# k-Pareto Optimality-Based Sorting with Maximization of Choice and Its Application to Genetic Optimization

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## Abstract

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## 1. Introduction

- Based on the proposed ranking metric Pareto-optimality, we study three genetic algorithms in detail: PO-count, PO-prob, and PO-prob*; see Section 4 for the detailed description. All three algorithms are based on NSGA-II and differ from the latter in the selection procedure by:
- PO-count: counting PO, which consists of counting the number of dominating solutions (counting PO);
- PO-prob: approximating PO via a probabilistic procedure (probabilistic PO);
- PO-prob*: sequentially combining probabilistic calculation of PO and Pareto dominance sorting from NSGA-II.

- We compare the proposed methods with NSGA-II and NSGA-III using the $0/1$ knapsack problem with the number of objectives varying from 2 to 25. Our experimental results with random and tournament selection show the following:
- Ranking by counting PO provides almost identical results as ranking by Pareto dominance;
- Using probabilistic PO ranking allows us to increase the hypervolume of the resulting solutions for many-objective and most multi-objective optimization problems;
- With the increase in the number of the objectives, probabilistic PO yields a set of solutions that are almost never dominated by the solutions of other algorithms;
- In general, PO-prob and NSGA-III algorithms yield fewer extreme solutions when the number of objectives increases;
- We demonstrate that probabilistic PO ranking is computationally much more efficient. It allows for reducing the time complexity of the sorting procedure from $O\left({N}^{2}M\right)$ to $O\left(NMlog\right(N\left)\right)$, where N is the population size and M is the number of objectives.

- Our algorithms are implemented as an extension of the Python library for evolutionary computation DEAP (https://deap.readthedocs.io/en/master/, accessed on 28 January 2022). They are available as open source (https://github.com/marharyta-aleksandrova/kPO, accessed on 28 January 2022).
- We also discuss how Pareto optimality can contribute to the problem of interpretability, see Section 6.

## 2. Related Work

#### 2.1. Searchability Deterioration of Pareto Dominance-Based Sorting

**points**to guide the selection process. These can be knee-points [9,15] and reference points either provided by a user or identified automatically, for example, the reference points in NSGA-III [16]. Apart from directing the selection process towards the true Pareto-frontier, these reference points also contribute to the diversity of the solution set as they are often widely distributed in the search space.

**indicators**to select the best solutions. Among recent works in this area, we can mention [10,17], which incorporate several ranking methods simultaneously, and [18] which used a novel convergence indicator. Computing the values of these indicators increases the execution time of the optimization process.

**reducing the dimensionality**. This can be achieved, for example, by using a transfer matrix [19] or by considering single-objective optimization problems via scalarization [12]. Such methods simplify complex relationships between objectives. This can lead to under-coverage of the search space.

**ensemble**several methods. In the approach from [20], one population is evolved using conventional non-dominated sorting, and another is evolved using an approximate non-dominated sorting procedure. This combination is supposed to improve both diversity and convergence at the same time.

**problem redefinition)**. It was implemented within the the framework of NSGA-III and is based on the idea of the niche-preservation operation of this algorithm.

**re-define**the way the

**dominance**is calculated. According to [11], we can define two main subgroups here: value-based dominance and number-based dominance. Value-based dominance methods modify the Pareto dominance by changing the objective values of the solutions when comparing them, for example, $\u03f5$-dominance [22]. Number-based methods try to compare a solution to another one by counting the number of objectives where it is better than, the same as, or worse than another; $(1-k)$-dominance [23] and L-optimality [24] are prominent examples. Alternative methods also include the fuzzification of Pareto-dominance, see [25,26], and the definition of dominance within a subgroup of a population, for example, $\theta $-dominance, which was used to improve the convergence of NSGA-III in [27].

#### 2.2. Computational Complexity

**algorithmic improvements**. For example, efficient non-dominated sorting (ENS) [8] reduces the time complexity by comparing solutions only with those that have already been assigned to a front. Similarly, authors of [28] proposed assigning solutions to the fronts in the ascending order of the sum of objectives. There are also approaches that are based on usage of

**alternative data structures**, such as trees [29]. The last group of methods performs

**approximate**non-dominated sorting. The first algorithm from this group, ANS, was proposed in 2016 [30]. In this algorithm, the dominance relationship between two solutions is determined by a maximum of three objective comparisons on top of a population sorted according to one of the objectives. It was shown by the authors of [30] and further confirmed in other research works [31] that this approach is not only more computationally efficient, but it also leads to better search performance. This idea was further developed in [32], where no more than two objectives were compared.

