Spacing Metrics Research Articles

야간 시간대 보행자의 범죄 두려움 감소를 위한 CPTED 적용 방안

Fear of crime is a deeply rooted source of anxiety in modern society, making it essential to reduce this fear to improve the quality of life for residents. Previous studies have indicated that pedestrians, especially during nighttime hours, often experience a significant level of fear of crime. Therefore, this study focuses on identifying vulnerable points for fear of crime during nighttime pedestrian activity in Jangyang 2-dong, Gwangjin-gu, Seoul. It also proposes crime prevention strategies based on Crime Prevention Through Environmental Design (CPTED) principles, considering road environments and inter-individual spacing metrics.
 Firstly, using K-means Clustering on reported crime data for Jangyang 2-dong, nighttime hours were defined as 21:00 to 01:00. Subsequently, social network analysis were applied to identify areas with low centrality, categorizing them into Cluster 1 (Jangyangbyeong-ro 6, Dukseom-ro 44, Dukseom-ro 50) and Cluster 2 (Jayang-ro 3, Dukseom-ro 58) as locations vulnerable to fear of crime. Finally, tailored crime prevention designs were proposed for each cluster. This study is significant for its use of social network analysis on safety data to identify fear of crime vulnerable areas and for suggesting crime prevention measures tailored to the characteristics of the region from a CPTED perspective.

A multi-objective optimization framework for functional arrangement in smart floating cities

Before the terms “smart city” and “floating city” were introduced, the world's population had increased and land shortage across the world was already widely recognized. As a first challenge, the previous studies have developed the concept of a smart city as a creative answer, following that, several scientists proposed the floating city concept in the literature as a solution to the increased sea levels. Moreover, engineers, architects, and designers deal with city planning, for smart and floating settlements as a difficult design challenge, and evolutionary algorithms could be employed to address this complex problem by optimizing residents' needs. As a continuation of our previous studies on this topic, this time, we develop a multi-objective continuous genetic algorithm with differential evolution (DE) mutation strategy (MO_CGADE) and a multi-objective ensemble differential evolution algorithm (MO_EDE) to solve the problem on hand. Then, we compare the performance of the MO_CGADE and MO_EDE algorithms with the non-dominated sorting genetic algorithm (NSGAII) to maximize two conflicted objective functions, namely, scenery, and walkability in the proposed smart floating city model created in the Grasshopper Algorithmic Modeling Environment. The parametric model that we create in the Grasshopper software includes 64 decision variables, area constraints and objective functions to be optimized by MO_CGADE, MO_EDE, and NSGAII algorithms. Computational results show that MO_CGADE and MO_EDE algorithms generate better Pareto ranking results than the traditional NSGAII algorithm in terms of cardinality, distribution spacing, and coverage metrics.

A New Hybrid Algorithm Based on Improved MODE and PF Neighborhood Search for Scheduling Task Graphs in Heterogeneous Distributed Systems

Multi-objective task graph scheduling is a well-known NP-hard problem that plays a significant role in heterogeneous distributed systems. The solution to the problem is expected to optimize all scheduling objectives. Pretty large state-of-the-art algorithms exist in the literature that mostly apply different metaheuristics for solving the problem. This study proposes a new hybrid algorithm comprising an improved multi-objective differential evolution algorithm (DE) and Pareto-front neighborhood search to solve the problem. The novelty of the proposed hybrid method is achieved by improving DE and hybridizing it with the neighborhood search method. The proposed method improves the performance of differential evolution by applying appropriate solution representation as well as effective selection, crossover, and mutation operators. Likewise, the neighborhood search algorithm is applied to improve the extracted Pareto-front and speed up the evolution process. The effectiveness and performance of the developed method are assessed over well-known test problems collected from the related literature. Meanwhile, the values of spacing and hyper-volume metrics are calculated. Moreover, the Wilcoxon signed method is applied to carry out pairwise statistical tests over the obtained results. The obtained results for the makespan, reliability, and flow-time of 50, 18, and 41, respectively, by the proposed hybrid algorithm in the study confirmed that the developed algorithm outperforms all proposed methods considering the performance and quality of objective values.

