Abstract
Loss of selection pressure in the presence of many objectives is one of the pertinent problems in evolutionary optimization. Therefore, it is difficult for evolutionary algorithms to find the bestfitting candidate solutions for the final Pareto optimal front representing a multiobjective optimization problem, particularly when the solution space changes with time. In this study, we propose a multiobjective algorithm called enhanced dynamic nondominated sorting genetic algorithm III (EdyNSGAIII). This evolutionary algorithm is an improvement of the earlier proposed dyNSGAIII, which used principal component analysis and Euclidean distance to maintain selection pressure and integrity of the final Pareto optimal front. EdyNSGAIII proposes a strategy to select a group of superperforming mutated candidates to improve the selection pressure at high dimensions and with changing time. This strategy is based on an earlier proposed approach on the use of mutated candidates, which are randomly chosen from the mutation and crossover stages of the original NSGAII algorithm. In our proposed approach, these mutated candidates are used to improve the diversity of the solution space when the rate of change in the objective function space increases with respect to time. The improved algorithm is tested on RPOOT problems and a realworld hydrothermal model, and the results show that the approach is promising.
1. Introduction
Handling an increasing number of objectives over time is one of the challenges of evolutionary algorithms (EAs). In particular, handling problems with many objectives and complicated Pareto optimal front can impact the performance of EAs negatively. The idea of Pareto dominance with respect to any two solutions a and b of a manyobjective optimization problem is guided by the following (where solution a dominates solution b): (i) solution a completely dominates solution b after the natural selection process; (ii) both solutions are infeasible, but solution a has a smaller constraint violation than solution b with respect to the objective function; and (iii) solution a is feasible, while solution b is infeasible [1]. Pareto dominance is straightforward to implement when the number of objectives is not greater than 3. However, with more than 3 objectives, it becomes difficult to select the final set of solutions that satisfy all objectives describing the problem.
When we consider Pareto optimality in the presence of changing time, there is an additional uncertainty that is introduced since a given solution selected at instant t_{1} may not satisfy the problem at instant t_{2}. Also, there is a possibility that a given solution at t_{1} will still be effective at t_{2}. However, the changing solution space over time might cause these fit solutions to be lost because of loss of selection pressure at high dimensions. Therefore, improving the performance of EAs to effectively handle Pareto optimality over time in the presence of many conflicting objectives is imperative. In this study, we present a multiobjective optimization algorithm called enhanced dynamic nondominated sorting genetic algorithm III (EdyNSGAIII). It is originally based on the NSGAIII algorithm [2] and is an improvement on an earlier proposed dynamic multiobjective evolutionary algorithm (DMOEA) called dyNSGAIII [3]. The aim of the proposed improvement is to enhance the performance of dyNSGAIII on the optimization of robust Pareto optimality over time (RPOOT) problems and a multiobjective realworld power scheduling problem. RPOOT problems consist of dynamic Pareto set (PS) and Pareto front (PF) with respect to time [4]. The proposed EdyNSGAIII algorithm uses a strategy to select and maintain a group of topperforming, mutated, and randomly chosen candidate solutions in the population in the presence of multiple objectives. In particular, these mutated solutions help to maintain the diversity of the Pareto front when there is a frequent change in objective function(s), constraint functions, and/or variable boundaries representing a given dynamic manyobjective optimization problem (DMOP). These candidate solutions help to maintain the selection pressure of the EA as much as possible even in high dimensionality.
The main contribution of this study is to use randomly generated mutated solutions from a reference pointbased search strategy to improve the selection pressure of dyNSGAIII when the number of objectives is 3 or greater and/or when the DMOP has several dynamic parameters and constraints.
The rest of the study is organized as follows: Section 2 describes the use of evolutionary algorithms in multiobjective optimization. Section 3 outlines the proposed methodology for implementing EdyNSGAIII for dynamic multiobjective optimization and discusses the test results obtained from using EdyNSGAIII to optimize 4 selected instances of 3objective, dynamic manyobjective optimization problems (DMOPs). Obtained results are compared with an earlier proposed dyNSGAIII algorithm. We then compare the performance of EdyNSGAIII with 2 other wellperforming DMOEAs for dimensionality reduction in the presence of many objectives. We also present a multiobjective model of a power generation scheduling problem and optimize it using both EdyNSGAIII and dyNSGAIII. The results are also discussed. Section 4 concludes the study.
