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Algorithms
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Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition)
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Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition)

A special issue of Algorithms (ISSN 1999-4893). This special issue belongs to the section "Combinatorial Optimization, Graph, and Network Algorithms".

Deadline for manuscript submissions: 31 March 2026 | Viewed by 4034

Special Issue Editor

Prof. Dr. Massimiliano Caramia

E-Mail Website
Guest Editor
Department of Enterprise Engineering, University of Rome "Tor Vergata", 00133 Roma, Italy
Interests: scheduling; graph theory; optimization; mathematical modeling; supply chain optimization; logistics; transportation; production systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Decision-making in real-world applications often require the consideration of more than one objective to find an effective solution. When (conflicting) objectives are associated with either a single decision-maker or cooperative decision-makers, this typically leads to multi-objective optimization. Here, optional solutions do not have the same image value, as happens in single-objective optimization, but are non-dominated, equivalent, and allow the definition of the Pareto front. When objectives are associated with different non-cooperative decision-makers, we fall into the game theory arena; furthermore, when objectives and/or decision-makers have a hierarchy among them, this asks to cope with nested optimization problems and, therefore, multi-level optimization.

All these problems are computationally difficult to solve, and their resolution typically involves reformulating the latter into several single-objective problems or one single-objective problem by introducing additional (non-linear) constraints. Moreover, to limit the computational burden, before their resolution, it is worthwhile to reduce the number of objectives to a very limited (significative) number by applying proper methodologies.

The aim of this Special Issue is to collect original manuscripts dealing with multi-objective and multi-level optimization; we sought original papers presenting innovative applications and/or contributing to the theory.

Prof. Dr. Massimiliano Caramia
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Algorithms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • multi-objective optimization
  • multi-level optimization
  • decision-making

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Related Special Issue

Published Papers (7 papers)