Bi-sided facility location problems: an efficient algorithm for k-centre, k-median, and travelling salesman problems

This study introduces a general framework, called Bi-sided facility location, for a wide range of problems in the area of combined facility location and routing problems such as locating test centres and designing the network of supermarkets. It is based on a multi-objective optimisation model to enhance the service quality which the clients received, called client-side, and enhance the interconnection quality and eligibility among the centres, called center-side. Well-known problems such as k-median and k-centre for the client-side and the travelling salesman problem for the centre-side are taken into account in this paper. After discussing the complexity of this kind of combination, we propose a heuristic approximation algorithm to find approximation Pareto-optimal solutions for the problem. The algorithm is an efficient local search utilising geometric objects such as the Voronoi diagram and Delaunay triangulation as well as algorithms for computing approximation travelling salesman tour. In addition to the comprehensive theoretical analysis of the proposed models and algorithm, we apply the algorithm to different instances and benchmarks, and compare it with NSGA-II based on set coverage and spacing metrics. The results confirm the efficiency of the algorithm in terms of running time and providing a diverse set of efficient trade-off solutions. Highlights Introducing a general bi-side location model considering centres and clients’ utilities Discussing and proving the NP-hardness of the model in the general framework Considering two instances; k-centre and k-median for client-side and TSP for centre-side Proposing an efficient geometric-based algorithm for solving the problems Implementing, testing, and comparing the proposed algorithm on several benchmarks

A new MOPSO algorithm for solving mathematical test functions and control engineering problems

In this article, a new multiobjective particle swarm optimization (MOPSO) algorithm is introduced to improve the performance of a sliding mode based robust fuzzy proportional–integral–derivative (PID) controller. In this regard, the non-dominated solution having minimum number of neighbors is considered as the global best position, while the sigma values of the members are employed to determine the personal best position. A modified multiple-crossover operator is combined with the operators of the particle swarm optimization to significantly increase the convergence speed of the algorithm. To limit the size of the archive, a dynamical elimination scheme defined in the Euclidean space is introduced. Besides, iteration-based linear relations are implemented to adaptively compute the inertia weight and learning coefficients. To evaluate the effectiveness of the introduced MOPSO algorithm, the requirements are conducted by means of three benchmark functions with regard to generational distance, spacing, and maximum spread metrics. This analysis demonstrates that the proposed algorithm operates better through comparison with well-known elitist multiobjective evolutionary algorithms. Moreover, the MOPSO algorithm is applied for optimal design of a hybrid robust fuzzy PID controller for a pneumatic system with two bellows. Conflicting objective functions are considered as the normalized values of overshoot and settling time of the displacement between the bellows that should be simultaneously minimized. The feasibility and efficiency of the strategy are assessed in comparison with the conventional controllers.

Multi-Objective chimp Optimizer: An innovative algorithm for Multi-Objective problems

The Multi-Objective Chimp Optimization Algorithm (MOChOA), a multi-objective variation of the recently proposed ChOA, is developed in this research to address multi-objective optimization issues in various engineering problems. The optimization community has recently been offered numerous evolutionary and meta-heuristic optimization strategies for tackling optimization challenges. When analyzing multi-objective optimization (MOO) problems, these approaches frequently produce poor solutions because they do not accurately estimate the Pareto optimal solutions or increase the distribution across all objectives. The significant and impressive performance of the ChOA has been a great inspiration for this paper to study on its multi-objective version called MOChOA. In this approach, a memory structure has been exploited as an archive to store the non-dominated solutions alongside a leader selection strategy and grid mechanism. The leader selection procedure consists of analyzing the archive contents to choose the best candidates for the leader of independent groups, which are drivers, barriers, chasers, and attackers. The strategy also guarantees the exploration of search space. Twelve test problems with various shapes and different dimensions have opted to evaluate MOChOA. The results of coverage, generational distance, averaged Hausdorff distance, spacing, diversity, and other metrics are considered in the performance evaluation of the algorithm. According to results, MOChOA can provide competitive results and outperform well-known intelligent algorithms such as ANSGAIII, ARMOEA, dMOPSO, EFRRR, IBEA, SPEA2SDE, SPEAR, and tDEA. Alongside the remarkable performance of the proposed algorithm, its light-weight structure causes a fast execution. The proposed MOChOA can be applied in various multi-objective optimization application, including engineering, science, chemical processes, economics and logistics when appropriate decisions must be made in the face of competing objectives.

