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Multi-Objective Optimization: Techniques and Applications
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Multi-Objective Optimization: Techniques and Applications

A special issue of Applied Sciences (ISSN 2076-3417). This special issue belongs to the section "Computing and Artificial Intelligence".

Deadline for manuscript submissions: 31 May 2025 | Viewed by 8467

Special Issue Editors

Dr. Douglas Alexandre Gomes Vieira

E-Mail Website
Guest Editor
Graduate Program in Mathematical Modeling, Federal Center of Technological Education of Minas Gerais, Belo Horizonte 30421-169, Brazil
Interests: math
Dr. Lisboa Adriano Chaves

E-Mail Website
Guest Editor
Graduate Program in Mathematical Modeling, Federal Center of Technological Education of Minas Gerais, Belo Horizonte 30421-169, Brazil
Interests: multiobjective optimization; machine learning; digital twin

Special Issue Information

Dear Colleagues,

This Special Issue presents a broad array of methodologies and applications for multiobjective optimization and decision-making. This includes innovative algorithms such as deterministic, linear, convex, non-linear, stochastic, and combinatorial algorithms, among others. Real-world applications in fields like artificial intelligence, machine learning, supply chain optimization, logistics, risk analysis, resource allocation, deficit allocation, portfolio management, sustainability, and renewable energy, among others, are of interest.

Dr. Douglas Alexandre Gomes Vieira
Dr. Lisboa Adriano Chaves
Guest Editors

Manuscript Submission Information

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Keywords

  • multi-objective optimization
  • decision making
  • deterministic optimization
  • combinatorial optimization
  • linear optimization

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Published Papers (5 papers)

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Research

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21 pages, 2008 KiB  
Article
Addressing the Global Logistics Performance Index Rankings with Methodological Insights and an Innovative Decision Support Framework
by Željko Stević, Nazlı Ersoy, Enes Emre Başar and Mahmut Baydaş
Appl. Sci. 2024, 14(22), 10334; https://doi.org/10.3390/app142210334 - 10 Nov 2024
Viewed by 1756
Abstract
This study examines the Logistics Performance Index (LPI) rankings developed by the World Bank from a methodological perspective and proposes an alternative decision support framework. LPI serves as an interactive tool that helps countries identify challenges, innovative solutions, and opportunities in their trade [...] Read more.
This study examines the Logistics Performance Index (LPI) rankings developed by the World Bank from a methodological perspective and proposes an alternative decision support framework. LPI serves as an interactive tool that helps countries identify challenges, innovative solutions, and opportunities in their trade and logistics sectors. In this study, the efficiency of logistics operations in 118 countries was evaluated using an integrated multi-criteria decision-making (MCDM) model objectively weighted by the Entropy method. Countries were ranked using the MCRAT, SAW, TOPSIS, and FUCA methods. According to the findings, large datasets provide more robust insights for sensitivity analyses, and wider weighting coefficient combinations make the data more meaningful. In addition, it is suggested to use low-compensation methods instead of classical additive methods for LPI. Unlike other studies in literature, this research applied an innovative sensitivity analysis to test the robustness of the model and comprehensively examined the effects of weighting techniques based on over 2500 different MCDM results. The findings suggest that the FUCA method should be recommended to decision-makers for calculating LPI rankings due to its simplicity, practicality, low compensatory power, and low sensitivity. This study offers methodological improvements when evaluating logistics performance and provides significant contributions to decision-making processes. The findings are expected to provide a valuable resource for policymakers and businesses in understanding a country’s position in global competition, as well as serving as a reference for researchers evaluating the logistics performance of countries. Full article
(This article belongs to the Special Issue Multi-Objective Optimization: Techniques and Applications)
►▼ Show Figures