2. ManyObjective Optimization Using Evolutionary Algorithms
Several realworld problems are best modelled as manyobjective optimization problems such as power system stability [5], induction motor design [6], radarabsorbing material design [7], and jobshop scheduling [8]. Therefore, several researchers have proposed various methods of improving the performance of EAs to effectively balance convergence and diversity in the presence of many conflicting objectives with rapidly changing Pareto fronts. In [9], the robust ranking and selection approach for handling intractable complex optimization considering randomness was reformulated as a multiobjective ranking and selection problem. In this case, a single ranking and selection problem is decomposed into several objectives. In [10], a cache approach was proposed to improve the balance between convergence and diversity in the presence of high dimensionality, thus improving selection pressure, while [11] proposed a twostage strategy and parallel cell coordinate system (PCCS) to remedy the same problem.
In [12], the test case generation problem was modelled as a dynamic manyobjective optimization problem instead of the popular singleobjective problem. A dynamic manyobjective sorting algorithm (DynaMOSA) was proposed to solve the proposed optimization problem, and it was observed that this approach outperformed singleobjective optimizers by 28%. Further research has focused on improving evolutionary and natureinspired algorithms to tackle challenges of many objectives characterizing realworld problems [13], minimizing impact of expensive feature evaluations [14], and improving selection pressure [15]. A hybridized minimal cost evolutionary deterministic algorithm (HMCEDA) was one of the first evolutionary algorithms used to solve DMOPs [16]. The algorithm was used to optimize 5 DMOPs with slowly changing environments. Dynamic particle swarm optimization (DPSO) algorithm has also been proposed in [17] to solve DMOPs using Pareto dominance with hyperplane distribution. DPSO handles changes in the search environment by adjusting the velocity parameter of PSO for previous nondominated candidates.
The immune optimization approach has been used in recent years as a solution to DMOPs. In particular, the interaction between B and T cells was used to create an artificial immune system (AIS) approach for tracking DMOPs [18]. Also, an immune surveillance approach was proposed for solving such problems in [19]. A recent approach combined AISbased clonal selection with generalized differential evolution 3 (GDE3) algorithm to solve DMOPs. In [20], a distributed population evolution strategy based on the message passing interface (MPI) approach was used to optimize several DMOPs of the generalized dynamic benchmark generator (GDBG). The proposed algorithm was based on a multipopulation, cloudbased variant of the differential evolution algorithm called Cloudde. It is clear to see that more realworld problems are modelled as manyobjective optimization problems. Therefore, it is important to improve the capability of EAs to effectively solve this problem. The next section details the proposed approach to preserve the quality of selected solutions and pull selected solutions as close as possible to the Pareto front.
3. Proposed Methodology
In this section, we will discuss the elite archive approach that we apply to improve the capability of dyNSGAIII to handle dynamic search spaces involving 3 or more objectives. In dyNSGAIII, we used adaptive principal component analysis (PCA) mutation with npoint crossover to improve the ability of NSGAIII to track the PF in dynamic environments [3]. Adaptive PCA selects features with the highest covariance in highdimensional environments, while npoint crossover increases the degree of randomness of selection of offspring of successive generations. Therefore, this results in a sustained balance of convergence and diversity in a search space involving up to 3 objectives. For up to 2 objectives with few parameter and constraint variations, the selection strategy by dyNSGAIII is maintained. However, for 3 or more objectives, a selection strategy similar to the elite NSGAIII approach is used. However, in our proposed algorithm (Algorithm 1), we increase the number of reference points and elite solutions to compensate for the effects of both changing space and time as the number of objectives increases. In particular, we consider two situations: when there is a small change in the problem with respect to time and when there is a large change. In both situations, we would examine the effect of introducing varying proportions of mutated solutions into the OF space on the ability of the DMOEA to arrive at the best possible solution to the DMOP. It is important to consider these two scenarios with respect to time because while frequent changes in small magnitude can result in not enough time to find a suitable solution or set of solutions to the problem, less frequent changes in comparatively larger magnitude can result in the best solutions being traded for less fit candidates. Therefore, it is important to ensure the best possible tradeoff between both cases.

