Multiobjective evolutionary algorithms based on target region preferences
Introduction
Many real-world applications deal with multiobjective optimization problems (MOPs), in which several objectives are to be optimized simultaneously. Because of the conflicting nature of the objectives, a single solution that reaches the optimum for every objective does not exist. Instead, a set of trade-off solutions termed Pareto optimal solutions constitute the solution set, which is called Pareto set (PS). The corresponding image in the objective space is referred to as Pareto front (PF).
Metaheuristics are a broad family of non-deterministic optimization methods that may provide a sufficiently good solution to complex optimization problems [1]. Multiobjective Metaheuristics (MOMHs) have demonstrated a great success in solving MOPs, among which multiobjective evolutionary algorithms (MOEAs) are by far very popular and widely used. Other alternatives include particle swarm optimization (PSO), artificial immune system (AIS), ant colony optimization (ACO), etc.
Since the ultimate goal of multi-objective optimization is to assist the decision maker (DM) in finding the most preferred solution, the integration of preferences becomes indispensable. Preference-based MOMHs (PMOMHs) have attracted widespread attention in both academic researches [[2], [3], [4]] and engineering applications [[5], [6], [7]]. With preference information before (a-priori) or during (interactive) the optimization process, PMOMHs obtain a subset of Pareto optimal solutions that are of interest to the DM without obtaining the whole PS, thus alleviating the selection burden of the DM.
The preferences can be articulated through reference point [8,9], Desirability functions [10,11], aspiration set [12], weight functions [13,14], trade-off constraints [15,16], weight vectors [17,18], pairwise comparison [19,20], outranking relations [21,22] and so on. A user-defined region in the objective space in the shape of a rectangle or a circle (which is termed target region) is an intuitive and flexible way to express preferences. It is more direct and intuitive than the trade-off constraints or weight vectors, by which the DM has no idea which part of the PF will be gained. It is more flexible than a single reference point, since it offers a controllable range for every objective. In fact, a reference point can be interpreted as a special region with a zero range.
Recently, a target region-based multiobjective evolutionary algorithm (TMOEA) framework has been proposed [23]. It aims at finding a more fine-grained resolution of a target region explicitly specified by a DM. Following this framework, three new algorithms, i.e., T-NSGA-II, T-SMS-EMOA, T-R2-EMOA (where T stands for target region) were devised based on NSGA-II [24], SMS-EMOA [25] and R2-EMOA [26], respectively. In this paper, we continue this work [23] with a further analysis of TMOEAs on extensive numerical experiments. We also compare with related state-of-the-art PMOMHs to demonstrate the advantages of TMOEAs. An application to real-world problems is also shown to validate the capabilities of TMOEAs in practice. The mission planning of agile earth observation satellites is a constrained mixed integer problem, a reference point-based MOEA was proposed to obtain user-preferred solutions [27]. In this paper, we investigate the same problem instances, but target regions are adopted to express the user preferences.
Compared to our earlier works [23,27], new contributions of this paper can be summarized as follows:
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Detailed implementations of the TMOEAs are presented and computational complexity of the TMOEAs is analyzed.
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Performance comparison is conducted among T-NSGA-II, T-SMS-EMOA and T-R2-EMOA on the ZDT and DTLZ test suite with multiple target regions.
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Characteristics and advantages of the TMOEAs are illustrated by comparing them with DF-SMS-EMOA [11], RVEA [17], AS-EMOA [28] and R-NSGA-II [8].
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An application case study of satellite mission planning is shown to validate the benefit of incorporating target region preferences in real-life multiobjective optimization problems.
The rest of this paper is organized as follows. A summary of state-of-the-art results in the PMOMH field is given in section 2. TMOEAs are introduced together with complexity analysis in section 3. In section 4, numerical experiments are performed to compare three TMOEAs, as well as four state-of-the-art PMOMHs. An application in satellite mission planning is given in section 5 and conclusions are drawn in section 6.
