Traditional Multi-objective Evolutionary Algorithms Research Articles

A Knowledge-Guided Competitive Co-Evolutionary Algorithm for Feature Selection

In real-world applications, feature selection is crucial for enhancing the performance of data science and machine learning models. Typically, feature selection is a complex combinatorial optimization problem and a multi-objective optimization problem. Its primary goals are to reduce the dimensionality of the dataset and enhance the performance of machine learning algorithms. The selection of features in high-dimensional datasets is challenging due to the intricate relationships between features, which pose significant challenges to the performance and computational efficiency of algorithms. This paper introduces a Knowledge-Guided Competitive Co-Evolutionary Algorithm (KCCEA) for feature selection, especially for high-dimensional features. In the proposed algorithm, we make improvements to the foundational dominance-based multi-objective evolutionary algorithm in two aspects. First, the use of feature correlation as knowledge to guide evolution enhances the search speed and quality of traditional multi-objective evolutionary algorithm solutions. Second, a dynamically allocated competitive–cooperative evolutionary mechanism is proposed, integrating the improved knowledge-guided evolution with traditional evolutionary algorithms, further enhancing the search efficiency and diversity of solutions. Through rigorous empirical testing on various datasets, the KCCEA demonstrates superior performance compared to basic multi-objective evolutionary algorithms, providing effective solutions to multi-objective feature selection problems while enhancing the interpretability and effectiveness of prediction models.

A Multi-Task Decomposition-Based Evolutionary Algorithm for Tackling High-Dimensional Bi-Objective Feature Selection

Evolutionary algorithms have been widely applied for solving multi-objective optimization problems, while the feature selection in classification can also be treated as a discrete bi-objective optimization problem if attempting to minimize both the classification error and the ratio of selected features. However, traditional multi-objective evolutionary algorithms (MOEAs) may have drawbacks for tackling large-scale feature selection, due to the curse of dimensionality in the decision space. Therefore, in this paper, we concentrated on designing an multi-task decomposition-based evolutionary algorithm (abbreviated as MTDEA), especially for handling high-dimensional bi-objective feature selection in classification. To be more specific, multiple subpopulations related to different evolutionary tasks are separately initialized and then adaptively merged into a single integrated population during the evolution. Moreover, the ideal points for these multi-task subpopulations are dynamically adjusted every generation, in order to achieve different search preferences and evolutionary directions. In the experiments, the proposed MTDEA was compared with seven state-of-the-art MOEAs on 20 high-dimensional classification datasets in terms of three performance indicators, along with using comprehensive Wilcoxon and Friedman tests. It was found that the MTDEA performed the best on most datasets, with a significantly better search ability and promising efficiency.

Towards Running Time Analysis of Interactive Multi-Objective Evolutionary Algorithms

Evolutionary algorithms (EAs) are widely used for multi-objective optimization due to their population-based nature. Traditional multi-objective EAs (MOEAs) generate a large set of solutions to approximate the Pareto front, leaving a decision maker (DM) with the task of selecting a preferred solution. However, this process can be inefficient and time-consuming, especially when there are many objectives or the DM has subjective preferences. To address this issue, interactive MOEAs (iMOEAs) combine decision making into the optimization process, i.e., update the population with the help of the DM. In contrast to their wide applications, there has existed only two pieces of theoretical works on iMOEAs, which only considered interactive variants of the two simple single-objective algorithms, RLS and (1+1)-EA. This paper provides the first running time analysis (the essential theoretical aspect of EAs) for practical iMOEAs. Specifically, we prove that the expected running time of the well-developed interactive NSGA-II (called R-NSGA-II) for solving the OneMinMax, OneJumpZeroJump problems are all asymptotically faster than the traditional NSGA-II. Meanwhile, we present a variant of OneMinMax, and prove that R-NSGA-II can be exponentially slower than NSGA-II. These results provide theoretical justification for the effectiveness of iMOEAs while identifying situations where they may fail. Experiments are also conducted to validate the theoretical results.