Scale dependent spatial structuring of mountain river large bed elements maximizes flow resistance

Macro-roughness elements such as boulders and bedrock outcrops, collectively referred to as large bed elements (LBEs), are key features influencing hydrodynamics and morphodynamics in mountain rivers. Where LBEs are abundant and account for a substantial portion of total flow resistance, existing geomorphic theory, previous physical experiments, and limited field observations support the theory that LBE configurations adjust to maximize flow resistance. However, methods to explicitly map individual features along entire river segments are lacking, thus limiting analysis of LBE spatial structure in boulder-bedded rivers. In addressing these gaps, this study sought to develop a procedure for mapping LBEs from 3D point-clouds, explore LBE spatial structure in a real boulder-bedded river, and test the hypothesis that LBE configurations were organized to maximize flow resistance. The mapping procedure applied a ground classification algorithm to produce a roughness surface model, from which LBEs were extracted by a marker controlled watershed algorithm. Implementing the procedure, 42,176 LBEs were mapped in 13.2-km of the mountainous Yuba River (Northern California). Scale and discharge-dependent LBE concentration and spacing metrics were quantified for multiple laterally and/or hierarchically nested spatial domains and classified to differentiate three flow-resistance based hydrodynamic regimes: isolated roughness, wake interference, and skimming flow. Of these regimes, wake interference corresponds to a state of maximum resistance, so hypothesis testing involved determining if this regime was dominant. Results confirmed 25 of 28 segment- and reach-scale LBE concentrations were in the wake interference regime. However, spacing metrics classified 24 of these same spatial domains in the skimming flow regime. Concentration metrics, which quantify LBE density in a given spatial area, differ from spacing metrics, which represent LBE proximity to one another. While comparison of segment and reach-scale regime classifications by each metric concluded concentration was superior to spacing for regime classification purposes, these disparities leave open questions about this extremal model of geomorphic adjustment. Lastly, lateral variability of metrics across the river corridor had implications for discharge-dependent resistance.

Elitist Rao Algorithms and R-Method for Optimization of Energy Systems

Identifying, examining, and optimizing the impact of different parameters in a renewable energy system can significantly help determine its efficiency. Furthermore, since these systems have several objectives such as power output, system efficiency, investment cost, economic, and ecological factors, it is often not preferable to present the optimum system parameters considering just one objective. This article proposes three improved versions of recently developed Rao algorithms named elitist Rao algorithms to find optimum system parameters of renewable energy systems. In addition, a new multi-attribute decision-making method named R-method is proposed for selecting the best solution from the Pareto-fronts obtained using the elitist Rao algorithms. The proposed algorithms are tested using 30 single-objective unconstrained benchmark functions, and the significance of improvement over basic Rao algorithms is validated using the Friedman statistical test. Later, the proposed algorithms’ performances are tested in multi- and many-objective optimization scenarios of a solar-assisted Stirling heat engine system and a turbocharged direct injection diesel engine system. Furthermore, the proposed algorithms’ effectiveness is presented in terms of hypervolume, coverage, and spacing metrics. Also, the performances of the proposed algorithms in single-, multi-, and many-objective optimization are compared with the other algorithms from the literature and found to be superior or competitive.