For Algorithm 1, we use the maximum Euclidean distance with respect to initial population and the final covariance matrix to evaluate the fitness of the elite population representing the nondominated PF. The covariance matrix is obtained according towhere is matrix of points within ndimensional Euclidean space and is mean value.
The solution space from which the set of elites is selected is centered with respect to selected reference points using PCA according towhere is the matrix of eigenvectors.
This approach maintains the elite population effectively for problems that have 2 objectives. However, for more than 3 objectives, we modify the number of reference points by , where is the number of objectives. To determine the rate of change in the given problem solution space, we consider the Minkowski distance of solution from its associated reference point at time and , respectively. If the change in is large, then rate of change is large for solution . Otherwise, the rate of change is considered to be small. For a small rate of change in the presence of 3 or more objectives, we infer that more competition is required to obtain the fittest candidates. Therefore, we introduce a larger proportion of mutated solutions according to the number of reference points. For instance, if a small change is detected, we introduce mutated solutions for every reference point. Otherwise, if the change is large, we increase the number of reference points by to guide the population to Pareto optimality. The assumption here is that for large changes in the solution space, premature convergence is unlikely.
When the objectives are more than 2, the number of reference points and elite solutions is increased by a factor of , where is the number of objectives. We recognize that parent selection can result in loss of selection pressure and diversity among the solutions. However, the archived approach has demonstrated improved performance [21, 22].
From steps 5 and 6 of Algorithm 1, we attempt to cater for changes in both space and time as the number of objectives and dimensions increases. This is because NSGAIII typically loses selection pressure as the number of objectives increases. Steps 5 and 6 of the algorithm are the main contribution of this research to improve the performance of earlier proposed dyNSGAIII algorithm. We believe that this modified selection and orientation of reference points in the objective function space would push fit and relevant solutions towards the Pareto optimal front. The Minkowski distance is used as the metric to ensure the proper spread of candidate solutions. The Minkowski distance between two points x and y is obtained as follows:where is the order of the norm.
The Minkowski distance metric ensures that the elite population P_{e} contains not only the candidates closest to the ideal point but also those close to these ideal candidates. It is a generalized form of the Euclidean distance and Manhattan distance metrics. This would ensure that the archive not only contains the best candidates, but also a number of “second” best candidates. Thus, the fitness of the archive is maintained. The purpose of such a strategy is to ensure that solutions that may have been discarded are preserved until later generations till the stopping criterion (max_gen) is reached. The suitability and durability period of candidate solutions are described according to the following equations.
Equations (4) and (5) are additional indices, which ensure that the elite solution archive remains fit with the increasing number of objectives under conditions of changing space and time.
Table 1 details the simulation parameters and settings for the dynamic environment. Table 2 gives details of the test cases used for determining the capability of EdyNSGAIII to obtain robust solutions over time.
From Table 1, parameter settings are selected to be the same as those used for the original dyNSGAIII. We want to ensure that the simulation environment is maintained to get the best performance from the EA.
3.1. EdyNSGAIII Optimization of RPOOT Problems
In this section, we discuss the performance of our proposed algorithm on selected RPOOT problems. The selected problems are multiobjective with complicated PF.
From Table 2, test instances F_{7}, F_{8}, F_{10,} and F_{11} are robust Pareto optimality over time (RPOOT) test problems from [4]. They all consist of 3 timedependent objective functions with dynamic Pareto set and Pareto front, respectively. Complexities of selected test problems are meant to test the ability of DMOEA to handle changing characteristics of the problems in the presence of changing time.
According to [16], F_{7} is a type I DMOP with nonconvex Pareto optimal front (POF), and F_{8} is a type II DMOP with nonconvex POF. Both DMOPs represent a continuous search space for the DMOEA with the rate of change in these problems being either gradual or sudden with time. From [23], F_{10} and F_{11} are described as unconstrained dynamic functions (UDFs). F_{10} has a simple trigonometric Pareto set, which is horizontal shifting with time. Its Pareto front is continuous with curvature change from convex to concave with angular shift with time. The search space has a dimension of , where is the number of dimensions of the search space. F_{11} is characterized by a trigonometric Pareto set with no associated dynamicity. The Pareto front is 3dimensional and concave with timeshifting center and radius of the concave front. The search space has dimension of . The Pareto front is discontinuous over the entire search space.