Figure 1

Figure 1
<p>Flowchart of the proposed methodology.</p> Full article ">Figure 2
<p>Importance of criteria weights between 2010 and 2023.</p> Full article ">Figure 3
<p>The six-year average of the rankings from the four MCDM methods and the LPI for different countries.</p> Full article ">Figure 4
<p>Mean and standard deviation values between the fixed factors and methods.</p> Full article ">Figure 5
<p>Mean standard deviations based on MCDM methods.</p> Full article ">Figure 6
<p>Mean standard deviations based on MCDM methods and LPI ranking.</p> Full article ">
16 pages, 3774 KiB  
Article
An Adaptive Multi-Objective Genetic Algorithm for Solving Heterogeneous Green City Vehicle Routing Problem
by Wanqiu Zhao, Xu Bian and Xuesong Mei
Appl. Sci. 2024, 14(15), 6594; https://doi.org/10.3390/app14156594 - 28 Jul 2024
Cited by 2 | Viewed by 1656
Abstract
Intelligent scheduling plays a crucial role in minimizing transportation expenses and enhancing overall efficiency. However, most of the existing scheduling models fail to comprehensively account for the requirements of urban development, as exemplified by the vehicle routing problem with time windows (VRPTW), which [...] Read more.
Intelligent scheduling plays a crucial role in minimizing transportation expenses and enhancing overall efficiency. However, most of the existing scheduling models fail to comprehensively account for the requirements of urban development, as exemplified by the vehicle routing problem with time windows (VRPTW), which merely specifies the minimization of path length. This paper introduces a new model of the heterogeneous green city vehicle routing problem with time windows (HGCVRPTW), addressing challenges in urban logistics. The HGCVRPTW model considers carriers with diverse attributes, recipients with varying tolerance for delays, and fluctuating road congestion levels impacting carbon emissions. To better deal with the HGCVRPTW, an adaptive multi-objective genetic algorithm based on the greedy initialization strategy (AMoGA-GIS) is proposed, which includes the following three advantages. Firstly, considering the impact of initial information on the search process, a greedy initialization strategy (GIS) is proposed to guide the overall evolution during the initialization phase. Secondly, the adaptive multiple mutation operators (AMMO) are introduced to improve the diversity of the population at different evolutionary stages according to their success rate of mutation. Moreover, we built a more tailored testing dataset that better aligns with the challenges faced by the HGCVRPTW. Our extensive experiments affirm the competitive performance of the AMoGA-GIS by comparing it with other state-of-the-art algorithms and prove that the GIS and AMMO play a pivotal role in advancing algorithmic capabilities tailored to the HGCVRPTW. Full article
(This article belongs to the Special Issue Multi-Objective Optimization: Techniques and Applications)
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Figure 1

Figure 1
<p>The diagram of AMoGA-GIS.</p> Full article ">Figure 2
<p>The representation of solution. The orange represents the tasks of the first vehicle V1, yellow represents the tasks of the second vehicle V2, and green represents the tasks of the third vehicle V3.</p> Full article ">Figure 3
<p>The crossover operation. The gene positions selected for crossover under probabilistic selection are highlighted with an orange background, where relevant genes from <b><span class="html-italic">x</span></b><sub>2</sub> are assigned to <b><span class="html-italic">x</span></b><sub>1</sub>. The positions marked in red font indicate the genes to be legalized and the missing genes in <b><span class="html-italic">x</span></b><sub>1</sub> are identified in <b><span class="html-italic">x</span></b><sub>2</sub> and assigned to <b><span class="html-italic">x</span></b><sub>1</sub>, resulting in the final legalized offspring <b><span class="html-italic">c</span></b><sub>1</sub>.</p> Full article ">Figure 4
<p>The mutation operations for allocation sequence. The orange represents the tasks of the first vehicle V1, yellow represents the tasks of the second vehicle V2, and green represents the tasks of the third vehicle V3. The arrows looping between two blocks indicate the exchange of the blocks, while an arrow pointing from one block to a gap signifies the insertion of the block into that gap.</p> Full article ">Figure 5
<p>The mutation operations for task allocation quantity. The orange represents the tasks of the first vehicle V1, yellow represents the tasks of the second vehicle V2, and green represents the tasks of the third vehicle V3. The white represents the allocation between three vehicles. The arrows looping between two blocks indicate the exchange of the blocks, while an arrow pointing from one block to a gap signifies the insertion of the block into that gap.</p> Full article ">Figure 6
<p>The distribution of tasks. The red dots represent warehouses, and the blue dots represent points that require planning and allocation.</p> Full article ">Figure 7
<p>The comparison results between NSGAIII and AMoGA-GIS.</p> Full article ">Figure 8
<p>The component analysis results of AMoGA-GIS.</p> Full article ">Figure 9
<p>The distribution of solutions to different problems.</p> Full article ">
20 pages, 7157 KiB  
Article
Multi-Objective Ship Route Optimisation Using Estimation of Distribution Algorithm
by Roman Dębski and Rafał Dreżewski
Appl. Sci. 2024, 14(13), 5919; https://doi.org/10.3390/app14135919 - 6 Jul 2024
Viewed by 1138
Abstract
The paper proposes an innovative adaptation of the estimation of distribution algorithm (EDA), intended for multi-objective optimisation of a ship’s route in a non-stationary environment (tidal waters). The key elements of the proposed approach—the adaptive Markov chain-based path generator and the dynamic programming-based [...] Read more.
The paper proposes an innovative adaptation of the estimation of distribution algorithm (EDA), intended for multi-objective optimisation of a ship’s route in a non-stationary environment (tidal waters). The key elements of the proposed approach—the adaptive Markov chain-based path generator and the dynamic programming-based local search algorithm—are presented in detail. The experimental results presented indicate the high effectiveness of the proposed algorithm in finding very good quality approximations of optimal solutions in the Pareto sense. Critical for this was the proposed local search algorithm, whose application improved the final result significantly (the Pareto set size increased from five up to nine times, and the Pareto front quality just about doubled). The proposed algorithm can also be applied to other domains (e.g., mobile robot path planning). It can be considered a framework for (simulation-based) multi-objective optimal path planning in non-stationary environments. Full article
(This article belongs to the Special Issue Multi-Objective Optimization: Techniques and Applications)
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Figure 1

