inspired optimization (PIO) based on consolidation parameters for simplifying the structure of the standard PIO. Second, the self-organizing map (SOM) is combined with the improved PIO for better control of the decision spaces, and thus, contributes to building a good neighborhood relation for the improved PIO. Finally, the elite learning strategy and the special crowding distance calculation mechanisms are used to prevent premature convergence and obtain solutions with uniform distribution, respectively. We evaluate the performance of the proposed MMOPIO in comparison to five state-of-the-art multi-objective optimization algorithms on some test instances, and demonstrate the superiority of MMOPIO in solving multimodal multi-objective optimization problems.

Solving the Multi-Objective Optimal Power Flow Problem Using the Multi-Objective Firefly Algorithm with a Constraints-Prior Pareto-Domination Approach

Known as a multi-objective, large-scale, and complicated optimization problem, the multi-objective optimal power flow (MOOPF) problem tends to be introduced with many constraints. In this paper, compared with the frequently-used penalty function-based method (PFA), a novel constraint processing approach named the constraints-prior Pareto-domination approach (CPA) is proposed for ensuring non-violation of various inequality constraints on dependent variables by introducing the Pareto-domination principle based on the sum of constraint violations. Moreover, for solving the constrained MOOPF problem, the multi-objective firefly algorithm with CPA (MOFA-CPA) is proposed and some optimization strategies, such as the crowding distance calculation and non-dominated sorting based on the presented CPA, are utilized to sustain well-distributed Pareto front (PF). Finally, in order to demonstrate the feasible and effective improvement of MOFA-CPA, a comparison study between MOFA-CPA and MOFA-PFA is performed on two test systems, including three bi-objective optimization cases and three tri-objective optimization cases. The simulation results demonstrate the capability of the MOFA-CPA for obtaining PF with good distribution and superiority of the proposed CPA for dealing with inequality constraints on dependent variables. In addition, some quality indicators are used to evaluate the convergence, distribution, and uniformity of the PFs found by the MOFA-CPA and MOFA-PFA.

A fast two-objective differential evolution for the two-objective coverage problem of WSNs

The aim of this article is to study the two-objective coverage problem of wireless sensor networks (WSNs) by means of differential evolution algorithm. Firstly, in order to reduce the computing redundancy of multi-objective optimization, namely to reduce the number of individuals which participate in non-dominated solution sorting, we introduced a fast two-objective differential evolution algorithm (FTODE). The FTODE contains a fast non-dominated solution sorted and a uniform crowding distance calculation method. The fast sorting method just handles the highest rank individuals but not all individuals in the current population. Meanwhile, during the individuals sorted, it can choose some of individuals into next generation and reduce the time complexity. The uniform crowding distance calculation can enhance the diversity of population due to it will retain the outline of optimal solution set by choosing the individual uniformly. Secondly, we use the FTODE framework to research the two-objective coverage problem of WSNs. The two objectives are formulated as: the minimum number of sensor used and the maximum coverage rate. For this specific problem, decimal integer encoding are used and a recombination operation is introduced into FTODE, which performs after initialization and guarantees at least one critical target’s sensor is divided into different disjoint sets. Finally, the simulation experiment shows that the FTODE provides competitive results in terms of time complexity and performance, and it also obtains better solutions than comparison algorithms on the two-objective coverage problem of WSNs.

A Fast Two-objective Differential Evolutionary Algorithm based on Pareto-optimal Set

The two-objective differential evolution with Pareto-optimal set, which is researched in this paper. Firstly, it is found that there are some redundant computations in the classic multi-objective evolutionary algorithm, such as the NSGA-II. Then, based on the concept of Pareto-optimal set, the non-dominated solution sorted and its potential features, the authors propose a ranking method for solution that only handles the highest rank individuals in current population. The highlight of the proposed method is that during the ranking process, the individuals can be chosen into the next generation meanwhile. When the individuals of next generation population are obtained the algorithm is broken out. Both the number of individuals for sorting process and the time complexity are reduced. Furthermore, a method of uniform crowding distance calculation is provided in this work. Finally, the authors incorporate the introduced ranking method and uniform crowding distance method into differential evolution, a fast two-objective differential evolution algorithm is obtained. For verifying the proposed method, they use the classical optimal problems ZDTl~ZDT4 and ZDT6 for tesing. Simulation results show that the authors' method has greatly improved in terms of time complexity and performance than other algorithms.