A Hybrid Initialization and Effective Reproduction-Based Evolutionary Algorithm for Tackling Bi-Objective Large-Scale Feature Selection in Classification

Evolutionary algorithms have been widely used for tackling multi-objective optimization problems, while feature selection in classification can also be seen as a discrete bi-objective optimization problem that pursues minimizing both the classification error and the number of selected features. However, traditional multi-objective evolutionary algorithms (MOEAs) can encounter setbacks when the dimensionality of features explodes to a large scale, i.e., the curse of dimensionality. Thus, in this paper, we focus on designing an adaptive MOEA framework for solving bi-objective feature selection, especially on large-scale datasets, by adopting hybrid initialization and effective reproduction (called HIER). The former attempts to improve the starting state of evolution by composing a hybrid initial population, while the latter tries to generate more effective offspring by modifying the whole reproduction process. Moreover, the statistical experiment results suggest that HIER generally performs the best on most of the 20 test datasets, compared with six state-of-the-art MOEAs, in terms of multiple metrics covering both optimization and classification performances. Then, the component contribution of HIER is also studied, suggesting that each of its essential components has a positive effect. Finally, the computational time complexity of HIER is also analyzed, suggesting that HIER is not time-consuming at all and shows promising computational efficiency.

A Survey on Learnable Evolutionary Algorithms for Scalable Multiobjective Optimization

Recent decades have witnessed great advancements in multiobjective evolutionary algorithms (MOEAs) for multiobjective optimization problems (MOPs). However, these progressively improved MOEAs have not necessarily been equipped with scalable and learnable problem-solving strategies for new and grand challenges brought by the scaling-up MOPs with continuously increasing complexity from diverse aspects, mainly including expensive cost of function evaluations, many objectives, large-scale search space, time-varying environments, and multi-task. Under different scenarios, divergent thinking is required in designing new powerful MOEAs for solving them effectively. In this context, research studies on learnable MOEAs with machine learning techniques have received extensive attention in the field of evolutionary computation. This paper begins with a general taxonomy of scaling-up MOPs and learnable MOEAs, followed by an analysis of the challenges that these MOPs pose to traditional MOEAs. Then, we synthetically overview recent advances of learnable MOEAs in solving various scaling-up MOPs, focusing primarily on four attractive directions (i.e., learnable evolutionary discriminators for environmental selection, learnable evolutionary generators for reproduction, learnable evolutionary evaluators for function evaluations, and learnable evolutionary transfer modules for sharing or reusing optimization experience). The insight of learnable MOEAs is offered to readers as a reference to the general track of the efforts in this field.

An efficient many objective optimization algorithm with few parameters

During the past two decades, numerous many-objective optimization evolutionary algorithms (MaOEAs) have been proposed to tackle the challenges traditional multi-objective evolutionary algorithms face, that is to deal with abundant non-dominated solutions and low selection pressure. Specifically, a series of sophisticated selection strategies have been adopted, some of which need additional pre-defined parameters or a significant amount of extra computation time, which retards their applications in real life. In this paper, we propose an efficient indicator-based MaOEA with few parameters, SDE+-MOEA, that uses an adaptive combination of commonly used selection methods to improve search efficiency based on SDE+. SDE+ addresses situations where SDE cannot distinguish individuals with the same SDE values. The adaptive selection method dynamically selects between one-time and iterative selection methods at different stages of evolution to improve the search efficiency. Furthermore, apart from the four parameters for generating offspring, our proposed SDE+-MOEA does not introduce additional parameters. We conduct experimental studies to compare SDE+-MOEA with 11 state-of-the-art algorithms, including 2REA, ISDE+, θ-DEA, LMPFE, CVEA3, SRA, MOEA/DD, IBEA, Two_Arch2, NSGA-III, and SPEA2SDE using four representative performance indicators (HV, SP, PD and GD) on MaF benchmark with 5, 8, 10, and 15 objectives. Experimental studies demonstrate that, compared to the algorithms, SDE+-MOEA achieves better HV and competitive convergence and uniformity performance, while requiring few parameters. It means that SDE+-MOEA can find a solution set with better convergence and uniformity to help decision-makers understand the solved problems. Furthermore, SDE+-MOEA loses little spreadability since a better convergence often leads to a worse spread.