Multi-Objective Quantum-Inspired Seagull Optimization Algorithm

Objective solutions of multi-objective optimization problems (MOPs) are required to balance convergence and distribution to the Pareto front. This paper proposes a multi-objective quantum-inspired seagull optimization algorithm (MOQSOA) to optimize the convergence and distribution of solutions in multi-objective optimization problems. The proposed algorithm adopts opposite-based learning, the migration and attacking behavior of seagulls, grid ranking, and the superposition principles of quantum computing. To obtain a better initialized population in the absence of a priori knowledge, an opposite-based learning mechanism is used for initialization. The proposed algorithm uses nonlinear migration and attacking operation, simulating the behavior of seagulls for exploration and exploitation. Moreover, the real-coded quantum representation of the current optimal solution and quantum rotation gate are adopted to update the seagull population. In addition, a grid mechanism including global grid ranking and grid density ranking provides a criterion for leader selection and archive control. The experimental results of the IGD and Spacing metrics performed on ZDT, DTLZ, and UF test suites demonstrate the superiority of MOQSOA over NSGA-II, MOEA/D, MOPSO, IMMOEA, RVEA, and LMEA for enhancing the distribution and convergence performance of MOPs.

Prioritization of model smell refactoring using a covariance matrix-based adaptive evolution algorithm

The refactoring process enhances the software design by modifying the structure of design parts impaired with model smells without altering the overall software behavior. However, handling these smells without proper prioritization will not produce the anticipated effects. In this paper, we solve the prioritization of the model smell refactoring using a multi-objective optimization (MOO) algorithm called the multi-objective (MO) covariance matrix adaptation evolution strategy (MO CMA-ES). Our formulation relies on the refactoring of unified modeling language (UML) class diagrams to mitigate the negative effect of design smells. We treat the prioritization problem as a real-valued MOO where we propose novel data encoding procedures. We use two conflicting objectives, quality, and maintainability , to balance the refactoring. We first build a new solution representation that guarantees smell fixing and eliminates the rejection limitation. Furthermore, we suggest a custom mapping scheme to properly encode real-valued quantities using unique representations. For performance evaluation purposes, we developed a large custom dataset with more than 30,000 class records, using seven popular open-source software projects. A novel relative coverage metric is proposed to mitigate the limitations associated with the standard coverage. For benchmarking purposes, we also consider an improved version of the nondominated sorting genetic algorithm (NSGA-II(. The reported performance scores confirm the superiority of the MO CMA-ES algorithm over NSGA-II. The former successfully identified the refactoring sequences that lead to the best improvements in software quality and maintainability while it is able to fix all identified design smells. These improvements are quantified in terms of hypervolume, coverage, spacing metrics, and execution time. The MO CMA-ES attained the highest average maximum quality score of 1149 while keeping the average maintainability at the lowest score of 13.8. In all experiment settings, the proposed solution leads to longer refactoring sequences at no additional computational cost.

Three Hybrid Scatter Search Algorithms for Multi-Objective Job Shop Scheduling Problem

The Job Shop Scheduling Problem (JSSP) consists of finding the best scheduling for a set of jobs that should be processed in a specific order using a set of machines. This problem belongs to the NP-hard class problems and has enormous industrial applicability. In the manufacturing area, decision-makers consider several criteria to elaborate their production schedules. These cases are studied in multi-objective optimization. However, few works are addressed from this multi-objective perspective. The literature shows that multi-objective evolutionary algorithms can solve these problems efficiently; nevertheless, multi-objective algorithms have slow convergence to the Pareto optimal front. This paper proposes three multi-objective Scatter Search hybrid algorithms that improve the convergence speed evolving on a reduced set of solutions. These algorithms are: Scatter Search/Local Search (SS/LS), Scatter Search/Chaotic Multi-Objective Threshold Accepting (SS/CMOTA), and Scatter Search/Chaotic Multi-Objective Simulated Annealing (SS/CMOSA). The proposed algorithms are compared with the state-of-the-art algorithms IMOEA/D, CMOSA, and CMOTA, using the MID, Spacing, HV, Spread, and IGD metrics; according to the experimental results, the proposed algorithms achieved the best performance. Notably, they obtained a 47% reduction in the convergence time to reach the optimal Pareto front.