Figure 1
<p>Conceptual diagram of the first ship route (<math display="inline"><semantics> <mover accent="true"> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>˜</mo> </mover> </semantics></math>) optimisation problem ([<span class="html-italic">Prob1</span>]) under consideration. (<b>left</b>) the tidal stream at time <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>A</mi> </msub> <mo>+</mo> <mn>3</mn> <mi>h</mi> </mrow> </semantics></math>; the ship—represented as the black triangle—sails <span class="html-italic">with</span> the current (as indicated by arrows). (<b>right</b>) the tidal stream at time <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>A</mi> </msub> <mo>+</mo> <mn>9</mn> <mi>h</mi> </mrow> </semantics></math>; the ship has to sail <span class="html-italic">against</span> the current. The colour bar shows absolute values, <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi mathvariant="bold">v</mi> <mi>t</mi> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>, of the tidal stream.</p> Full article ">Figure 2
<p>Solution space representation. (<b>left</b>) multi-stage graph <math display="inline"><semantics> <msub> <mi>G</mi> <mn>2</mn> </msub> </semantics></math>. (<b>right</b>) 3D-graph <math display="inline"><semantics> <msub> <mi>G</mi> <mn>3</mn> </msub> </semantics></math> (for the sake of clarity of the drawing, the coordinate axes have been moved).</p> Full article ">Figure 3
<p><math display="inline"><semantics> <msub> <mi>T</mi> <mi>d</mi> </msub> </semantics></math> function (see Equation (<a href="#FD6-applsci-14-05919" class="html-disp-formula">6</a>)) shown for s = 1, 10, 20, …; <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mi mathvariant="italic">MAX</mi> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The second optimisation problem under consideration ([<span class="html-italic">Prob2</span>])—it is presented as [<span class="html-italic">Prob1</span>] in <a href="#applsci-14-05919-f001" class="html-fig">Figure 1</a> but with the tidal stream given by Equation (<a href="#FD12-applsci-14-05919" class="html-disp-formula">A5</a>). (<b>left</b>) the tidal stream at time <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>A</mi> </msub> <mo>+</mo> <mn>3</mn> <mi>h</mi> </mrow> </semantics></math>. (<b>right</b>) the tidal stream at time <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>A</mi> </msub> <mo>+</mo> <mn>9</mn> <mi>h</mi> </mrow> </semantics></math>. The colour bar shows absolute values <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi mathvariant="bold">v</mi> <mi>t</mi> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> of the tidal stream.</p> Full article ">Figure 5
<p>[<span class="html-italic">Prob1</span>]: Pareto set (the paths in the design space) obtained with the proposed algorithm. (<b>top</b>) the colour of the path illustrates the relative ship velocity (STW, in m/s). (<b>bottom</b>) the colour of the path illustrates the absolute ship velocity (SOG).</p> Full article ">Figure 6
<p>[<span class="html-italic">Prob1</span>] Pareto set (the design space) in consecutive steps of the algorithm. (<b>top left</b>) step is 1, Pareto set size is 50. (<b>top right</b>) step is 30, Pareto set size is 267. (<b>bottom left</b>) step is 100, Pareto set size is 438. (<b>bottom right</b>) step is 200, Pareto set size is 512. The colour illustrates the relative speed (in m/s) of the ship.</p> Full article ">Figure 7
<p>[<span class="html-italic">Prob2</span>]: Pareto set (the design space) in consecutive steps of the algorithm. (<b>top left</b>) step is 1, Pareto set size is 59. (<b>top right</b>) step is 30, Pareto set size is 458. (<b>bottom left</b>) step is 100, Pareto set size is 772. (<b>bottom right</b>) step is 200, Pareto set size is 907. The colour illustrates the relative speed (in m/s) of the ship.</p> Full article ">Figure 8
<p>[<span class="html-italic">Prob1</span>]: Pareto front (the criterion space) in consecutive steps of the algorithm. The colour of the Pareto front illustrates the value of the adapted Hypervolume (HV) metric. (<b>top left</b>) step is 1, Pareto set size is 50, and HV is 0.0. (<b>top right</b>) step is 30, Pareto set size is 267, and HV is 0.61. (<b>bottom left</b>) step is 100, Pareto set size is 438, and HV is 0.82. (<b>bottom right</b>) step is 200, Pareto set size is 512, and HV is 1.0.</p> Full article ">Figure 9
<p>[<span class="html-italic">Prob2</span>]: Pareto front (the criterion space) in consecutive steps of the algorithm. The colour of Pareto front illustrates the value of adapted Hypervolume (HV) metric. (<b>top left</b>) step is 1, Pareto set size is 59, and HV is 0.0. (<b>top right</b>) step is 30, Pareto set size is 458, and HV is 0.75. (<b>bottom left</b>) step is 100, Pareto set size is 772, and HV is 0.96. (<b>bottom right</b>) step is 200, Pareto set size is 907, and HV is 1.0.</p> Full article ">Figure 10
<p>[<span class="html-italic">Prob1</span>]: Pareto front size and quality (measured by the adapted HV metric value) in consecutive steps of the algorithm. (<b>left</b>) Pareto front size in consecutive steps, and the colour illustrates the HV metric value. (<b>right</b>) HV metric value, and the colour illustrates the Pareto front size.</p> Full article ">Figure 11
