Incorporating Human Preferences in Decision Making for Dynamic Multi-Objective Optimization in Model Predictive Control

We present a new two-step approach for automatized a posteriori decision making in multi-objective optimization problems, i.e., selecting a solution from the Pareto front. In the first step, a knee region is determined based on the normalized Euclidean distance from a hyperplane defined by the furthest Pareto solution and the negative unit vector. The size of the knee region depends on the Pareto front’s shape and a design parameter. In the second step, preferences for all objectives formulated by the decision maker, e.g., 50–20–30 for a 3D problem, are translated into a hyperplane which is then used to choose a final solution from the knee region. This way, the decision maker’s preference can be incorporated, while its influence depends on the Pareto front’s shape and a design parameter, at the same time favorizing knee points if they exist. The proposed approach is applied in simulation for the multi-objective model predictive control (MPC) of the two-dimensional rocket car example and the energy management system of a building.

A Controlled Strengthened Dominance Relation for Evolutionary Many-Objective Optimization

Maintaining a balance between convergence and diversity is particularly crucial in evolutionary multiobjective optimization. Recently, a novel dominance relation called "strengthened dominance relation" (SDR) is proposed, which outperforms the existing dominance relations in balancing convergence and diversity. In this article, two points that influence the performance of SDR are studied and a new dominance relation, which is mainly based on SDR, is proposed (CSDR). An adaptation strategy is presented to dynamically adjust the dominance relation according to the current generation number. The CSDR is embedded into NSGA-II to substitute the Pareto dominance, labeled as NSGA-II/CSDR. The performance of our proposed method is validated by comparing it with five state-of-the-art algorithms on commonly used benchmark problems. NSGA-II/CSDR outperforms other algorithms in the most test instances considering both convergence and diversity.

Optimal Current Allocation Strategy for Hybrid Hierarchical HVDC System with Parallel Operation of High-Voltage and Low-Voltage DC Lines

For long-distance and bulk-power delivery of new energy, high-voltage direct current (HVDC) is a more effective way than high-voltage alternative current (HVAC). In view of the current capacity disparity between line commutated converter (LCC) and voltage source converter (VSC), a hybrid hierarchical HVDC topology with parallel operation of 800 kV and 400 kV DC lines is investigated. The optimal current allocation method for hybrid hierarchical HVDC is proposed distinct from the same rated current command configuration method of high-voltage and low-voltage converters in traditional topology. Considering the transmission loss reduction of the HVDC system, a multi-order fitting function of transmission loss including LCC converter stations, VSC converter stations and DC lines is established. To minimize the transmission loss and the voltage deviation of key DC nodes comprehensively, a multi-objective genetic algorithm and maximum satisfaction method are utilized to obtain the optimal allocation value of rated current command for high-voltage and low-voltage converters. Through the optimization model, an improved constant current controller based on the current allocation strategy is designed. The hybrid hierarchical HVDC system model is built in PSCAD software, and simulation results verify the effectiveness of the proposed topology and optimal current allocation strategy.

A Modified Coronavirus Herd Immunity Optimizer for the Power Scheduling Problem

The Coronavirus herd immunity optimizer (CHIO) is a new human-based optimization algorithm that imitates the herd immunity strategy to eliminate of the COVID-19 disease. In this paper, the coronavirus herd immunity optimizer (CHIO) is modified to tackle a discrete power scheduling problem in a smart home (PSPSH). PSPSH is a combinatorial optimization problem with NP-hard features. It is a highly constrained discrete scheduling problem concerned with assigning the operation time for smart home appliances based on a dynamic pricing scheme(s) and several other constraints. The primary objective when solving PSPSH is to maintain the stability of the power system by reducing the ratio between average and highest power demand (peak-to-average ratio (PAR)) and reducing electricity bill (EB) with considering the comfort level of users (UC). This paper modifies and adapts the CHIO algorithm to deal with such discrete optimization problems, particularly PSPSH. The adaptation and modification include embedding PSPSH problem-specific operators to CHIO operations to meet the discrete search space requirements. PSPSH is modeled as a multi-objective problem considering all objectives, including PAR, EB, and UC. The proposed method is examined using a dataset that contains 36 home appliances and seven consumption scenarios. The main CHIO parameters are tuned to find their best values. These best values are used to evaluate the proposed method by comparing its results with comparative five metaheuristic algorithms. The proposed method shows encouraging results and almost obtains the best results in all consumption scenarios.

