Dynamic multiobjective optimization problems: test cases, approximations, and applications

After demonstrating adequately the usefulness of evolutionary multiobjective optimization (EMO) algorithms in finding multiple Pareto-optimal solutions for static multiobjective optimization problems, there is now a growing need for solving dynamic multiobjective optimization problems in a similar manner. In this paper, we focus on addressing this issue by developing a number of test problems and by suggesting a baseline algorithm. Since in a dynamic multiobjective optimization problem, the resulting Pareto-optimal set is expected to change with time (or, iteration of the optimization process), a suite of five test problems offering different patterns of such changes and different difficulties in tracking the dynamic Pareto-optimal front by a multiobjective optimization algorithm is presented. Moreover, a simple example of a dynamic multiobjective optimization problem arising from a dynamic control loop is presented. An extension to a previously proposed direction-based search method is proposed for solving such problems and tested on the proposed test problems. The test problems introduced in this paper should encourage researchers interested in multiobjective optimization and dynamic optimization problems to develop more efficient algorithms in the near future.     

A novel cooperative coevolutionary dynamic multi-objective optimization algorithm using a new predictive model

Dynamic multi-objective optimization problem (DMOP) is quite challenging and it dues to that there are multiple conflicting objects changing over with time or environment. In this paper, a novel cooperative coevolutionary dynamic multi-objective optimization algorithm (PNSCCDMO) is proposed. The main idea of a new cooperative coevolution based on non-dominated sorting is that it allows the decomposition process of the optimization problem according to the search space of decision variables, and each species subcomponents will cooperate to evolve for better solutions. This way derives from nature and can improve convergence significantly. A modified linear regression prediction strategy is used to make rapid response to the new changes in the environment. The effectiveness of PNSCCDMO is validated against various of DMOPs compared with the other four algorithms, and the experimental result indicates PNSCCDMO has a good capability to track the Pareto front as it is changed with time in dynamic environments.     

A hybrid evolutionary multiobjective optimization strategy for the dynamic power supply problem in magnesia grain manufacturing

The supply trajectory of electric power for submerged arc magnesia furnace determines the yields and grade of magnesia grain during the manufacture process. As the two production targets (i.e., the yields and the grade of magnesia grain) are conflicting and the process is subject to changing conditions, the supply of electric power needs to be dynamically optimized to track the moving Pareto optimal set with time. A hybrid evolutionary multiobjective optimization strategy is proposed to address the dynamic multiobjective optimization problem. The hybrid strategy is based on two techniques. The first one uses case-based reasoning to immediately generate good solutions to adjust the power supply once the environment changes, and then apply a multiobjective evolutionary algorithm to accurately solve the problem. The second one is to learn the case solutions to guide and promote the search of the evolutionary algorithm, and the best solutions found by the evolutionary algorithm can be used to update the case library to improve the accuracy of case-based reasoning in the following process. Due to the effectiveness of mutual promotion, the hybrid strategy can continuously adapt and search in dynamic environments. Two prominent multiobjective evolutionary algorithms are integrated into the hybrid strategy to solve the dynamic multiobjective power supply optimization problem. The results from a series of experiments show that the proposed hybrid algorithms perform better than their component multiobjective evolutionary algorithms for the tested problems.     

记忆增强的动态多目标分解进化算法

现实世界中的一些多目标优化问题经常受动态环境影响而不断发生变化,要求优化算法不断地及时跟踪时变的Pareto 最优解集.提出了一种记忆增强的动态多目标分解进化算法.将动态多目标优化问题分解为若干个动态单目标优化子问题并同时优化这些子问题,以便快速逼近Pareto 最优解集.给出了一个改进的环境变化检测算子,以便更好地检测环境变化.设计了一种基于子问题的串式记忆方法,利用过去类似环境下搜索到的最优解来有效地响应新的环境变化.在8 个标准的测试问题上,将新算法与其他3 种记忆增强的动态进化多目标优化算法进行了实验比较.结果表明,新算法比其他3 种算法具有更快的运行速度、更强的记忆能力与鲁棒性能,并且新算法所获得的解集还具有更好的收敛性与分布性.     

动态多目标优化的预测遗传算法

为了在动态环境中很好地跟踪最优解,考虑动态优化问题的特点,提出一种新的多目标预测遗传算法.首先对 Pareto 前沿面进行聚类以求得解集的质心;其次应用该质心与参考点描述 Pareto 前沿面;再次通过预测方法给出预测点集,使得算法在环境变化后能够有指导地增加种群多样性,以便快速跟踪最优解;最后应用标准动态测试问题进行算法测试,仿真分析结果表明所提出算法能适应动态环境,快速跟踪 Pareto 前沿面.     