<p>[<span class="html-italic">Prob2</span>]: Pareto front size and quality (measured by the adapted HV metric value) in consecutive steps of the algorithm. (<b>left</b>) Pareto front size in consecutive steps, and the colour illustrates the HV metric value. (<b>right</b>) HV metric value, and the colour illustrates the Pareto front size.</p> Full article ">Figure 12
<p>[<span class="html-italic">Prob1</span>]: Pareto set (the design space) in consecutive steps of the algorithm without the local search. (<b>top left</b>) step is 1, and Pareto set size is 14. (<b>top right</b>) step is 30, and Pareto set size is 30. (<b>bottom left</b>) step is 100, and Pareto set size is 54. (<b>bottom right</b>) step is 200, and Pareto set size is 76. The colour illustrates the relative speed (in m/s) of the ship.</p> Full article ">Figure 13
<p>[<span class="html-italic">Prob2</span>]: Pareto set (the design space) in consecutive steps of the algorithm without the local search. (<b>top left</b>) step is 1, and Pareto set size is 10. (<b>top right</b>) step is 30, and Pareto set size is 32. (<b>bottom left</b>) step is 100, and Pareto set size is 65. (<b>bottom right</b>) step is 200, and Pareto set size is 89. The colour illustrates the relative speed (in m/s) of the ship.</p> Full article ">Figure 14
<p>[<span class="html-italic">Prob1</span>]: Pareto front (the criterion space) in consecutive steps of the algorithm without the local search. The colour of Pareto front illustrates the value of the adapted Hypervolume (HV) metric. (<b>top left</b>) step is 1, Pareto set size is 14, and HV is 0.0. (<b>top right</b>) step is 30, Pareto set size is 30, and HV is 0.55. (<b>bottom left</b>) step is 100, Pareto set size is 54, and HV is 0.78. (<b>bottom right</b>) step is 200, Pareto set size is 76, and HV is 1.0.</p> Full article ">Figure 15
<p>[<span class="html-italic">Prob2</span>]: Pareto front (the criterion space) in consecutive steps of the algorithm without the local search. The colour of Pareto front illustrates the value of the adapted Hypervolume (HV) metric. (<b>top left</b>) step is 1, Pareto set size is 10, and HV is 0.0. (<b>top right</b>) step is 30, Pareto set size is 32, and HV is 0.35. (<b>bottom left</b>) step is 100, Pareto set size is 65, and HV is 0.75. (<b>bottom right</b>) step is 200, Pareto set size is 89, and HV is 1.0.</p> Full article ">Figure 16
<p>[<span class="html-italic">Prob1</span>]: Pareto front size and quality (measured by the adapted HV metric value) in consecutive steps of the algorithm without the local search. (<b>left</b>) Pareto front size in consecutive steps, and the colour illustrates the HV metric value. (<b>right</b>) HV metric value, and the colour illustrates the Pareto front size.</p> Full article ">Figure 17
<p>Same as <a href="#applsci-14-05919-f016" class="html-fig">Figure 16</a> but for [<span class="html-italic">Prob2</span>].</p> Full article ">
13 pages, 3835 KiB  
Article
An Improved Evolutionary Multi-Objective Clustering Algorithm Based on Autoencoder
by Mingxin Qiu, Yingyao Zhang, Shuai Lei and Miaosong Gu
Appl. Sci. 2024, 14(6), 2454; https://doi.org/10.3390/app14062454 - 14 Mar 2024
Cited by 1 | Viewed by 1088
Abstract
Evolutionary multi-objective clustering (EMOC) algorithms have gained popularity recently, as they can obtain a set of clustering solutions in a single run by optimizing multiple objectives. Particularly, in one type of EMOC algorithm, the number of clusters k is taken as one of [...] Read more.
Evolutionary multi-objective clustering (EMOC) algorithms have gained popularity recently, as they can obtain a set of clustering solutions in a single run by optimizing multiple objectives. Particularly, in one type of EMOC algorithm, the number of clusters k is taken as one of the multiple objectives to obtain a set of clustering solutions with different k. However, the numbers of clusters k and other objectives are not always in conflict, so it is impossible to obtain the clustering solutions with all different k in a single run. Therefore, evolutionary multi-objective k-clustering (EMO-KC) has recently been proposed to ensure this conflict. However, EMO-KC could not obtain good clustering accuracy on high-dimensional datasets. Moreover, EMO-KC’s validity is not ensured as one of its objectives (SSDexp, which is transformed from the sum of squared distances (SSD)) could not be effectively optimized and it could not avoid invalid solutions in its initialization. In this paper, an improved evolutionary multi-objective clustering algorithm based on autoencoder (AE-IEMOKC) is proposed to improve the accuracy and ensure the validity of EMO-KC. The proposed AE-IEMOKC is established by combining an autoencoder with an improved version of EMO-KC (IEMO-KC) for better accuracy, where IEMO-KC is improved based on EMO-KC by proposing a scaling factor to help effectively optimize the objective of SSDexp and introducing a valid initialization to avoid the invalid solutions. Experimental results on several datasets demonstrate the accuracy and validity of AE-IEMOKC. The results of this paper may provide some useful information for other EMOC algorithms to improve accuracy and convergence. Full article
(This article belongs to the Special Issue Multi-Objective Optimization: Techniques and Applications)
►▼ Show Figures