A configuration space evolutionary algorithm with local minimizer for weighted circles packing problem

The weighted circles packing (WCP) problem typically aims for arranging some circular objects into a larger rectangular (or circular) container with the constraint of non-overlapping. The WCP problem derives from the layout design of the very large-scale integration (VLSI) and the plant machine equipment layout. It has wide applications in engineering practice. The problem considers two different objectives: minimizing the area of the envelope container and minimizing the sum of weighted distances associated with the given circular objects. Based on the quasi-physical strategy, the WCP problem becomes an unconstrained optimization problem. Then, a novel multi-objective evolutionary algorithm (MOEA) called configuration space evolutionary algorithm with local minimizer (CSE_LM) is proposed to solve the problem. Unlike the traditional MOEAs, the unique aspects of the CSE_LM mainly include a special evolutionary mechanism of configuration library (population), a convergence strategy of introducing a gradually decreasing configuration library radius, and a nearest and farthest candidate solution method to maintain diversity and uniformity of configurations. Furthermore, a novel mechanism of introducing the minimization of an extrusive elastic potential energy as an additional objective in the layout system is developed to handle the interference among circular objects. Experiment results for four typical instances show the validity of the CSE_LM for the WCP.

A framework for inverse surrogate modeling for fitness estimation applied to Multi-Objective Evolutionary Algorithms

Many-Objective Optimization Problems, or MaOPs, are complex optimization problems with more than three objective functions. Traditional Multi-Objective Evolutionary Algorithms (MOEAs) have shown poor scalability in solving this problem. Neuroevolution techniques can be used to overdraw that problem. The use of machine learning techniques to enhance optimization algorithms applied to MaOPs has drawn attention due to their capacity to add domain knowledge during the search process. One method of this kind is inverse surrogate modeling for fitness estimation, which uses machine learning models to enhance MOEAs, mapping the objective function values to the decision variables. Some inverse modeling methods in literature have performed well in various optimization problems. Despite the promising results, new strategies using the generated knowledge during the search can be further improved. The main goal of this work is to propose a framework that uses an inverse modeling approach coupled with an MOEA. The inverse surrogate modeling is used to sample solutions that are combined with the evolutionary search of the MOEA. The framework is evaluated in different scenarios, including multi-objective benchmark problems and real-world problems. The results show that the proposed framework had statistically better or equivalent results in 86% scenarios in the benchmark problems and 100% scenarios in the real-world problem.

Tourism Route Recommendation Based on A Multi-Objective Evolutionary Algorithm Using Two-Stage Decomposition and Pareto Layering

Tourism route planning is widely applied in the smart tourism field. The Pareto-optimal front obtained by the traditional multi-objective evolutionary algorithm exhibits long tails, sharp peaks and disconnected regions problems, which leads to uneven distribution and weak diversity of optimization solutions of tourism routes. Inspired by these limitations, we propose a multi-objective evolutionary algorithm for tourism route recommendation (MOTRR) with two-stage and Pareto layering based on decomposition. The method decomposes the multi-objective problem into several subproblems, and improves the distribution of solutions through a two-stage method. The crowding degree mechanism between extreme and intermediate populations is used in the two-stage method. The neighborhood is determined according to the weight of the subproblem for crossover mutation. Finally, Pareto layering is used to improve the updating efficiency and population diversity of the solution. The two-stage method is combined with the Pareto layering structure, which not only maintains the distribution and diversity of the algorithm, but also avoids the same solutions. Compared with several classical benchmark algorithms, the experimental results demonstrate competitive advantages on five test functions, hypervolume (HV) and inverted generational distance (IGD) metrics. Using the experimental results of real scenic spot datasets from two famous tourism social networking sites with vast amounts of users and large-scale online comments in Beijing, our proposed algorithm shows better distribution. It proves that the tourism routes recommended by our proposed algorithm have better distribution and diversity, so that the recommended routes can better meet the personalized needs of tourists.

An evolutionary algorithm based on independently evolving sub-problems for multimodal multi-objective optimization

Multimodal multi-objective problems (MMOPs) arise frequently in the real world, in which multiple Pareto optimal solution (PS) sets correspond to the same objective set. Traditional multi-objective evolutionary algorithms (MOEAs) show poor performance in solving MMOPs due to a lack of diversity maintenance in the decision space. Thus, recently, many multimodal multi-objective evolutionary algorithms (MMEAs) have been proposed. However, for most existing MMEAs, they generally have an over-convergence phenomenon, leading to the deterioration of the diversity of the decision space. To address these issues, this paper proposes a MMEAs based on independently evolving sub-problems. The sub-problem independent evolution method fits the definition of MMOPs because sub-problems form meaningful niches when they converge gradually, ensuring the convergence of the objective space, and enhancing the diversity of the decision space. In the environmental selection phase, we propose a two-stage environmental selection strategy that can guarantee both the convergence of the objective space and the distribution of the decision space. Finally, we refer to the k-nearest neighbor deletion strategy in the decision space to guarantee the distributivity of each equivalent PS. The experimental results show that our algorithm has higher competitive performance than seven other state-of-the-art MMEAs on two series of test functions.