Parameter Settings and Representations in Pareto-Based Optimization for Job Shop Scheduling

A multiobjective optimization problem involves multiple objectives, often conflicting, to be met or optimized. A Pareto front provides a set of best solutions to determine the trade-offs between the various objectives. New evolutionary approaches demonstrated its ability to build well-delineated Pareto fronts in diverse multiobjective optimization problems, including multicriteria optimization in job shop scheduling with regular and nonregular objective functions. Good parameter settings and appropriate representations can enhance the behavior of an evolutionary algorithm. The present article shows a study of the influence of distinct parameter combinations as well as different chromosome representations. Details of implementation and results are discussed.     

A distributed Cooperative coevolutionary algorithm for multiobjective optimization

Recent advances in evolutionary algorithms show that coevolutionary architectures are effective ways to broaden the use of traditional evolutionary algorithms. This paper presents a cooperative coevolutionary algorithm (CCEA) for multiobjective optimization, which applies the divide-and-conquer approach to decompose decision vectors into smaller components and evolves multiple solutions in the form of cooperative subpopulations. Incorporated with various features like archiving, dynamic sharing, and extending operator, the CCEA is capable of maintaining archive diversity in the evolution and distributing the solutions uniformly along the Pareto front. Exploiting the inherent parallelism of cooperative coevolution, the CCEA can be formulated into a distributed cooperative coevolutionary algorithm (DCCEA) suitable for concurrent processing that allows inter-communication of subpopulations residing in networked computers, and hence expedites the computational speed by sharing the workload among multiple computers. Simulation results show that the CCEA is competitive in finding the tradeoff solutions, and the DCCEA can effectively reduce the simulation runtime without sacrificing the performance of CCEA as the number of peers is increased.     

AN ANALYSIS OF MULTIOBJECTIVE EVOLUTIONARY ALGORITHMS FOR OPTIMIZATION PROBLEMS WITH TIME CONSTRAINTS

Many multiobjective optimization problems in the engineering field are required to be solved within more or less severe time restrictions. Because the optimization criteria, the parameters, and/or constraints might change with time, the optimization solutions must be recalculated when a change takes place. The time required by the optimization procedure to arrive at the new solutions should be bounded accordingly with the rate of change observed in these dynamic problems. This way, the faster the optimization algorithm is to obtain solutions, the wider is the set of dynamic problems to which that algorithm can be applied. Here, we analyze the performance of the nondominated sorting algorithm (NSGA-II), strength Pareto evolutionary algorithm (SPEA2), and single front genetic algorithms (SFGA, and SFGA2) on two different multiobjective optimization problems, with two and three objectives, respectively. For these two studied problems, the single front genetic algorithms have obtained adequate quality in the solutions in very little time. Moreover, for the second and more complex problem approached, SFGA2 and NSGA-II obtain the best hypervolume in the found set of nondominated solutions, but SFGA2 employs much less time than NSGA-II. These results may suggest that single front genetic algorithms, especially SFGA2, could be appropiate to deal with optimization problems with high rates of change, and thus stronger time constraints.     

Quantum immune clonal coevolutionary algorithm for dynamic multiobjective optimization

The existing algorithms to solve dynamic multiobjective optimization (DMO) problems generally have difficulties in non-uniformity, local optimality and non-convergence. Based on artificial immune system, quantum evolutionary computing and the strategy of co-evolution, a quantum immune clonal coevolutionary algorithm (QICCA) is proposed to solve DMO problems. The algorithm adopts entire cloning and evolves the theory of quantum to design a quantum updating operation, which improves the searching ability of the algorithm. Moreover, coevolutionary strategy is incorporated in global operation and coevolutionary competitive operation and coevolutionary cooperative operation are designed to improve the uniformity, the diversity and the convergence performance of the solutions. The results on test problems and performance metrics compared with ICADMO and DBM suggest that QICCA has obvious effectiveness and advantages which shows great capability of evolving convergent, diverse and uniformly distributed Pareto fronts.     

可求解多目标优化问题的演化博弈优化算法

借鉴演化博弈的思想和选择机制,提出了一种新的基于演化博弈的优化算法(EGOA)用于多目标问题的求解.算法框架具备对该类问题的通用性.为了对算法性能进行评估,采用了一组多目标优化问题(MOPs)的测试函数进行实验.实验结果表明,使用本算法搜索得到的演化稳定策略集合能够很好地逼近多目标优化问题的帕累托前沿,与一些经典的演化算法相比具有良好的问题求解能力.     