Indices for measuring the performance of EdyNSGAIII include the duration of candidate survival (t_{r}), IGD (d_{r}), and spacing between nondominated solutions (s_{r}). t_{r} is used to estimate how long fit solutions can remain relevant after successive generations. Therefore, the greater the survival time, the more likely ideal solutions are to be part of the final Pareto front. d_{r} indicates the spread of potential solutions of the given dynamic multiobjective optimization problem (DMOP). The smaller the magnitude of d_{r}, the better the spread of solutions in the objective function space. s_{r} measures the spacing between the final nondominated set of solutions of the DMOP. This will ensure that the Pareto front is covered as much as possible by the final solution set. Mathematically, the indices are expressed as follows:where is the period for which selected candidates remain suitable solutions for the DMOP and is the period for which the ith Pareto solution is feasible.where is the number of Pareto solutions and is the inverted generational distance of the kth Pareto solution.where is the least Euclidean distance between survival indices of the kth mutated solution in nondominated front and the Pareto front .
3.2. Results on EdyNSGAIII Optimization of RPOOT Problems
From research done in [3], the original dyNSGAIII performed well on RPOOT problems. However, it struggled to find suitable solutions when the Pareto front was complicated, and the number of objectives was 3 or more. Therefore, in this study, we compare the performance of EdyNSGAIII with dyNSGAIII for dynamic multiobjective optimization problems with 3 objectives and changing Pareto set and Pareto front.
From Table 3, it can be seen that the improved EdyNSGAIII outperformed dyNSGAIII for the selected DMOPs with complicated and discontinuous Pareto front and Pareto set. The selected DMOPs also take into consideration the effect of changing time. However, we observe from t_{r} and d_{r} that EdyNSGAIII is able to maintain fit solutions with the archive approach throughout successive generations of the search. For each algorithm, simulations were run 20 times to ensure proper convergence. s_{r} also indicates that candidate solutions are well spread across the Pareto front in the case of EdyNSGAIII compared with dyNSGAIII.
To further compare the performance of dyNSGAIII and EdyNSGAIII, we compare the percentage of runs in which each algorithm was able to associate specified reference points with at least one population member before the max_gen stopping criterion is attained. The idea behind this is to determine how effectively both algorithms use reference points to guide the population towards Pareto optimality. For M = 3, we set max_gen = 500. From the results obtained in Table 4, it can be seen that EdyNSGAIII is capable of associating a higher percentage of reference points with population candidates when the stopping criterion is satisfied. Figure 1 shows the ideal concave Pareto front for the F_{7} 3objective type I DMOP.
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From Figure 1, it can be seen that EdyNSGAIII has a better approximation of the PF for F_{7} as seen in Figure 1(b). The candidate solutions provide better coverage of the PF compared with dyNSGAIII (Figure 1(c)). The computational complexity of EdyNSGAIII is , which is approximately the same as that of dyNSGAIII. M is the dimensionality of the objective function, and N is the population size.
From the results obtained, we observe that the proposed archive approach for handling DMOPs with more than 2 objectives is effective in handling up to three objectives for problems involving timechanging Pareto set and Pareto front. These attributes characterize many reallife problems, which are generally dynamic in nature.
3.3. Comparative Performance of EdyNSGAIII on Selected ManyObjective DTLZ5 (x, y) Problems
In this section, we will compare the performance of EdyNSGAIII with two other DMOEAs on their ability to eliminate redundant objectives in a case where the number of objectives is large. We will consider the DTLZ5 (x, y) problem set [24], where x represents the number of nondominated objectives and y is the total number of objectives representing the DMOP. This DMOP tests the ability of EdyNSGAIII to adapt to complex search environments, which characterize problems with many objectives. In particular, we test the ability of the algorithm to select nondominated objectives out of the total number of objectives, thereby eliminating redundant objectives. This results in a simplification of the search space while simultaneously preserving the solutions required to solve the DMOP.