An Improved Multi-Objective Particle Swarm Optimization With TOPSIS and Fuzzy Logic for Optimizing Trapezoidal Labyrinth Weir

Labyrinth Weir (LW) is a popular control structure that passes a significantly higher flow rate compared to the linear weirs. In order to approach the optimal design of a trapezoidal LW, a multi-objective problem is defined to concurrently minimize the LW consumed concrete volume and maximize its discharge capacity. Simultaneously, a Radial Basis function Neural Networks (RBFNN) is designed and used for estimating LW discharge coefficient (Cd) according to the existing experimental results. An improved multi-objective particle swarm optimization (MOPSO) algorithm named TOPSIS Fuzzy MOPSO (TFMOPSO) is proposed to solve the LW optimization problem. This algorithm utilizes the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) to rank the solutions, while a fuzzy inference system is developed to select the algorithm strategy for finding two leaders among the non-dominated solutions. The performance of the proposed TFMOPSO has been tested on the optimization problem of the LW of the Ute dam. The results of TFMOPSO, along with three other state-of-the-art multi-objective algorithms, are explored in terms of hypervolume, coverage, and spacing metrics. It is demonstrated that the TFMOPSO outperforms other algorithms and studies for solving the LW multi-objective optimization problem for the case of Ute dam. Also, RBFNN is found to be one of the most appropriate approaches among studied algorithms in estimating the discharge coefficient of LW, while Pareto optimal solutions from TFMOPSO exhibit a significant improvement compared to the original design of Ute dam LW.

A Comparison of Extended Dijkstra and ACO Algorithm for Shortest Path Problem in Dynamic Networks with Delay Times

Shortest path problem is a typical routing optimization problem that is generally involved with a multi-criteria decision-making process. Therefore, the main objective of this paper is to find the shortest path in discrete-time dynamic networks based on bi-criteria of time and reliability by considering the effect of delay times that varies according to different departure time scenarios. Firstly, the well-known single-criterion Dijkstra’s algorithm is extended to fit the conditions of a bi-criteria problem. The solutions obtained from the extended Dijkstra was then compared with a proposed ant colony optimization (ACO) algorithm via a set of multi-objective performance metrics including CPU time, error ratio, spacing and diversity metrics. The analysis was made based on three network scales ranged from small (20-100 nodes), to medium (500-1900 nodes) and large (2000-10000 nodes). The computational results obtained from the analysis suggested that the extended Dijkstra’s algorithm has a higher efficiency in medium and large scaled networks. Furthermore, the comparison of the proposed ACO versus Dijkstra’s algorithm proved the preference of ACO for networks with larger-scaled (nodes over 5000), while, for smaller and medium-scaled networks (nodes 20-2000), the extended Dijkstra’s algorithm has a dominantly better performance in CPU time as compared to proposed ACO.

Balanced multi-objective optimization algorithm using improvement based reference points approach