## 3. $\mathit{k}$-Pareto Optimality-Based Sorting for Genetic Optimization

#### 3.1. Genetic Optimization Overview

**Definition 1.**

- First, we create an initial population of solutions (or individuals) of a predefined size, $pop\_size$, and evaluate the fitness of every individual with respect to a predefined fitness function, step 1. In the case of multi- or many-objective optimization, this is a function with multi-dimensional output.
- Second, we select a subset of individuals for “procreation” based on a chosen parent selection criterion, step 2. Two popular criteria are random and tournament selection. In random selection, the parents are chosen randomly; in tournament selection, the choice is made based on fitness among a set of randomly selected individuals of a predefined size.
- Next, on the step 3, the chosen parents are “mated” to create a required number of children or offspring. This operation is also known as crossover. Often, the number of children is equal to the size of the original population. The generated offspring can also be mutated. The latter process is controlled by the value of the mutation probability, $mut\_prob$. After evaluating the fitness values for the newly created offspring, the two sets of solutions are joined.
- Finally, on the selection step, step 4, the combined population is sorted according to a specific criterion, and the best $pop\_size$ individuals are advanced to the next generation. The evolution continues until a certain criterion is fulfilled. Often, the process is controlled by the maximum number of generations, $n\_gen$.

#### 3.2. Formal Definition of k-Pareto Optimality

#### 3.3. Illustrative Example

## 4. Experimental Setup

## 5. Experimental Results

#### 5.1. Characteristics of the Proposed Approach

#### 5.1.1. Evolution of Solutions for ${n}_{k}=2$

**random selection**. Figure 4c,d show three generations of solutions for PO-count and PO-prob: $gen=30$, $gen=150$, and $gen=300$. As PO-prob* performs identically to PO-prob until $gen$ reaches 350, its intermediate solutions are not plotted. Figure 4a,b show the first front of the final generation $gen=500$ for all algorithms.

**tournament selection**, PO-prob behaves similarly to other algorithms and produces non-fragmented fronts, see Figure 5. At the same time, tournament selection results in worse coverage of the Pareto-frontier than random selection, compare Figure 4 and Figure 5. The maximum achieved values of objective 2 with tournament selection are around $0.92\times {10}^{4}$. From Figure 4d, we can see that the corresponding values for random selection were achieved for $gen=100$. At this point, the front is not yet fragmented. This might indicate that extreme and middle solutions are advanced further during different periods in the evolution process, and an adaptive selection procedure might be required for PO-prob. We aim to investigate this question further in our future work.

#### 5.1.2. Visualization of the First Front for ${n}_{k}>2$

#### 5.1.3. Fraction of Non-Dominated Solutions

#### 5.1.4. Characteristics: Main Findings

- For two knapsakcs, ${n}_{k}=2$:
- –
- PO-prob performs better in identifying extreme solutions. In the case of random selection, it also results in fragmented coverage of the Pareto-frontier. This tendency can, however, be repaired by using traditional non-dominated sorting during later generations, as in PO-prob*.
- –
- NSGA-II performs the best in covering the middle part of the Pareto-frontier. The behavior and performance of NSGA-III and PO-count are similar to that of NSGA-II.

- When $n\_k>2$, the tendency changes:
- –
- PO-prob does not result in fragmented sets of solutions; this is observed for ${n}_{k}\ge 3$ in our experiments.
- –
- PO-prob and NSGA-III result in fewer extreme solutions than other algorithms; this is observed for ${n}_{k}\ge 4$ in our experiments.

- Ranking based on probabilistic PO results in a very low number of incomparable solutions, which is not the case for PO-count and PD sorting. This demonstrates the ability of the probabilistic PO sorting to distinguish between the solutions and find the direction for further evolution.