Hybrid multiobjective evolutionary algorithm considering combination timing for multi-type vehicle routing problem with time windows

Vehicle routing problem (VRP) has been a classical combinatorial optimization challenge, which has been extensively investigated in recent decades. As a VRP typical variant, multi-type vehicle routing problem with time windows (MT-VRPTW) has been concerned with multiple types of heterogeneous fleets of vehicles, within the delivery time window and limited capacity constraints. For such complicated multiobjective optimization problems, hybrid multiobjective evolutionary algorithm (HMOEA) has become an effective method, however, the research on hybrid algorithms and combination timing is still challenging. This study proposes a hybrid multiobjective evolutionary algorithm based on combination timing (HMOEA-CT) to solve MT-VRPTW with the criteria of minimizing the number of vehicles and wasting time simultaneously. The HMOEA-CT combines a fast convergence in multiple directions (FCMDs)-based global search and a problem-specific difference (PSD)-based local search in different evolutionary stages. The FCMD is assumed to be the global exploration search strategy, which converges faster at the center and edge regions of the Pareto front. The PSD is used to improve the local exploitation search ability by directing poor-performing individuals closer to better-performing individuals. Furthermore, the PSD cooperates with FCMD at the right time according to the different evolutionary stages to improve efficiency. Experimental results on Solomon benchmark problems demonstrate the competitive performance comparison of the HMOEA-CT with traditional multiobjective evolutionary algorithms.

Multi-Objective Optimization Using Cooperative Garden Balsam Optimization with Multiple Populations

Traditional multi-objective evolutionary algorithms (MOEAs) consider multiple objectives as a whole when solving multi-objective optimization problems (MOPs). In this paper, the hybridization of garden balsam optimization (GBO) is presented to solve multi-objective optimization, applying multiple populations for multiple objectives individually. Moreover, in order to improve the diversity of the solutions, both crowding distance computations and epsilon dominance relations are adopted when updating the archive. Furthermore, an efficient selection procedure called co-evolutionary multi-swarm garden balsam optimization (CMGBO) is proposed to ensure the convergence of well-diversified Pareto regions. The performance of the used algorithm is validated on 12 test functions. The algorithm is employed to solve four real-world problems in engineering. The achieved consequences corroborate the advantage of the proposed algorithm with regard to convergence and diversity.

Micro multi-strategy multi-objective artificial bee colony algorithm for microgrid energy optimization

Multi-objective evolutionary algorithm (MOEA) has become a common and effective method to solve real-world multi-objective optimization problems. However, in some practical problems, such as the microgrid energy optimization problem (MEOP), the algorithm needs to run on the micro controller to control each distributed power supply in real time. Due limitation of hardware resources on the micro controller, the MOEAs are not suitable. The emerging micro population MOEAs are suitable for this scenario. But the micro population MOEA is vulnerable to lost diversity, resulting in its performance decline. Therefore, this paper proposes a new micro multi-strategy multi-objective ABC algorithm to solve MEOP, called μMMABC. Multi-strategy ABC optimizer is used to divide the population into multiple subgroups and produce offspring in parallel to balance the exploration and exploitation. In addition, an adaptive updating mechanism is proposed to renew the population adaptively. The mechanism can adaptively select more convergent and diverse solutions at different stages to balance the exploration and exploitation of the algorithm. Furthermore, in order to improve the performance of μMMABC on problems with irregular Pareto fronts, the reference point reconstruction with intermediate strategy is also proposed. Some benchmark test suites are used to test the performance of μMMABC. Finally, it is used to solve the MEOP. The experimental results show that the proposed algorithm is more competitive and effective than the traditional MOEAs and other micro population MOEAs in solving the MEOP.

Multiobjective particle swarm optimization with direction search and differential evolution for distributed flow-shop scheduling problem.