Figure 1

Figure 1
<p>Architecture of the proposed AE-IEMOKC.</p> Full article ">Figure 2
<p>Schematic diagram of the centroid-based chromosome encoding method based on valid initialization.</p> Full article ">Figure 3
<p>Clustering results obtained by GKA, MOKGA, and AE-IEMOKC on the Iris, Wine, and Seeds datasets when <span class="html-italic">k</span> = 3: (<b>a</b>) The clustering results obtained by GKA (<b>b</b>) The clustering results obtained by MOKGA. (<b>c</b>) The clustering results obtained by AE-IEMOKC.</p> Full article ">Figure 4
<p>Pareto fronts obtained by the four algorithms on the Wine dataset, where solutions containing the invalid clusters are removed due to their invalidity: (<b>a</b>) Comparison of SSD. (<b>b</b>) Comparison of <span class="html-italic">f</span><sub>1</sub>.</p> Full article ">Figure 5
<p>The ARI and ACC of the Pareto fronts obtained by IEMO-KC and AE-IEMOKC on the Iris dataset: (<b>a</b>) Comparison of ARI. (<b>b</b>) Comparison of ACC.</p> Full article ">Figure 6
<p>The clustering results obtained by IEMO-KC and AE-IEMOKC on the Iris dataset: (<b>a</b>) The clustering results obtained by IEMO-KC. (<b>b</b>) The clustering results obtained by AE-IEMOKC.</p> Full article ">Figure 7
<p>Averaged running time of EMO-KC, IEMO-KC1, IEMO-KC, and AE-IEMOKC on the five datasets.</p> Full article ">