Section snippets
Preference-based multiobjective metaheuristics
Preference-based multiobjective metaheuristics (PMOMHs) can be regarded as a collaboration of MOMH and multiple criteria decision making (MCDM). Preference information provided by the DM is utilized to guide the search towards preferred parts of the PF, instead of approximating the whole PF. The advantages of incorporating preferences into MOMH include the following:
- (1)
When it comes to many-objective optimization problems (MaOP, there are more than three objectives), it is usually difficult to
Introduction to TMOEAs
Target region-based multiobjective evolutionary algorithms (TMOEAs) aim at a more fine-grained resolution of the target region(s) without exploring the whole set of Pareto optimal solutions. The target region, or region of interest (ROI), is explicitly specified by the DM. TMOEAs can guide the search towards this region and achieve a well-converged and well-distributed set of Pareto optimal solutions within it. Fig. 1 shows an example of the result solutions by TMOEAs. T-NSGA-II, T-SMS-EMOA and
Comparison among three TMOEAs
The TMOEAs are implemented based on the MOEAFramework [73],3 which is a free and open source Java library for developing and experimenting with MOEAs. ZDT1-4 and DTLZ1-3 with three and four objectives are chosen as the test problems. For each problem, one target region and multiple (two or three) target regions are both tested. The Simulated Binary Crossover (SBX) operator and Polynomial Mutation (PM) operator
Problem description
Earth observation satellite (EOS) mission planning is now widely investigated with the development of aerospace technologies. Agile earth observation satellite (AEOS), in which the onboard camera can turn around three axes (roll, pitch, yaw) for satellite imaging [76], is addressed in this paper. The mission of an EOS is to acquire images of specified areas on the earth surface (targets), in response to observation requests from customers. Given the satellite trajectory and target location,
Conclusion
Target region-based multiobjective evolutionary algorithms (TMOEAs) are addressed in this paper, both in academic benchmarks and real-world application. After a short review of the state-of-the-art preference region and reference point based approaches, TMOEAs are introduced with T-NSGA-II, T-SMS-EMOA and T-R2-EMOA as three instantiations. The performances of the three algorithms are compared in ZDT and DTLZ benchmarks. Results show that T-SMS-EMOA has the best overall performance regarding
Acknowledgement
Longmei Li acknowledges financial support from China Scholarship Council (CSC). Heike Trautmann and Michael Emmerich acknowledge support by the European Research Center for Information Systems (ERCIS).
References (77)
- et al.
Chapter four-preference incorporation in evolutionary multiobjective optimization: a survey of the state-of-the-art
Adv. Comput.
(2015) - et al.
Reservoir flood control operation using multi-objective evolutionary algorithm with decomposition and preferences
Appl. Soft Comput.
(2017) - et al.
Guidance in evolutionary multi-objective optimization
Adv. Eng. Software
(2001) - et al.
Generalized decomposition and cross entropy methods for many-objective optimization
Inf. Sci.
(2014) - et al.
Evolutionary multiobjective optimization using an outranking-based dominance generalization
Comput. Oper. Res.
(2010) - et al.
A comparative study of different approaches using an outranking relation in a multi-objective evolutionary algorithm
Comput. Oper. Res.
(2013) - et al.
SMS-EMOA: multiobjective selection based on dominated hypervolume
Eur. J. Oper. Res.
(2007) - et al.
On reference point free weighted hypervolume indicators based on desirability functions and their probabilistic interpretation
Proced. Technol.
(2014) - et al.
g-dominance: reference point based dominance for multiobjective metaheuristics
Eur. J. Oper. Res.
(2009) - et al.
A novel preference articulation operator for the evolutionary multi-objective optimisation of classifiers in concealed weapons detection
Inf. Sci.
(2015)
Reference point based multi-objective optimization through decomposition
Selecting and scheduling observations of agile satellites
Aero. Sci. Technol.
Structural design using multi-objective metaheuristics. Comparative study and application to a real-world problem
Struct. Multidiscip. Optim.
Consideration of partial user preferences in evolutionary multiobjective optimization
MCDA and multiobjective evolutionary algorithms
Preference incorporation to solve many-objective airfoil design problems
Preference-based evolutionary algorithm for airport runway scheduling and ground movement optimisation
Reference point based multi-objective optimization using evolutionary algorithms
The r-dominance: a new dominance relation for interactive evolutionary multicriteria decision making
Evol. Comput. IEEE Trans.
Preference-based Pareto optimization in certain and noisy environments
Eng. Optim.
Integration of preferences in hypervolume-based multiobjective evolutionary algorithms by means of desirability functions
Evol. Comput. IEEE Trans.
An aspiration set EMOA based on averaged hausdorff distances
Weighted preferences in evolutionary multi-objective optimization
Directed multiobjective optimization based on the weighted hypervolume indicator
J. Multi-Criteria Decis. Anal.
A framework for incorporating trade-off information using multi-objective evolutionary algorithms
Evolutionary many-objective optimization of hybrid electric vehicle control: from general optimization to preference articulation
IEEE Trans. Emerg. Top. Comput. Intell.
An interactive evolutionary metaheuristic for multiobjective combinatorial optimization
Manag. Sci.