As a classic problem of distributed scheduling, the distributed flow-shop scheduling problem (DFSP) involves both the job allocation and the operation sequence inside the factory, and it has been proved to be an NP-hard problem. Many intelligent algorithms have been proposed to solve the DFSP. However, the efficiency and quality of the solution cannot meet the production requirements. Therefore, this paper proposes a bi-objective particle swarm optimization with direction search and differential evolution to solve DFSP with the criteria of minimizing makespan and total processing time. The direction search strategy explores the particle swarm in multiple directions of the Pareto front, which enhances the strong convergence ability of the algorithm in different areas of Pareto front and improves the solution speed of the algorithm. The search strategy based on differential evolution is the local search strategy of the algorithm, which can prevent the multiobjective particle swarm optimization from converging prematurely and avoid falling into local optimum, so that a better solution can be found. The combination of these two strategies not only increases the probability of particles moving in a good direction, but also increases the diversity of the particle swarm. Finally, experimental results on benchmark problems show that, compared with traditional multiobjective evolutionary algorithms, the proposed algorithm can accelerate the convergence speed of the algorithm while guaranteeing that the obtained solutions have good distribution performance and diversity.

Weighted Indicator-Based Evolutionary Algorithm for Multimodal Multiobjective Optimization

Multimodal multiobjective problems (MMOPs) arise frequently in the real world, in which multiple Pareto-optimal solution (PS) sets correspond to the same point on the Pareto front. Traditional multiobjective evolutionary algorithms (MOEAs) show poor performance in solving MMOPs due to a lack of diversity maintenance in the decision space. Thus, recently, many multimodal MOEAs (MMEAs) have been proposed. However, for most existing MMEAs, the convergence performance in the objective space does not meet expectations. In addition, many of them cannot always obtain all equivalent Pareto solution sets. To address these issues, this study proposes an MMEA based on a weighted indicator, termed MMEA-WI. The algorithm integrates the diversity information of solutions in the decision space into an objective space performance indicator to maintain the diversity in the decision space and introduces a convergence archive to ensure a more effective approximation of the Pareto-optimal front (PF). These strategies can readily be applied to other indicator-based MOEAs. The experimental results show that MMEA-WI outperforms some state-of-the-art MMEAs on the chosen benchmark problems in terms of the inverted generational distance (IGD) and IGD in the decision space (IGDX) metrics.

A Hybrid Strategic Oscillation with Path Relinking Algorithm for the Multiobjective k-Balanced Center Location Problem

This paper presents a hybridization of Strategic Oscillation with Path Relinking to provide a set of high-quality nondominated solutions for the Multiobjective k-Balanced Center Location problem. The considered location problem seeks to locate k out of m facilities in order to serve n demand points, minimizing the maximum distance between any demand point and its closest facility while balancing the workload among the facilities. An extensive computational experimentation is carried out to compare the performance of our proposal, including the best method found in the state-of-the-art as well as traditional multiobjective evolutionary algorithms.

An Approach for the Generation of Multi-Objective Algorithms Applied to the Integration and Test Order Problem

Multi-Objective Evolutionary Algorithms (MOEAs) have been successfully applied to solve hard real software engineering problems. However, to choose and design a MOEA is considered a difficult task, since there are several parameters and components to be configured. These aspects directly impact the generated solutions and the performance of MOEAs. In this sense, this paper proposes an approach for the automatic generation of MOEAs applied to the Integration and Test Order (ITO) problem. Such a problem refers to the generation of optimal sequences of units for integration testing. The approach includes a set of parameters and components of different MOEAs, and is implemented with two design algorithms: Grammatical Evolution (GE) and Iterated Racing (irace). Evaluation results are presented, comparing the MOEAs generated by both design algorithms. Furthermore, the generated MOEAs are compared to two well-known MOEAs used in the literature to solve the ITO problem. Results show that the MOEAs generated with GE and irace perform similarly, and both outperform traditional MOEAs. The approach can reduce efforts spent to design and configure MOEAs, and serves as basis for implementing solutions to other software engineering problems.