Solving Many-Objective Optimization Problems by a Pareto-Based Evolutionary Algorithm With Preprocessing and a Penalty Mechanism

It is known that the Pareto-based approach is not well suited for optimization problems with a large number of objectives, even though it is a class of mainstream methods in multiobjective optimization. Typically, a Pareto-based algorithm comprises two parts: 1) a Pareto dominance-based criterion and 2) a diversity estimator. The former guides the selection toward the optimal front, while the latter promotes the diversity of the population. However, the Pareto dominance-based criterion becomes ineffective in solving optimization problems with many objectives (e.g., more than 3) and, thus, the diversity estimator will determine the performance of the algorithm. Unfortunately, the diversity estimator usually has a strong bias toward dominance resistance solutions (DRSs), thereby failing to push the population forward. DRSs are solutions that are far away from the Pareto-optimal front but cannot be easily dominated. In this article, we propose a new Pareto-based algorithm to resolve the above issue. First, to eliminate the DRSs, we design an interquartile range method to preprocess the solution set. Second, to balance convergence and diversity, we present a penalty mechanism of alternating operations between selection and penalty. The proposed algorithm is compared with five state-of-the-art algorithms on a number of well-known benchmarks with 3-15 objectives. The experimental results show that the proposed algorithm can perform well on most of the test functions and generally outperforms its competitors.

A many-objective evolutionary algorithm based on vector angle distance scaling

In the past two decades, multi-objective evolutionary algorithms (MOEAs) have achieved great success in solving two or three multi-objective optimization problems. As pointed out in some recent studies, however, MOEAs face many difficulties when dealing with many-objective optimization problems(MaOPs) on account of the loss of the selection pressure of the non-dominant candidate solutions toward the Pareto front and the ineffective design of the diversity maintenance mechanism. This paper proposes a many-objective evolutionary algorithm based on vector guidance. In this algorithm, the value of vector angle distance scaling(VADS) is applied to balance convergence and diversity in environmental selection. In addition, tournament selection based on the aggregate fitness value of VADS is applied to generate a high quality offspring population. Besides, we adopt an adaptive strategy to adjust the reference vector dynamically according to the scales of the objective functions. Finally, the performance of the proposed algorithm is compared with five state-of-the-art many-objective evolutionary algorithms on 52 instances of 13 MaOPs with diverse characteristics. Experimental results show that the proposed algorithm performs competitively when dealing many-objective with different types of Pareto front.

Smart Home Battery for the Multi-Objective Power Scheduling Problem in a Smart Home Using Grey Wolf Optimizer

The power scheduling problem in a smart home (PSPSH) refers to the timely scheduling operations of smart home appliances under a set of restrictions and a dynamic pricing scheme(s) produced by a power supplier company (PSC). The primary objectives of PSPSH are: (I) minimizing the cost of the power consumed by home appliances, which refers to electricity bills, (II) balance the power consumed during a time horizon, particularly at peak periods, which is known as the peak-to-average ratio, and (III) maximizing the satisfaction level of users. Several approaches have been proposed to address PSPSH optimally, including optimization and non-optimization based approaches. However, the set of restrictions inhibit the approach used to obtain the optimal solutions. In this paper, a new formulation for smart home battery (SHB) is proposed for PSPSH that reduces the effect of restrictions in obtaining the optimal/near-optimal solutions. SHB can enhance the scheduling of smart home appliances by storing power at unsuitable periods and use the stored power at suitable periods for PSPSH objectives. PSPSH is formulated as a multi-objective optimization problem to achieve all objectives simultaneously. A robust swarm-based optimization algorithm inspired by the grey wolf lifestyle called grey wolf optimizer (GWO) is adapted to address PSPSH. GWO has powerful operations managed by its dynamic parameters that maintain exploration and exploitation behavior in search space. Seven scenarios of power consumption and dynamic pricing schemes are considered in the simulation results to evaluate the proposed multi-objective PSPSH using SHB (BMO-PSPSH) approach. The proposed BMO-PSPSH approach’s performance is compared with that of other 17 state-of-the-art algorithms using their recommended datasets and four algorithms using the proposed datasets. The proposed BMO-PSPSH approach exhibits and yields better performance than the other compared algorithms in almost all scenarios.