Multiobjective optimization immune algorithm in dynamic environments and its application to greenhouse control

A novel multiobjective optimization immune algorithm in dynamic environments, as associated with Pareto optimality and immune metaphors of germinal center in the immune system, is proposed to deal with a class of dynamic multiobjective optimization problems which the dimension of the objective space may change over time. Several immune operators, depending on both somatic maturation and T-cell regulation, are designed to adapt to the changing environment so that the algorithm can achieve a reasonable tradeoff between convergence and diversity of population, among which an environmental recognition rule related to the past environmental information is established to identify an appearing environment. Preliminary experiments show that the proposed algorithm cannot only obtain great superiority over two popular algorithms, but also continually track the time-varying environment. Comparative analysis and practical application illustrate its potential.     

A prediction strategy based on center points and knee points for evolutionary dynamic multi-objective optimization

In real life, there are many dynamic multi-objective optimization problems which vary over time, requiring an optimization algorithm to track the movement of the Pareto front (Pareto set) with time. In this paper, we propose a novel prediction strategy based on center points and knee points (CKPS) consisting of three mechanisms. First, a method of predicting the non-dominated set based on the forward-looking center points is proposed. Second, the knee point set is introduced to the predicted population to predict accurately the location and distribution of the Pareto front after an environmental change. Finally, an adaptive diversity maintenance strategy is proposed, which can generate some random individuals of the corresponding number according to the degree of difficulty of the problem to maintain the diversity of the population. The proposed strategy is compared with four other state-of-the-art strategies. The experimental results show that CKPS is effective for evolutionary dynamic multi-objective optimization.     

Solution of multiobjective optimization problems: coevolutionary algorithm based on evolutionary game theory

When attempting to solve multiobjective optimization problems (MOPs) using evolutionary algorithms, the Pareto genetic algorithm (GA) has now become a standard of sorts. After its introduction, this approach was further developed and led to many applications. All of these approaches are based on Pareto ranking and use the fitness sharing function to keep diversity. On the other hand, the scheme for solving MOPs presented by Nash introduced the notion of Nash equilibrium and aimed at solving MOPs that originated from evolutionary game theory and economics. Since the concept of Nash Equilibrium was introduced, game theorists have attempted to formalize aspects of the evolutionary equilibrium. Nash genetic algorithm (Nash GA) is the idea to bring together genetic algorithms and Nash strategy. The aim of this algorithm is to find the Nash equilibrium through the genetic process. Another central achievement of evolutionary game theory is the introduction of a method by which agents can play optimal strategies in the absence of rationality. Through the process of Darwinian selection, a population of agents can evolve to an evolutionary stable strategy (ESS). In this article, we find the ESS as a solution of MOPs using a coevolutionary algorithm based on evolutionary game theory. By applying newly designed coevolutionary algorithms to several MOPs, we can confirm that evolutionary game theory can be embodied by the coevolutionary algorithm and this coevolutionary algorithm can find optimal equilibrium points as solutions for an MOP. We also show the optimization performance of the co-evolutionary algorithm based on evolutionary game theory by applying this model to several MOPs and comparing the solutions with those of previous evolutionary optimization models. This work was presented, in part, at the 8th International Symposium on Artificial Life and Robotics, Oita, Japan, January 24#x2013;26, 2003.     

基于多区域中心点预测的动态多目标优化算法

现实生活中存在很多动态多目标优化问题(DMOPs),这类问题要求算法在环境变化后快速收敛到新的Pareto前沿,并保持解集的多样性,随着Pareto前沿复杂程度的增加,这一问题更加突出.鉴于此,提出一种基于多区域中心点预测的动态多目标优化算法(MCPDMO).首先,根据环境变化的严重程度将种群划分为多个子区域,使得个体的分配更加适应动态变化的环境;然后,分别计算每个子区域的中心点,对不同子区域在不同时刻的中心点建立时间序列,并利用差分模型预测新环境的最优解集,以提高算法对不同环境变化的响应能力;最后,为验证算法的有效性,与3种动态多目标优化算法在10个标准测试函数上进行仿真实验.实验结果表明,所提出算法在具有复杂Pareto前沿的动态问题上表现出更优的收敛性和分布性.     

Multiobjective evolutionary algorithms for electric power dispatch problem

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.     

Solving Multiobjective Optimization Problems Using an Artificial Immune System

In this paper, we propose an algorithm based on the clonal selection principle to solve multiobjective optimization problems (either constrained or unconstrained). The proposed approach uses Pareto dominance and feasibility to identify solutions that deserve to be cloned, and uses two types of mutation: uniform mutation is applied to the clones produced and non-uniform mutation is applied to the not so good antibodies (which are represented by binary strings that encode the decision variables of the problem to be solved). We also use a secondary (or external) population that stores the nondominated solutions found along the search process. Such secondary population constitutes the elitist mechanism of our approach and it allows it to move towards the true Pareto front of a problem over time. Our approach is compared with three other algorithms that are representative of the state-of-the-art in evolutionary multiobjective optimization. For our comparative study, three metrics are adopted and graphical comparisons with respect to the true Pareto front of each problem are also included. Results indicate that the proposed approach is a viable alternative to solve multiobjective optimization problems.     