The DTLZ5 (x, y) DMOP being considered include DTLZ5 (3,10) and DTLZ5 (5,10). We compare the performance of EdyNSGAIII with two other DMOEAs: ParetoPCANSGAII [24] and Cloudde [20]. These DMOEAs were selected based on their promising performance with dimensionality reduction. In particular, EdyNSGAIII and ParetoPCANSGAII use PCA for dimensionality reduction.
From the results obtained in Table 5, ParetoPCANSGAII and EdyNSGAIII are the best performers for the lowest normalized Minkowski distance of selected solutions from the ideal PF (as indicated by the metrics highlighted in dark gray).
ParetoPCANSGAII has the lowest normalized Minkowski distance for DTLZ5 (3,10) after 100 iterations (0.3187) for objective F8, while EdyNSGAIII has the lowest overall Minkowski distance for objective F3 after 100 iterations. This is important because it is a test of the algorithm’s ability to push solutions as close as possible to the ideal PF despite the challenge of losing selection pressure in the presence of many objectives. Overall, EdyNSGAIII achieved the lowest overall Minkowski distance for 5 objectives, while ParetoPCANSGAII achieved the lowest Minkowski distance for 4 objectives across all instances of DTLZ5 (x, y), for all iterations. It is important to note the role of PCA in dimensionality reduction for EdyNSGAIII and ParetoPCANSGAII, which is likely the reason for their superior performance.
We also examine the ability of the algorithms to find nondominated objectives as an indication of their capability to perform effective dimensionality reduction. Here, we consider specific instances of the DTLZ5 (x, y) test suite, across all iterations. From the results in Tables 5 and 6, we observe that EdyNSGAIII achieves dimensionality reduction for 7 of 14 of the iterations for DTLZ5 (3, 10) and (5, 10) DMOPs (indicated by results highlighted in lighter shade of gray). The other 7 are distributed between Cloudde and ParetoPCANSGAII. This makes EdyNSGAIII the best performer in terms of dimensionality reduction, which indicates that it is a promising candidate for tackling the problem of the curse of dimensionality associated with many DMOEAs as they try to optimize problems with many objectives. We observe that for all 3 DMOEAs, their search capability generally improves as the number of iterations increases. This is demonstrated by the decreasing value of the normalized Minkowski distance as the number of iterations increases between 100 and 1000 for both instances of DTLZ5 (3, 10) and (5, 10) considered.
3.4. Performance of EdyNSGAIII on MultiObjective Hybrid Power Generation Scheduling Model
In Section 3.1, we examined the performance of EdyNSGAIII on the mathematical model of several timevarying multiobjective problems. Preliminary results show that EdyNSGAIII performed better than an earlier proposed dyNSGAIII algorithm. We will now test the capability of our proposed algorithm to handle a multiobjective, realworld power generation scheduling model consisting of multiple energy sources.
The original problem was formulated as a hydrothermal power generation scheduling problem in [25]. The objective was to generate enough electricity to satisfy demand while minimizing generation cost and environmental emissions. This would also involve an optimal balance between the number of hydroelectric and thermal generating units required to satisfy these objectives and related constraints. The problem formulation is dynamic in nature due to the fact that the demand for power changes with time. Therefore, the DMOEA is expected to track new optimal solutions whenever problem parameters change. The dynamic problem formulation is detailed in (2) of [25] with a transmission loss term specified in (9). Parameters for the hydrothermal system are included in the appendix section of [25].
The dynamic nature of the problem formulation is simulated by considering an overall time window of 96 hours (4 days). We assume here that the DMOP and its associated parameters and constraints continually vary over time windows of varying durations as follows: 5 minutes, 10 minutes, 15 minutes, and 30 minutes. Therefore, we consider the adaptability of EdyNSGAIII to constant changes in the DMOP over 1,152, 576, 384, and 192 different time slots, respectively.