#### 5.2. Performance

#### 5.2.1. Hypervolume

#### 5.2.2. Fraction of Dominated Solutions

**Results for ${n}_{k}=2$ with random selection.**Analyzing the results from Table 4, we can see that for two knapsacks and random selection, PO-prob is dominated the least number of times. NSGA-II, NSGA-III, and PO-count, on average, dominate no more than 13.52% of the solutions of PO-prob. PO-prob*, however, dominates on average 23.46% of the solutions of PO-prob. At the same time, PO-prob almost never dominates other algorithms. The only corresponding non-zero value is 8.37%, which represents the fraction of solutions of PO-prob* dominated by PO-prob. This means that solution spaces of PO-prob and other algorithms are distinct and do not intersect much. This supports our previous observation, see Figure 4.

**Results for ${n}_{k}=2$ with tournament selection.**For the same number of knapsacks, ${n}_{k}=2$, and tournament selection, we can see that the probability-based algorithms are dominated more often than the counting-based algorithms, see Table 5. The average number of dominated solutions through all algorithms, $\theta $, is 44.52% for PO-prob and 52.31% for PO-prob*. At the same time, the values of $\theta $ for NSGA-II, NSGA-III, and PO-count stay relatively close to the corresponding values for random selection.

**Results for ${n}_{k}=7$.**However, this pattern changes when the number of objectives increases. Already for ${n}_{k}=7$, NSGA-II and PO-count are substantially outperformed by other algorithms, both for random and tournament selection. More than 60% of solutions of these two algorithms are dominated by the probability-based algorithms PO-prob and PO-prob*. The level of domination in the inverse direction is less than 1%.

**Results for ${n}_{k}=25$.**When the number of knapsacks increases even further, we can notice that all algorithms tend to produce more distinct sets of solutions, as the domination fractions reduce. However, the general pattern stays the same. The solutions of NSGA-II and PO-count are more often dominated by the solutions of other algorithms. The solutions of PO-prob are almost never dominated. The next best performance is demonstrated by NSGA-III, with 9.59% and 3.60% of solutions dominated by PO-prob for random and tournament selection, respectively. PO-prob* has approximately 20% of solutions dominated by PO-prob and 10% of solutions dominated by NSGA-III.

**Distribution of $\theta $**. To further analyze how the domination fraction changes for different numbers of knapsacks, we demonstrate the distribution of $\theta $ for different values of ${n}_{k}$ in Figure 10. Recall that $\theta $ shows the fraction of dominated solutions averaged over different dominating algorithms, and its values are present in bold in Table 4 and Table 5.

#### 5.2.3. Performance: Main Findings

- The performance of PO-count is very close to that of NSGA-II, both in terms of hypervolume and the fraction of dominated solutions.
- Our results support the finding from [39]. We show that, contrary to expectations, NSGA-III results in lower values of hypervolume than NSGA-II for large numbers of objectives. However, we also show that for large values of ${n}_{k}$, the solutions produced by NSGA-III are rarely dominated by those produced by NSGA-II. At the same time, NSGA-III does dominate some fraction of solutions produced by NSGA-II. This shows that NSGA-III can be beneficial for many-objective optimization problems. This observation was not reported in [39].
- Finally, our experiments clearly demonstrate the advantages of probability-based algorithms for many-objective optimization problems. Both PO-prob and PO-prob* result in solutions that cover larger hypervolumes and are less often dominated by the solutions of other algorithms.

#### 5.3. Time Complexity

**Cumulative sorting duration.**Figure 11a shows cumulative sorting duration as a function of the number of generations. As the size of the population remains the same for every generation, cumulative sorting duration increases linearly for all algorithms except PO-prob*. This algorithm is a combination of PO-prob for generations 0–350 and NSGA-II afterwards. That is why for PO-prob* we observe a curve with two linear pieces with corresponding inclinations.

**Sorting duration as a function of $pop\_size$.**In Figure 11b, we report the dependency of sorting time on the population size for values of $pop\_size$ ranging from 50 to 500. The reported values are averages of 100 independent executions of one iteration of the corresponding genetic algorithm. We do not show the results for PO-prob* here, as it is an aggregation of two other algorithms.

## 6. Conclusions

**interpretation**. It represents the probability that a randomly chosen point will dominate the point under investigation, see Section 3. Thereby, the resulting algorithms are easier to manage and understand.