Many-objective BAT algorithm

In many objective optimization problems (MaOPs), more than three distinct objectives are optimized. The challenging part in MaOPs is to get the Pareto approximation (PA) with high diversity and good convergence. In Literature, in order to solve the issue of diversity and convergence in MaOPs, many approaches are proposed using different multi objective evolutionary algorithms (MOEAs). Moreover, to get better results, the researchers use the sets of reference points to differentiate the solutions and to model the search process, it further evaluates and selects the non-dominating solutions by using the reference set of solutions. Furthermore, this technique is used in some of the swarm-based evolutionary algorithms. In this paper, we have used some effective adaptations of bat algorithm with the previous mentioned approach to effectively handle the many objective problems. Moreover, we have called this algorithm as many objective bat algorithm (MaOBAT). This algorithm is a biologically inspired algorithm, which uses echolocation power of micro bats. Each bat represents a complete solution, which can be evaluated based on the problem specific fitness function and then based on the dominance relationship, non-dominated solutions are selected. In proposed MaOBAT, dominance rank is used as dominance relationship (dominance rank of a solution means by how many other solutions a solution dominated). In our proposed strategy, dynamically allocated set of reference points are used, allowing the algorithm to have good convergence and high diversity pareto fronts (PF). The experimental results show that the proposed algorithm has significant advantages over several state-of-the-art algorithms in terms of the quality of the solution.

Evolutionary Many-Objective Optimization Based on Adversarial Decomposition

The decomposition-based evolutionary algorithm has become an increasingly popular choice for posterior multiobjective optimization. Facing the challenges of an increasing number of objectives, many techniques have been developed which help to balance the convergence and diversity. Nevertheless, according to a recent study by Ishibuchi et al., due to the predefined search directions toward the ideal point, their performance strongly depends on the Pareto front (PF) shapes, especially the orientation of the PFs. To balance the convergence and diversity for decomposition-based methods and to alleviate their performance dependence on the orientation of the PFs, this paper develops an adversarial decomposition method for many-objective optimization, which leverages the complementary characteristics of different subproblem formulations within a single paradigm. More specifically, two populations are co-evolved by two subproblem formulations with different contours and adversarial search directions. To avoid allocating redundant computational resources to the same region of the PF, the two populations are matched into one-to-one solution pairs according to their working regions upon the PF. Each solution pair can at most contribute one principal mating parent during the mating selection process. When comparing nine state-of-the-art many-objective optimizers, we have witnessed the competitive performance of our proposed algorithm on 130 many-objective test problems with various characteristics, including regular and inverted PFs.

Evolutionary Many-Objective Algorithm Using Decomposition-Based Dominance Relationship

Decomposition-based evolutionary algorithms have shown great potential in many-objective optimization. However, the lack of theoretical studies on decomposition methods has hindered their further development and application. In this paper, we first theoretically prove that weight sum, Tchebycheff, and penalty boundary intersection decomposition methods are essentially interconnected. Inspired by this, we further show that highly customized dominance relationship can be derived from decomposition for any given decomposition vector. A new evolutionary algorithm is then proposed by applying the customized dominance relationship with adaptive strategy to each subpopulation of multiobjective to multiobjective framework. Experiments are conducted to compare the proposed algorithm with five state-of-the-art decomposition-based evolutionary algorithms on a set of well-known scaled many-objective test problems with 5 to 15 objectives. Simulation results have shown that the proposed algorithm can make better use of the decomposition vectors to achieve better performance. Further investigations on unscaled many-objective test problems verify the robust and generality of the proposed algorithm.