In this study, we analyze the performance of EdyNSGAIII using the hypervolume Sharpe ratio (HSR) indicator [26]. The HSR indicator is an instance of the Sharpe ratio indicator in which the expected return and return covariance matrix are expressed with respect to the hypervolume (HV) indicator. The Sharpe ratio balances risk by ensuring that a given EA selects an optimal PF based on an appropriate rewardtovolatility ratio. Like all quality indicators, used in the selection or performance assessment of EAs, the HSR indicator is weakly monotonic. For a nonempty set of assets , the global solution , which satisfies the rewardtovolatility ratio, is obtained according towhere is a vector representing expected returns of , is the covariance matrix of asset returns, is the investment vector such that , and is the return of baseline riskless asset.
Desirable properties of the HSR indicator compared with the conventional HV indicator include monotonicity, nondegeneracy, and optimal investment and placement of reference points [26]. Consequently, the HSR indicator requires less parameter settings than the HV indicator. Also, since it assigns zero investment to dominated solutions and strictly positive investment to nondominated solutions, the possibility of a strictly nondominated front is more likely. Further details on the formulation and performance of the HSR indicator can be found in [26].
3.5. Results on Performance of EdyNSGAIII on MultiObjective Hybrid Power Generation Scheduling Model
In this section, we will analyze the performance of the proposed algorithm on the optimization of the dynamic, hybrid power generation scheduling model introduced in Section 3.4. The purpose of this analysis is twofold. First, we want to observe how EdyNSGAIII selects nondominated solutions using the reference pointbased HSR indicator. The dynamic nature of the DMOP will enable us to examine how the PF changes with corresponding changes in the time window of the DMOP. Second, we want to observe the capability of the proposed DMOEA to be used in realtime decisionmaking based on how quickly it responds to changes in the problem objectives and constraints. This is important because EAs generally perform better on static problems compared with timevarying problems. However, many realworld problems are dynamic, and the robustness of DMOEAs is greatly dependent on their capability to track timevarying PFs.
Figures 2 and 3 show surface plots for normalized values representing the cost and emission objectives of the DMOP. We compare the performance of EdyNSGAIII with dyNSGAIII for 4 instances of feature evaluations (FEs): 500, 1,000, 5,000, and 10,000. The aim is to observe the capability of the DMOEAs to settle on the least expensive compromise between the cost and emission objectives for the generation model. From the figures, normalized values between 0 and 0.2 represent the best compromise between the two objectives. From the surface plots, we observe that dyNSGAIII exhibits higher peaks in the OF space compared with EdyNSGA, particularly as the number of feature evaluations increases. This means that dyNSGAIII is likely to find it difficult to settle on the least expensive compromise between the two objectives. We also observe that EdyNSGAIII concentrates the least expensive solutions in the center of the OF space, which is not the case with dyNSGAIII. The EdyNSGAIII algorithm has a relaxed landscape at the center of the OF space compared with dyNSGAIII. The latter is characterized by peaks, which shows that dyNSGAIII finds it
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challenging to settle on the least expensive set of solutions particularly when the number of feature evaluations is high. For 500 feature evaluations, dyNSGAIII is able to find a set of suitable solutions near the center of the OF space. However, as the number of feature evaluations increases, it becomes difficult for the algorithm to converge on a suitable set of reference points. It is important for DMOEAs to have the capability to track changes in the dynamic OF space since this is the basis for the realworld application of these algorithms. In this study, we consider two objectives with timevarying parameters and constraints. However, many realworld problems have more than two objectives. Therefore, satisfying the first aim of this analysis as detailed at the beginning of this section is vital.
In Figure 4, we analyze the ability of EdyNSGAIII to be used in realtime decisionmaking by finding a feasible compromise between the cost and emission objectives. In Figures 4(a) and 4(b), we see the contour plots of EdyNSGAIII for 500 and 10,000 feature evaluations, respectively. For both FE settings, we see that EdyNSGAIII is able to settle on cheap solutions as seen by the concentration in the center of the plot. It can be seen that
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more expensive solutions are concentrated at the edges of the plot. This is seen less in the plots of dyNSGAIII, particularly in the case of 10,000 FEs (Figure 4(d)). In this plot, there are almost no cheap solutions, which effectively balance cost and emission objectives of the DMOP. This difficulty to arrive at suitable solutions in the face of rapidly changing objectives, parameters, and constraints is one key reason why DMOEAs are limited in their application to optimization of realworld problems.