## 7. Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NSGA-II | Non-dominated Sorting Genetic Algorithm II [6] |

NSGA-III | Non-dominated Sorting Genetic Algorithm III [16] |

PD | Pareto dominance-based sorting, as used in NSGA-II |

PESA-II | Pareto Envelope-based Selection Algorithm II [5] |

(k-)PO | (k-)Pareto Optimality [14] |

PO-count | Pareto Optimality computed via counting |

PO-prob | Pareto Optimality computed using probabilistic approximation |

PO-prob* | PO-prob sequentially combined with Pareto dominance sorting |

SPEA2 | Strength Pareto Evolutionary Algorithm 2 [7] |

## Appendix A. 0/1 Multi-objective Knapsack Problem Formulation

## Appendix B. Visualization of the First front with Parallel Coordinates for n_k = 2

**Figure A1.**Profit of the knapsacks from the final front for ${n}_{k}=2$ with random selection. Different knapsacks are encoded by numbers on the x-axis. y-axis is shared.

**Figure A2.**Profit of the knapsacks from the final front for ${n}_{k}=2$ with tournament selection, $\ast {10}^{4}$. Different knapsacks are encoded by numbers on the x-axis. y-axis is shared.

## Appendix C. Average Distance to Diagonal

**Figure A3.**Average distance to diagonal for different number of knapsacks as a function number of generations. The legend and y-axis are shared among plots.

**random selection**. In Figure A3a, we can see that for two knapsacks, ${n}_{k}=2$, the curves for NSGA-II, NSGA-III, and PO-count are very close to each other, and the associated solutions are relatively close to the diagonal. On the other hand, for $gen\ge 200$, the solutions produced by PO-prob and PO-prob* are more than two times farther away from the diagonal. It means that these algorithms produce solutions that are much closer to extreme values and cover less the central part of the Pareto-frontier. This observation also agrees with the results presented in Section 5.1.1, see Figure 4b. As expected, after the number of generations reaches 350, the solutions of PO-prob* tend to be closer to the diagonal, as the NSGA-II selection procedure is used.

**tournament selection**, we observe a similar dependency between the average distance to the diagonal and the number of knapsacks. The average distance to the diagonal for PO-prob and PO-prob* decreases when $n\_k$ increases. This effect becomes visible faster than for random selection; compare Figure A3d,c. Moreover, we can notice that for all algorithms, tournament selection tends to produce solutions that are farther away from the diagonal for larger numbers of knapsacks. In the case of ${n}_{k}=2$, however, PO-prob with random selection produces more extreme solutions than with tournament selection, see Figure A3a,b. This observation also corresponds to the results presented in Figure 4b and Figure 5f of Section 5.1.1.

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**Figure 4.**Visualization of solutions for ${n}_{k}=2$ and random selection. y-axis is shared among plots. (

**a**) First front for $gen=500$, counting algorithms; (

**b**) First front for $gen=500$, probabilistic algorithms; (

**c**) Evolution of solutions for PO-count; (

**d**) Evolution of solutions for PO-prob.

**Figure 5.**Visualization of solutions for ${n}_{k}=2$ and tournament selection. y-axis is shared among plots. (

**a**) First front for $gen=500$, counting algorithms; (

**b**) First front for $gen=500$, probabilistic algorithms; (

**c**) Evolution of solutions for PO-count; (

**d**) Evolution of solutions for PO-prob.

**Figure 8.**Percentage of solutions in the first non-dominated front (sorting by Pareto dominance, PD) as a function of number of generations. The legend and y-axis are shared among plots.

**Figure 9.**Relative increase in hypervolume compared to NSGA-II as a function of the number of knapsacks ${n}_{k}$. The legend is shared among plots.

**Figure 10.**Average percentage of solutions dominated by other algorithms. The legend is shared among plots.

**Figure 11.**Time complexity for random selection and 10 knapsacks, ${n}_{k}=10$. (

**a**) Cumulative sorting duration as a function of number of generations; (

**b**) Sorting duration as a function of population size, $pop\_size$.

**Table 1.**Distribution of points between fronts for different sorting procedures: PD—Pareto dominance; PO-count—counting Pareto optimality; PO-prob—probabilistic Pareto optimality; PO-prob, $\u03f5$—probabilistic Pareto optimality with an adjustment for 0 probability.