Review

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25 pages, 1890 KiB  
Review
Multidisciplinary Optimization of Aircraft Aerodynamics for Distributed Propulsion Configurations
by Shaojun Luo, Tian Zi Eng, Zhili Tang, Qianrong Ma, Jinyou Su and Gabriel Bugeda
Appl. Sci. 2024, 14(17), 7781; https://doi.org/10.3390/app14177781 - 3 Sep 2024
Viewed by 1671
Abstract
The combination of different aerodynamic configurations and propulsion systems, namely, aero-propulsion, affects flight performance differently. These effects are closely related to multidisciplinary collaborative aspects (aerodynamic configuration, propulsion, energy, control systems, etc.) and determine the overall energy consumption of an aircraft. The potential benefits [...] Read more.
The combination of different aerodynamic configurations and propulsion systems, namely, aero-propulsion, affects flight performance differently. These effects are closely related to multidisciplinary collaborative aspects (aerodynamic configuration, propulsion, energy, control systems, etc.) and determine the overall energy consumption of an aircraft. The potential benefits of distributed propulsion (DP) involve propulsive efficiency, energy-saving, and emissions reduction. In particular, wake filling is maximized when the trailing edge of a blended wing body (BWB) is fully covered by propulsion systems that employ boundary layer ingestion (BLI). Nonetheless, the thrust–drag imbalance that frequently arises at the trailing edge, excessive energy consumption, and flow distortions during propulsion remain unsolved challenges. These after-effects imply the complexity of DP systems in multidisciplinary optimization (MDO). To coordinate the different functions of the aero-propulsive configuration, the application of MDO is essential for intellectualized modulate layout, thrust manipulation, and energy efficiency. This paper presents the research challenges of ultra-high-dimensional optimization objectives and design variables in the current literature in aerodynamic configuration integrated DP. The benefits and defects of various coupled conditions and feasible proposals have been listed. Contemporary advanced energy systems, propulsion control, and influential technologies that are energy-saving are discussed. Based on the proposed technical benchmarks and the algorithm of MDO, the propulsive configuration that might affect energy efficiency is summarized. Moreover, suggestions are drawn for forthcoming exploitation and studies. Full article
(This article belongs to the Special Issue Multi-Objective Optimization: Techniques and Applications)
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Figure 1

Figure 1
<p>Thrust–drag balance and load distribution in the aero-propulsive configuration of TWB (<b>left</b>) and BWB (<b>right</b>) (<math display="inline"> <semantics> <mrow> <mo>∞</mo> <mi>u</mi> </mrow> </semantics> </math> being the velocity at infinite, <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>u</mi> </mrow> </semantics> </math> the difference between the local velocity and the velocity at infinite, and <span class="html-italic">s</span> and <span class="html-italic">n</span> the longitudinal and traversal components of <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>u</mi> </mrow> </semantics> </math>, respectively).</p> Full article ">Figure 2
<p>Distributed propulsion systems in BWB.</p> Full article ">Figure 3
<p>Wake dissipation of kinetic energy with/without an extra aft-mounted engine.</p> Full article ">Figure 4
<p>Aero-propulsive interaction of BWB and DP.</p> Full article ">Figure 5
<p>Benefit of wake ingestion in various aero-propulsive configurations.</p> Full article ">Figure 6
<p>Conventional and reform (regional/distributed) propulsion configuration in TWB and BWB.</p> Full article ">Figure 7
<p>Control methods of airflow distortion in BLI S-shaped inlet diffusers.</p> Full article ">Figure 8
<p>A sort of parallel numerical implementation flowchart in optimization algorithms.</p> Full article ">
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