• We suggest an enhancement for exploration and exploitation factors of the Equilibrium Optimizer (EO) to balance between exploration and exploitation operators in the proposed multi-objective equilibrium optimizer. • We propose an improvement-based reference points method increasing the diversity between the population members in multi-objective optimization problems. • The CEC 2020, CEC 2009, DTLZ, and ZDT test problems are solved using a multi-objective equilibrium optimizer algorithm. • Our proposed algorithm outperforms the other algorithms in terms of the spread and inverted generational distance measures for different test problems that satisfy all characteristics for optimization problems. In this work, we explore a novel multi-objective optimization algorithm to identify a set of solutions that could be optimal for more than one task. The proposed approach is used to generate a set of solutions that balance the tradeoff between convergence and diversity in multi-objective optimization problems. Equilibrium Optimizer (EO) algorithm is a novel developed meta-heuristic algorithm inspired by the physics laws. In this paper, we propose a Multi-objective Equilibrium Optimizer Algorithm (MEOA) for tackling multi-objective optimization problems. We suggest an enhancement for exploration and exploitation factors of the EO algorithm to randomize the values of these factors with decreasing the initial value of the exploration factor with the iteration and increasing the exploitation factor to accelerate the convergence toward the best solution. To achieve good convergence and well-distributed solutions, the proposed algorithm is integrated with the Improvement-Based Reference Points Method (IBRPM). The proposed approach is applied to the CEC 2020, CEC 2009, DTLZ, and ZDT test functions. Also, the inverted generational and spread spacing metrics are used to compare the proposed algorithm with the most recent evolutionary algorithms. It's obvious from the results that the proposed algorithm is better in both convergence and diversity.

State-transition simulated annealing algorithm for constrained and unconstrained multi-objective optimization problems

In this article, a novel multi-objective optimization algorithm based on a state-transition simulated annealing algorithm (MOSTASA) is proposed, in which four state-transition operators for generating candidate solutions and the Pareto optimal solution is obtained by combining it with the concept of Pareto dominance and then storing it in a Pareto archive. To ensure the uniform distribution of the Pareto optimal solution, we define a crowded comparison operator to update the Pareto archive. Simulation experiments were conducted on several standard constrained and unconstrained multi-objective problems, in which convergence and spacing metrics were used to assess the performance of the MOSTASA. The test results manifest that the MOSTASA can converge to the true Pareto-optimal front, and the solution distribution is uniform. Compared to the performance of other multi-objective optimization algorithms, the proposed algorithm is more efficient and reliable.

An efficient controlled elitism non-dominated sorting genetic algorithm for multi-objective supplier selection under fuzziness

Abstract Supplier selection and order allocation constitute vital strategic decisions that must be made by managers within supply chain management environments. In this paper, we propose a multi-objective fuzzy model for supplier selection and order allocation in a two-level supply chain with multi-period, multi-source, and multi-product characteristics. The supplier evaluation objectives considered in this model include cost, delay, and electronic-waste (e-waste) minimization, as well as coverage and weight maximization. A signal function is used to model the price discount offered by the suppliers. Triangular fuzzy numbers are used to deal with the uncertainty of delay and e-waste parameters while the fuzzy Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) is used to obtain the weights of the suppliers. The resulting NP-hard problem, a Pareto-based meta-heuristic algorithm called controlled elitism non-dominated sorting genetic algorithm (CENSGA), is developed. The Non-dominated Sorting Genetic Algorithm (NSGA-II) and Multi-Objective Particle Swarm Optimization (MOPSO) are used to validate the applicability of the CENSGA algorithm and the Taguchi technique to tune the parameters of the algorithms. The results are analysed using graphical and statistical comparisons illustrating how the proposed CENSGA dominates NSGA-II and MOPSO in terms of mean ideal solution distance (MID) and spacing metrics.

Rao algorithms for multi-objective optimization of selected thermodynamic cycles

This work proposes multi-objective Rao algorithms. The basic Rao algorithms are modified for solving multi-objective optimization problems. The proposed algorithms have no algorithm-specific parameters and no metaphorical meaning. Based on the interaction of the population with best, worst, and randomly selected solutions, the proposed algorithms explore the search space. The proposed algorithms handle multiple objectives simultaneously based on dominance principles and crowding distance evaluation. In addition, multi-attribute decision-making method-based selection scheme for identifying the best solutions from the Pareto fronts is included. The proposed algorithm performances are investigated on a case study of solar-assisted Brayton heat engine system and a case study of Stirling heat engine system to see whether there can be any improvement in the performances of the considered systems. Furthermore, the efficiencies of the Rao algorithms are evaluated in terms of spacing, hypervolume, and coverage metrics. The results obtained by the proposed algorithms are compared with those obtained by the latest advanced optimization algorithms. It is observed that the results obtained by the proposed algorithms are superior. The performances of the considered case studies are improved by the application of the proposed optimization algorithms. The proposed optimization algorithms are simple, robust, and can be easily implemented to solve different engineering optimization problems.