Front | PD | PO-count | PO-prob | PO-prob, $\mathit{\u03f5}=0.1$ | |||
---|---|---|---|---|---|---|---|

Points | Val. | Points | Val. | Points | Val. | ||

1 | A, B, C | A, B, C | 0 | A, C | 0.00 | C | 0.05 |

2 | D, E | E | 1 | B | 0.08 | A | 0.07 |

3 | F | D, F | 2 | E | 0.11 | B | 0.08 |

4 | D, F | 0.28 | E | 0.11 | |||

5 | D, F | 0.28 |

Problem | $0/1$ knapsack, ${n}_{k}\in \{2-8,10,15,25\}$, 250 items |

Baseline | NSGA-II, NSGA-III |

Algorithms | PO-count, PO-prob, PO-prob* |

Implementation | python package DEAP |

Parent selection | random, binary tournament |

Crossover | uniform |

Parameters | $n\_gen=500$, $pop\_size=250$, $mut\_prob=0.01$ |

Runs | 30 |

${\mathit{n}}_{\mathit{k}}$ | NSGA-II, h-Vol. | NSGA-III | PO-Count | PO-Prob | PO-Prob* |
---|---|---|---|---|---|

Random selection | |||||

2 | $9.35\times {10}^{7}$ | –0.40 | –0.19 | 4.36 | 4.48 |

3 | $7.98\times {10}^{11}$ | –0.73 | –0.80 | 3.18 | 1.97 |

4 | $6.50\times {10}^{15}$ | –1.95 | –0.13 | 2.15 | 1.14 |

5 | $5.06\times {10}^{19}$ | –7.11 | –0.15 | –1.36 | –0.90 |

6 | $3.99\times {10}^{23}$ | –10.67 | 0.33 | –1.63 | –0.74 |

7 | $3.08\times {10}^{27}$ | –12.15 | –0.56 | –1.14 | 0.55 |

8 | $2.26\times {10}^{31}$ | –11.45 | 0.23 | 0.76 | 2.53 |

10 | $1.21\times {10}^{39}$ | –10.78 | –0.31 | 6.07 | 7.88 |

15 | $2.18\times {10}^{58}$ | –4.23 | –0.14 | 24.13 | 25.02 |

25 | $5.73\times {10}^{96}$ | –0.04 | 0.07 | 61.23 | 51.88 |

Tournament selection | |||||

2 | $9.01\times {10}^{07}$ | –0.09 | –0.33 | 1.14 | 0.67 |

3 | $7.55\times {10}^{11}$ | 0.12 | –0.29 | 3.11 | 2.66 |

4 | $6.12\times {10}^{15}$ | –0.79 | –0.33 | 1.16 | 0.48 |

5 | $4.73\times {10}^{19}$ | –3.49 | 0.22 | –1.42 | –0.77 |

6 | $3.74\times {10}^{23}$ | –7.18 | 0.24 | –2.13 | –0.54 |

7 | $2.88\times {10}^{27}$ | –9.07 | 0.39 | –2.02 | –0.31 |

8 | $2.16\times {10}^{31}$ | –10.54 | –0.51 | –2.01 | 0.25 |

10 | $1.17\times {10}^{39}$ | –10.02 | 0.20 | 0.90 | 4.35 |

15 | $2.21\times {10}^{58}$ | –7.90 | –0.76 | 11.67 | 14.14 |

25 | $6.49\times {10}^{96}$ | -11.12 | –1.86 | 30.35 | 28.59 |

Dominated algorithm, ${n}_{k}=2$ | |||||

NSGA-II | NSGA-III | PO-count | PO-prob | PO-prob* | |

NSGA-II | 44.24 | 39.21 | 13.52 | 47.38 | |

NSGA-III | 37.50 | 34.51 | 13.52 | 44.76 | |

PO-count | 43.34 | 45.01 | 13.52 | 46.94 | |

PO-prob | 0.00 | 0.00 | 0.00 | 8.37 | |

PO-prob* | 6.24 | 6.64 | 5.02 | 23.46 | |

mean, $\theta $ | 21.77 | 23.97 | 19.60 | 16.01 | 36.86 |

Dominated algorithm, ${n}_{k}=7$ | |||||

NSGA-II | NSGA-III | PO-count | PO-prob | PO-prob* | |