A Classification-Based Surrogate-Assisted Multiobjective Evolutionary Algorithm for Production Optimization under Geological Uncertainty

Summary Multiobjective optimization (MOO) is a popular procedure for waterflooding optimization under geological uncertainty that maximizes the expectation of net present value (NPV) over all possible uncertainty models and minimizes the variance simultaneously. However, the optimization process involves a large number of decision variables, and the problem is computationally expensive. Surrogate-assisted evolutionary algorithms (SAEAs), which have proved to be an effective way to solve expensive problems, design computationally inexpensive functions to approximate each objective function. On the basis of characterization, we have designed an efficient multiobjective evolutionary algorithm (MOEA) to effectively deal with computationally expensive simulation-based optimization problems. The uniqueness of this algorithm is that it incorporates a Pareto-rank-learning scheme with surrogate-assisted infill criterion. The first is to introduce a multiclass error-correcting output codes (ECOC) model that directly predicts the dominance relationship between candidate solutions and prescreens, and the second is to train a radial-basis function (RBF) network that predicts the objective functions of prescreened solutions to calculate the hypervolume (HV) improvement that maintains convergence and diversity. Compared with typical surrogate-based methods, the developed method provides a classifier first that can enhance the accuracy in high dimensions and reduce computational complexity. To the best of our knowledge, the proposed method compares with state-of-the-art surrogate frameworks for multiobjective production-optimization problems. In this paper, the proposed approach is applied to two 200D benchmark problems and two synthetic reservoir models. The results show that the proposed method can achieve more comprehensive and efficient reservoir management (RM) with a higher convergence speed compared with traditional MOEAs and surrogate-assisted optimization methods.

Configuration space evolutionary algorithm for multi-objective unequal-area facility layout problems with flexible bays

Facility layout problem (FLP) which deals with the layout of facilities within a given plant floor is an NP-hard combinatorial optimization problem. This paper studies multi-objective unequal-area facility layout problems (UA-FLPs) with the flexible bay structure (FBS), whose objectives refer to the material handling cost, the closeness relationship, the distance requirement and the aspect ratio of facilities. In recent years, some successes have been achieved by multi-objective evolutionary algorithms (MOEAs) for solving various kinds of optimization problems with multiple conflicting objectives. However, traditional MOEAs face a great challenge in the convergence and diversity of solutions for UA-FLPs. In this paper, a novel MOEA called the configuration space evolutionary (CSE) algorithm is developed to solve the UA-FLPs with multiple objectives. We consider a mating pool called a configuration (solution) bank in the CSE, and use evolutionary operations (selection, novel crossover and mutation) to produce new configurations of the pool. By introducing a measure of the radius dspace of the configuration bank, whose value is gradually reduced to narrow the search space, the convergence of solutions in the CSE is controlled. A method of the nearest and farthest candidate solution based on objective function normalization is combined with the fast non-dominated sorting to choose the Pareto-optimal solutions, which is good for the algorithm to keep diversity of the obtained solutions. The main contributions of this study lie in the use of a mechanism of evolution of population in the algorithm based on a configuration bank, and the use of a selection strategy based on the nearest and farthest candidate solution method, in order to improve the convergence and diversity of solutions. Experiments are carried out on eight different representative instances and performance metrics from the literature. Compared with the existing MOEAs, the CSE is able to find the better results and show better performance. The numerical experiments confirm the effectiveness of the CSE for solving multi-objective UA-FLPs.

Reference Point Based Multi-Objective Optimization of Reservoir Operation: a Comparison of Three Algorithms

Traditional multi-objective evolutionary algorithms treat each objective equally and search randomly in all solution spaces without using preference information. This might reduce the search efficiency and quality of solutions preferred by decision makers, especially when solving problems with complicated properties or many objectives. Three reference point based algorithms which adopt preference information in optimization progress, e.g., R-NSGA-II, r-NSGA-II and g-NSGA-II, have been shown to be effective in finding more preferred solutions in theoretical test problems. However, more efforts are needed to test their effectiveness in real-world problems. This study conducts a comparison of the above three algorithms with a standard algorithm NSGA-II on a reservoir operation problem to demonstrate their performance in improving the search efficiency and quality of preferred solutions. Under the same calculation times of the objective functions, Pareto optimal solutions of the four algorithms are used in the empirical comparison in terms of the approximation to the preferred solutions. Three performance indicators are then adopted for further comparison. Results show that R-NSGA-II and r-NSGA-II can improve the search efficiency and quality of preferred solutions. The convergence and diversity of their solutions in the concerned region are better than NSGA-II, and the closeness degree to the reference point can be increased by 42.8%, and moreover the number of preferred solutions can be increased by more than 3 times when part of objectives are preferred. By contrast, g-NSGA-II shows worse performance. This study exhibits the performance of three reference point based algorithms and provides insights in algorithm selection for multi-objective reservoir optimization problems.