Evolutionary algorithms and the Preference Ranking Organization Method for Enrichment Evaluations as applied to a multiobjective design of shell-and-tube heat exchangers

The results of the present work are directed at solving the shell-and-tube heat exchanger design optimization problem on the basis of a multiobjective approach by applying Evolutionary Algorithms. The considered objective functions are associated with equipment's heat transfer area and pumping power. The performance of the algorithms has been analyzed applying the Hypervolume, Coverage, Spacing and Pair-Wise Distance metrics. Their use allows one to compare the algorithms from the points of view of their efficiency, their diversity, and the quality of obtained solutions. The Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE) is applied to allow one to generate the solution from the Pareto frontier, according to the criteria and preferences defined by the decision maker.

Long-Term Hydropower Generation Scheduling of Large-Scale Cascade Reservoirs Using Chaotic Adaptive Multi-Objective Bat Algorithm

With growing concerns over renewable energy, the cascade hydropower reservoirs operation (CHRO), which balances the development of economic benefits and power supply security, plays an increasingly important role in hydropower systems. Due to conflicting objectives and complicated operation constraints, the CHRO problem considering the requirements of maximizing power generation benefit and firm power output is determined as a multi-objective optimization problem (MOP). In this paper, a chaotic adaptive multi-objective bat algorithm (CAMOBA) is proposed to solve the CHRO problem, and the external archive set is added to preserve non-dominant solutions. Meanwhile, population initialization based on the improved logical mapping function is adopted to improve population diversity. Furthermore, the self-adaptive local search strategy and mutation operation are designed to escape local minima. The CAMOBA is applied to the CHRO problem of the Qingjiang cascade hydropower stations in southern China. The results show that CAMOBA outperforms the multi-objective bat algorithm (MOBA) and non-dominated sorting genetic algorithms-II (NSGA-II) in different hydrological years. The spacing (SP) and hypervolume (HV) metrics verify the excellent performance of CAMOBA in diversity and convergence. In summary, the CAMOBA is demonstrated to get better scheduling solutions, providing an effective approach for solving the cascade hydropower reservoirs operation (CHRO).

A new stopping criterion for multi-objective evolutionary algorithms: application in the calibration of a hydrologic model

Multi-objective genetic algorithms have been successfully applied in a wide variety of problems. Although widely used, there are few theoretical guidelines for determining when to stop the search. Many users commonly use rules like stopping when there is no significant improvement during the last generations or when a certain number of generations are reached. In this paper, we propose a new stopping criterion approach and evaluate its performance with three widely used evolutionary algorithms in the calibration of a hydrologic model. The stopping criterion is based on the minimum number of generations required to achieve a determined number of non-dominated solutions in Pareto Front. The new stopping criterion was tested in the lumped hydrologic model IPH-II calibration, using the genetic algorithms NSGA-II, NSGA-III, and SPEA-II and two objective functions. The generational distance, spacing, and maximum spread metrics were used to assess the performance of the proposed stopping criterion in comparison to the standard criterion. Results show no significant loss in goodness of fit associated with the proposed stopping criterion, both in calibration and validation periods. Performance metrics have shown similar values when the standard and the proposed stopping criteria were compared. However, the average computational time to complete the optimization process was reduced up to 38.2% when the proposed stopping criterion was used. Thus, it can be concluded that the new stopping criterion reduces the iteration workload without compromising the accuracy of solution sets.