NSGA-II | 0.00 | 8.57 | 0.00 | 0.04 | |

NSGA-III | 39.72 | 40.51 | 0.08 | 19.00 | |

PO-count | 9.99 | 0.00 | 0.00 | 0.05 | |

PO-prob | 66.73 | 27.51 | 66.72 | 48.65 | |

PO-prob* | 61.52 | 1.41 | 63.31 | 0.07 | |

mean, $\theta $ | 44.49 | 7.23 | 44.78 | 0.04 | 16.94 |

Dominated algorithm, ${n}_{k}=25$ | |||||

NSGA-II | NSGA-III | PO-count | PO-prob | PO-prob* | |

NSGA-II | 0.00 | 8.28 | 0.00 | 0.15 | |

NSGA-III | 18.97 | 18.83 | 0.00 | 8.48 | |

PO-count | 6.40 | 0.00 | 0.00 | 0.20 | |

PO-prob | 36.55 | 9.59 | 36.52 | 18.76 | |

PO-prob* | 30.13 | 0.80 | 31.89 | 0.00 | |

mean, $\theta $ | 23.01 | 2.60 | 23.88 | 0.00 | 6.90 |

Dominated algorithm, ${n}_{k}=2$ | |||||

NSGA-II | NSGA-III | PO-count | PO-prob | PO-prob* | |

NSGA-II | 43.93 | 46.67 | 47.81 | 61.20 | |

NSGA-III | 35.99 | 43.17 | 45.71 | 59.33 | |

PO-count | 35.83 | 36.70 | 42.32 | 57.43 | |

PO-prob | 6.61 | 7.75 | 7.53 | 31.29 | |

PO-prob* | 7.78 | 9.95 | 10.41 | 42.26 | |

mean, $\theta $ | 21.55 | 24.58 | 26.95 | 44.52 | 52.31 |

Dominated algorithm, ${n}_{k}=7$ | |||||

NSGA-II | NSGA-III | PO-count | PO-prob | PO-prob* | |

NSGA-II | 0.00 | 11.21 | 0.00 | 0.09 | |

NSGA-III | 42.92 | 44.00 | 0.53 | 15.69 | |

PO-count | 12.16 | 0.00 | 0.01 | 0.03 | |

PO-prob | 63.32 | 14.72 | 63.73 | 39.24 | |

PO-prob* | 62.91 | 1.71 | 61.52 | 0.11 | |

mean, $\theta $ | 45.33 | 4.11 | 45.12 | 0.16 | 13.76 |

Dominated algorithm, ${n}_{k}=25$ | |||||

NSGA-II | NSGA-III | PO-count | PO-prob | PO-prob* | |

NSGA-II | 0.00 | 6.69 | 0.03 | 0.32 | |

NSGA-III | 17.35 | 20.83 | 0.03 | 10.60 | |

PO-count | 5.57 | 0.00 | 0.01 | 0.12 | |

PO-prob | 29.77 | 3.60 | 30.92 | 20.40 | |

PO-prob* | 25.71 | 0.44 | 27.97 | 0.23 | |

mean, $\theta $ | 19.60 | 1.01 | 21.60 | 0.07 | 7.86 |

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**MDPI and ACS Style**

Ruppert, J.; Aleksandrova, M.; Engel, T.
*k*-Pareto Optimality-Based Sorting with Maximization of Choice and Its Application to Genetic Optimization. *Algorithms* **2022**, *15*, 420.
https://doi.org/10.3390/a15110420

**AMA Style**

Ruppert J, Aleksandrova M, Engel T.
*k*-Pareto Optimality-Based Sorting with Maximization of Choice and Its Application to Genetic Optimization. *Algorithms*. 2022; 15(11):420.
https://doi.org/10.3390/a15110420

**Chicago/Turabian Style**

Ruppert, Jean, Marharyta Aleksandrova, and Thomas Engel.
2022. "*k*-Pareto Optimality-Based Sorting with Maximization of Choice and Its Application to Genetic Optimization" *Algorithms* 15, no. 11: 420.
https://doi.org/10.3390/a15110420