A New Dominance Method Based on Expanding Dominated Area for Many-Objective Optimization

The performance of the traditional Pareto-based evolutionary algorithms sharply reduces for many-objective optimization problems, one of the main reasons is that Pareto dominance could not provide sufficient selection pressure to make progress in a given population. To increase the selection pressure toward the global optimal solutions and better maintain the quality of selected solutions, in this paper, a new dominance method based on expanding dominated area is proposed. This dominance method skillfully combines the advantages of two existing popular dominance methods to further expand the dominated area and better maintain the quality of selected solutions. Besides, through dynamically adjusting its parameter with the iteration, our proposed dominance method can timely adjust the selection pressure in the process of evolution. To demonstrate the quality of selected solutions by our proposed dominance method, the experiments on a number of well-known benchmark problems with 5–25 objectives are conducted and compared with that of the four state-of-the-art dominance methods based on expanding dominated area. Experimental results show that the new dominance method not only enhances the selection pressure but also better maintains the quality of selected solutions.

Environment Sensitivity-Based Cooperative Co-Evolutionary Algorithms for Dynamic Multi-Objective Optimization

Dynamic multi-objective optimization problems (DMOPs) not only involve multiple conflicting objectives, but these objectives may also vary with time, raising a challenge for researchers to solve them. This paper presents a cooperative co-evolutionary strategy based on environment sensitivities for solving DMOPs. In this strategy, a new method that groups decision variables is first proposed, in which all the decision variables are partitioned into two subcomponents according to their interrelation with environment. Adopting two populations to cooperatively optimize the two subcomponents, two prediction methods, i.e., differential prediction and Cauchy mutation, are then employed respectively to speed up their responses on the change of the environment. Furthermore, two improved dynamic multi-objective optimization algorithms, i.e., DNSGAII-CO and DMOPSO-CO, are proposed by incorporating the above strategy into NSGA-II and multi-objective particle swarm optimization, respectively. The proposed algorithms are compared with three state-of-the-art algorithms by applying to seven benchmark DMOPs. Experimental results reveal that the proposed algorithms significantly outperform the compared algorithms in terms of convergence and distribution on most DMOPs.

Many-Objective Particle Swarm Optimization Using Two-Stage Strategy and Parallel Cell Coordinate System

It is a daunting challenge to balance the convergence and diversity of an approximate Pareto front in a many-objective optimization evolutionary algorithm. A novel algorithm, named many-objective particle swarm optimization with the two-stage strategy and parallel cell coordinate system (PCCS), is proposed in this paper to improve the comprehensive performance in terms of the convergence and diversity. In the proposed two-stage strategy, the convergence and diversity are separately emphasized at different stages by a single-objective optimizer and a many-objective optimizer, respectively. A PCCS is exploited to manage the diversity, such as maintaining a diverse archive, identifying the dominance resistant solutions, and selecting the diversified solutions. In addition, a leader group is used for selecting the global best solutions to balance the exploitation and exploration of a population. The experimental results illustrate that the proposed algorithm outperforms six chosen state-of-the-art designs in terms of the inverted generational distance and hypervolume over the DTLZ test suite.

An Interval Approximation Approach for a Multiobjective Programming Model with Fuzzy Objective Functions

Research in optimization under uncertainty is alive. It assumes different shapes and forms, all concurring to the general goal of designing effective and efficient tools for handling imprecision in an Optimization setting. In this paper we present a new approach for dealing with multiobjective programming problems with fuzzy objective functions. Similar to many approaches in the literature, our approach relies on the deffuzification of involved fuzzy quantities. Our improvement stem from the choice of a deffuzification operator that captures essential features of fuzzy parameters at hand rather than those that yield single values, leading to a loss of many useful information. Two oracles play a pivotal role in the proposed method. The first one returns a near interval approximation to a given fuzzy number. The other one delivers a Pareto Optimal solution of the resulting multiobjective program with interval coefficient. A numerical example is also provided for the sake of illustration.