Interactive evolutionary multiobjective optimization driven by robust ordinal regression
Bull. Pol. Acad. Sci. Tech. Sci.
A new approach to target region based multiobjective evolutionary algorithms
A fast and elitist multiobjective genetic algorithm: NSGA-II
IEEE Trans. Evol. Comput.
R2-EMOA: focused multiobjective search using R2-indicator-based selection
Preference incorporation to solve multi-objective mission planning of agile earth observation satellites
A multiobjective evolutionary algorithm guided by averaged hausdorff distance to aspiration sets
An investigation on preference order ranking scheme for multiobjective evolutionary optimization
Evol. Comput. IEEE Trans.
Preference-based solution selection algorithm for evolutionary multiobjective optimization
Evol. Comput. IEEE Trans.
Handling preferences in evolutionary multiobjective optimization: a survey
Preference incorporation in multi-objective evolutionary algorithms: a survey
A review of hybrid evolutionary multiple criteria decision making methods
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2022, Swarm and Evolutionary ComputationCitation Excerpt :In literature, some researchers model the investor’s preferences in order to find the most satisfactory portfolio from the investor’s perspective, such as Fernandez et al. [29] . Meanwhile, some researchers are interested in solving this problem through algorithmic, for example, target region based multi-objective evolutionary algorithms [30], can target a subset of Pareto optimal solutions that are well distributed in the area provided by the user. The knee based decision-making method proposed by He et al. [31], searches several preferred solutions from a large number of solutions in Pareto frontier, where each solution contains the best convergence performance at least in its neighborhood.
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2021, Swarm and Evolutionary ComputationCitation Excerpt :Both a single RP and a reference vector may reflect DMs requirements as well as aspiration levels [12] or expectations. Another RP-related approach is specifying directly a Region of Interest (ROI) or target region [13]. DM may express their satisfaction with solutions within ROI by means of density function [14], desirability function [15,16] or a combinations of the above two [17].
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2021, Swarm and Evolutionary ComputationCitation Excerpt :In order to solve this problem, Yu et al. [29] proposed a preferred multiobjective decomposition algorithm based on weight iteration (MOEA/D-PRE). Li et al. [30] proposed an idea in which a nonuniform mapping scheme using the originally evenly distributed reference points on a canonical simplex can be mapped to new positions close to the aspiration-level vector supplied by the DM (MOEA/D-STM) and some researchers have suggested other ways to guide the population toward ROI [31–35]. However, these algorithms lack the division of hierarchical solution set and the control of density.
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2020, Expert Systems with ApplicationsCitation Excerpt :Detailed surveys on interactive optimization methods can be found in Meignan, Knust, Frayret, Pesant, and Gaud (2015) and Xin et al. (2018), while a recent review on preference modeling and articulation can be found in Wang, Olhofer, and Jin (2017). Common types of preferential information found in the literature are: reference points (Qi, Li, Yu, & Miao, 2019; Dasdemir, Köksalan, & Öztürk, 2020); aspiration levels (Saborido, Ruiz, Luque, & Miettinen, 2019); classification of objectives (Miettinen & Mäkelä, 2006); target region (Li, Wang, Trautmann, Jing, & Emmerich, 2018); navigation (Vallerio, Hufkens, Van Impe, & Logist, 2015); solution score (Li, Chen, Savić, & Yao, 2018); pairwise comparison of solutions (Kadziński, Tomczyk, & Słowiński, 2020; Tomczyk & Kadziński, 2019); order of a small subset of solutions (Deb, Sinha, Korhonen, & Wallenius, 2010); classification of solutions (Greco, Matarazzo, & Słowiński, 2010); and selection of solutions (Fowler et al., 2010). In general, indirect methods that focus on holistic evaluation of solutions, such as pairwise comparison, ordering, classification, and selection, are more comprehensible for the DM, reducing his cognitive effort, without requiring any quantitative preferences, e.g., weights, trade-offs, aspiration levels and others (Greco, Mousseau, & Słowiński, 2008).
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2020, Swarm and Evolutionary ComputationCitation Excerpt :The former corresponds to the values of parameters of an assumed preference model. The example forms of such a direct preference information include objective weights [17], comparison thresholds [14], aspiration or reservation levels [56,70], parameters of a desirability function [68], a reference point [17,48,53,61,67], a target region [46], a size of territory preventing crowding [38], scores of solutions [44,62], or fitness intervals assigned to the solutions contained in a small population [60]. However, a direct specification of such parameter values – even if some imprecision or uncertainty is admitted – is considered to be cognitively demanding for the DM.
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