Dynamic multiobjective evolutionary algorithm: adaptive cell-based rank and density estimation

This paper proposes a new evolutionary approach to multiobjective optimization problems - the dynamic multiobjective evolutionary algorithm (DMOEA). In DMOEA, a novel cell-based rank and density estimation strategy is proposed to efficiently compute dominance and diversity information when the population size varies dynamically. In addition, a population growing and declining strategies are designed to determine if an individual will survive or be eliminated based on some qualitative indicators. Meanwhile, an objective space compression strategy is devised to continuously refine the quality of the resulting Pareto front. By examining the selected performance metrics on three recently designed benchmark functions, DMOEA is found to be competitive with or even superior to five state-of-the-art MOEAs in terms of maintaining the diversity of the individuals along the tradeoff surface, tending to extend the Pareto front to new areas, and finding a well-approximated Pareto optimal front. Moreover, DMOEA is evaluated by using different parameter settings on the chosen test functions to verify its robustness of converging to an optimal population size, if it exists. Simulations show that DMOEA has the potential of autonomously determining the optimal population size, which is found insensitive to the initial population size chosen.     

基于分解和多策略变异的多目标差分进化算法*

差分进化是一种有效的优化技术,已成功用于多目标优化问题。但也存在Pareto最优集合的收敛慢和多样性差等问题。针对上述不足,本文提出了一种基于分解和多策略变异的多目标差分进化算法(MODE/DMSM)。该算法利用基于分解的方法将多目标优化问题分解为多个单目标优化问题;通过高效的非支配排序方法选择具有良好收敛性和多样性的解来指导差分进化过程;采用了多策略变异方法来平衡进化过程中收敛性和多样性。在ZDT和DTLZ的10个测试函数上的仿真结果表明,本文算法在Parato最优集合的收敛性和多样性优于其他六种代表性多目标优化算法。     

Multiobjective Evolutionary Algorithm With Controllable Focus on the Knees of the Pareto Front

The optimal solutions of a multiobjective optimization problem correspond to a nondominated front that is characterized by a tradeoff between objectives. A knee region in this Pareto-optimal front, which is visually a convex bulge in the front, is important to decision makers in practical contexts, as it often constitutes the optimum in tradeoff, i.e., substitution of a given Pareto-optimal solution with another solution on the knee region yields the largest improvement per unit degradation. This paper presents a selection scheme that enables a multiobjective evolutionary algorithm (MOEA) to obtain a nondominated set with controllable concentration around existing knee regions of the Pareto front. The preference-based focus is achieved by optimizing a set of linear weighted sums of the original objectives, and control of the extent of the focus is attained by careful selection of the weight set based on a user-specified parameter. The fitness scheme could be easily adopted in any Pareto-based MOEA with little additional computational cost. Simulations on various two- and three-objective test problems demonstrate the ability of the proposed method to guide the population toward existing knee regions on the Pareto front. Comparison with general-purpose Pareto based MOEA demonstrates that convergence on the Pareto front is not compromised by imposing the preference-based bias. The performance of the method in terms of an additional performance metric introduced to measure the accuracy of resulting convergence on the desired regions validates the efficacy of the method.      

Multiobjective evolutionary algorithms: a comparative case studyand the strength Pareto approach

Evolutionary algorithms (EAs) are often well-suited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a single run. However, the few comparative studies of different methods presented up to now remain mostly qualitative and are often restricted to a few approaches. In this paper, four multiobjective EAs are compared quantitatively where an extended 0/1 knapsack problem is taken as a basis. Furthermore, we introduce a new evolutionary approach to multicriteria optimization, the strength Pareto EA (SPEA), that combines several features of previous multiobjective EAs in a unique manner. It is characterized by (a) storing nondominated solutions externally in a second, continuously updated population, (b) evaluating an individual's fitness dependent on the number of external nondominated points that dominate it, (c) preserving population diversity using the Pareto dominance relationship, and (d) incorporating a clustering procedure in order to reduce the nondominated set without destroying its characteristics. The proof-of-principle results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware-software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Pareto-optimal front and distributing the generated solutions over the tradeoff surface. Moreover, SPEA clearly outperforms the other four multiobjective EAs on the 0/1 knapsack problem