EdyNSGAIII attempts to remedy this problem by the use of a niche of mutated solutions introduced into the candidate space. The effect of this approach is further investigated by comparing the performance of EdyNSGAIII with dyNSGAIII using the HSR performance indicator. From Table 5, we see that EdyNSGAIII is the best performer for 4 settings of the feature evaluations. We observe that the inclusion of mutated solutions causes more efficient placement of reference points, and consequently, a nondominated PF is selected based on an appropriate rewardtovolatility ratio.
4. Conclusions
This study has proposed an archive approach using mutated solutions for assigning reference points based on an earlier proposed method for improving the capability of NSGAII and NSGAIII to handle MOPs. We extend this approach to dynamic multiobjective optimization problems by improving the capability of earlier proposed dyNSGAIII to keep track of fit solutions in changing space and time.
When the number of objectives describing a DMOP is 3 or more, the dimensionality of the objective function space becomes complicated. This poses a challenge to the DMOEA as it struggles to maintain suitable candidate solutions throughout the search period. In particular, NSGAIII experiences a loss of selection pressure when the number of objectives and consequently the number of dimensions increase. We also observed this behavior with dyNSGAIII.
Therefore, in this study, we use an archive to preserve fit solutions when the number of objectives is 3 or more. This approach of using an archive of elite solutions in cases where the number of objectives is 3 or more is the attribute that distinguishes EdyNSGAIII from the earlier proposed dyNSGAIII. To determine the rate of change in the given problem solution space, we consider the Minkowski distance (x, y) of solution a from its associated reference point at time and , respectively. If the change in (x, y) is large, then rate of change is large for solution a. Otherwise, the rate of change is considered to be small. For a small rate of change in the presence of 3 or more objectives, we infer that more competition is required to obtain the fittest candidates. Therefore, we introduce a larger proportion of mutated solutions according to the number of reference points. For instance, if a small change is detected, we introduce mutated solutions for every reference point. Otherwise, if the change is large, we increase the number of reference points by to guide the population to Pareto optimality. This strategy is what differentiates EdyNSGAIII from dyNSGAIII. We tested the performance of the improved EdyNSGAIII on four DMOPs with a complicated Pareto front. Preliminary results show that the proposed approach outperforms the original dyNSGAIII. Also, EdyNSGAIII was able to associate a larger percentage of final candidate solutions with at least one reference point compared with dyNSGAIII, which indicates that the search strategy uses reference points to guide the solution space towards Pareto optimality. We also compared the performance of dyNSGAIII and EdyNSGAIII on a realworld power generation scheduling model. We recognize the need to apply our DMOEA to optimize realworld problems. These algorithms are limited in practical applicability due to their inability to adapt to realtime changes in problem objectives, parameters, and constraints. Consequently, these algorithms get trapped in dynamic environments, which makes their performance unreliable. From the results obtained, the performance of EdyNSGAIII is promising. However, we observe that for a large number of feature evaluations (5,000–10,000), there is a tendency for our proposed algorithm to get trapped in local optima. The realworld problem considered here has two objectives. Therefore, we ask ourselves the question: what happens when the number of objectives is more than two in the presence of many FEs?
Future work would focus on testing the performance of EdyNSGAIII on DMOPs with more than 3 objectives with many dynamic feature evaluations. We observe that the proposed DMOEA optimizes multimodal, manyobjective RPOOT problems satisfactorily. However, adapting it to perform satisfactorily on dynamic, realworld problems is still a work in progress. Particularly, when the number of objectives is 10 or greater, the number of reference points increases significantly, which can result in an increase in the processing time of the proposed algorithm. We are currently working to find a tradeoff between accuracy and speed of the algorithm. Section 3.3 discusses the performance of EdyNSGAIII on dimensionality reduction DMOPs, which is a test of its ability to reduce the number of redundant objectives in manyobjective scenarios. However, more research is required to be able to train the algorithm to effectively handle such complex search environments. Satisfactory performance would demonstrate that EdyNSGAIII can be used to effectively optimize a variety of realworld problems.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported partially by South African National Research Foundation Grants (Nos. 141951, 132159, 137951, and 132797).