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Research

20 pages, 2080 KB  
Article
A Multi-Objective Evolutionary Computation Approach for Improving Neural Network-Based Surrogate Models in Structural Engineering
by Néstor López-González, Eduardo Rodríguez and David Greiner
Algorithms 2025, 18(12), 754; https://doi.org/10.3390/a18120754 (registering DOI) - 28 Nov 2025
Viewed by 151
Abstract
Surrogate models are widely used in science and engineering to approximate other methods that are usually computationally expensive. Here, artificial neural networks (ANNs) are employed as surrogate regression models to approximate the finite element method in the problem of structural analysis of steel [...] Read more.
Surrogate models are widely used in science and engineering to approximate other methods that are usually computationally expensive. Here, artificial neural networks (ANNs) are employed as surrogate regression models to approximate the finite element method in the problem of structural analysis of steel frames. The focus is on a multi-objective neural architecture search (NAS) that minimizes the training time and maximizes the surrogate accuracy. To this end, several configurations of the non-dominated sorting genetic algorithm (NSGA-II) are tested versus random search. The robustness of the methodology is demonstrated by the statistical significance of the hypervolume indicator. Non-dominated solutions (consisting of the set of best designs in terms of accuracy for each training time or in terms of training time for each accuracy) reveal the importance of multi-objective hyperparameter tuning in the performance of ANNs as regression surrogates. Non-evident optimal values were attained for the number of hidden layers, the number of nodes per layer, the batch size, and alpha parameter of the Leaky ReLU transfer function. These results are useful for comparing with state-of-the-art ANN regression surrogates recently attained in the recent structural engineering literature. This approach facilitates the selection of models that achieve the optimal balance between training speed and predictive accuracy, according to the specific requirements of the application. Full article
(This article belongs to the Special Issue Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition))
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24 pages, 4757 KB  
Article
MORA: A Multicriteria Optimal Resource Allocation and Decision Support Toolkit for Wildfire Management
by Theofanis Orphanoudakis, Christos Betzelos and Helen Catherine Leligou
Algorithms 2025, 18(11), 677; https://doi.org/10.3390/a18110677 - 23 Oct 2025
Viewed by 341
Abstract
Forest ecosystems are vital to sustainable development, contributing to economic, environmental and social well-being. However, the increasing frequency and severity of wildfires threaten these ecosystems, demanding more effective and integrated fire management (IFM) strategies. Current suppression efforts face limitations due to high resource [...] Read more.
Forest ecosystems are vital to sustainable development, contributing to economic, environmental and social well-being. However, the increasing frequency and severity of wildfires threaten these ecosystems, demanding more effective and integrated fire management (IFM) strategies. Current suppression efforts face limitations due to high resource demands and the need for timely, informed decision-making under uncertain conditions. This paper presents the SILVANUS project’s approach to developing an advanced Decision Support System (DSS) designed to assist incident commanders in optimizing resource allocation during wildfire events. Leveraging Geographic Information Systems (GIS), real-time data collection, AI-enhanced analytics and multicriteria optimization algorithms, the SILVANUS DSS component integrates diverse data sources to support dynamic, risk-informed decisions. The system operates within a cloud-edge infrastructure to ensure scalability, interoperability and secure data management. We detail the formalization of the resource allocation problem, describe the implementation of the DSS within the SILVANUS platform, and evaluate its performance in both controlled simulations and real-world pilot scenarios. The results demonstrate the system’s potential to enhance situational awareness and improve the effectiveness of wildfire response operations. Full article
(This article belongs to the Special Issue Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition))
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18 pages, 851 KB  
Article
Learning System-Optimal and Individual-Optimal Collision Avoidance Behaviors by Autonomous Mobile Agents
by Katsutoshi Hirayama, Kazuma Gohara, Jinichi Koue, Tenda Okimoto and Donggyun Kim
Algorithms 2025, 18(11), 671; https://doi.org/10.3390/a18110671 - 22 Oct 2025
Viewed by 417
Abstract
Automated collision avoidance is a central topic in multi-agent systems that consist of mobile agents. One simple approach to pursue system-wide performance is a centralized algorithm, which, however, becomes computationally expensive when involving a large number of agents. There have thus been proposed [...] Read more.
Automated collision avoidance is a central topic in multi-agent systems that consist of mobile agents. One simple approach to pursue system-wide performance is a centralized algorithm, which, however, becomes computationally expensive when involving a large number of agents. There have thus been proposed fully distributed collision avoidance algorithms that can naturally handle many-to-many encounter situations. The DSSA+ is one of those algorithms, which is heuristic and incomplete but has lower communication and computation overheads than other counterparts. However, the DSSA+ and some other distributed collision avoidance algorithms basically optimize the agents’ behavior only in the short term, not caring about the total efficiency in their paths. This may result in some agents’ paths with over-deviation or over-stagnation. In this paper, we present Distributed Stochastic Search algorithm with a deep Q-network (DSSQ), in which the agents can generate time-efficient collision-free paths while they learn independently whether to detour or change speeds by Deep Reinforcement Learning. A key idea in the learning principle of the DSSQ is to let the agents pursue their individual optimality. We have experimentally confirmed that a sequence of short-term system-optimal solutions found by the DSSA+ gradually becomes long-term individually optimal for every agent. Full article
(This article belongs to the Special Issue Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition))
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20 pages, 2181 KB  
Article
Optimal Unit Scheduling Considering Multi-Scenario Source–Load Uncertainty and Frequency Security
by Xiaodong Yang, Yue Chang, Lun Cheng, Yujing Su and Tao Wang
Algorithms 2025, 18(10), 595; https://doi.org/10.3390/a18100595 - 23 Sep 2025
Viewed by 323
Abstract
In response to the significant challenges posed by the volatility and low-inertia characteristics of new energy outputs to the safe operation of power systems, a day-ahead scheduling model for generating units that takes into account source–load uncertainty and frequency security is proposed. To [...] Read more.
In response to the significant challenges posed by the volatility and low-inertia characteristics of new energy outputs to the safe operation of power systems, a day-ahead scheduling model for generating units that takes into account source–load uncertainty and frequency security is proposed. To address source–load uncertainty, kernel density estimation and copula theory are employed to model the correlation between wind and solar energy and generate a set of typical daily scenarios. Considering the low-inertia characteristic, frequency security constraints are incorporated into the scheduling model, and an optimization model for generating units that takes into account multi-scenario uncertainty is constructed. By solving the expected value of the objective function, economic and safe scheduling of the system under uncertain environments is achieved. An experimental analysis and case studies verify the advantages and feasibility of the proposed model through a comparison of scheduling costs and frequency security. Full article
(This article belongs to the Special Issue Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition))
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24 pages, 1054 KB  
Article
Consensus-Based Automatic Group Decision-Making Method with Reliability and Subjectivity Measures Based on Sentiment Analysis
by Johnny Bajaña-Zajía, José Ramón Trillo, Francisco Javier Cabrerizo and Juan Antonio Morente-Molinera
Algorithms 2025, 18(8), 477; https://doi.org/10.3390/a18080477 - 3 Aug 2025
Viewed by 845
Abstract
The use of informal language on social media and the sheer volume of information make it difficult for a computer system to analyse it automatically. The aim of this work is to design a new group decision-making method that applies two new consensus [...] Read more.
The use of informal language on social media and the sheer volume of information make it difficult for a computer system to analyse it automatically. The aim of this work is to design a new group decision-making method that applies two new consensus methods based on sentiment analysis. This method is designed for application in the analysis of texts on social media. To test the method, we will use posts from the so called social network X. The proposed model differs from previous work in this field by defining a new degree of subjectivity and a new degree of reliability associated with user opinions. This work also presents two new consensus measures, one focused on measuring the number of words classified as positive and negative and the other on analysing the percentage of occurrence of those words. Our method allows us to automatically extract preferences from the transcription of the texts used in the debate, avoiding the need for users to explicitly indicate their preferences. The application to a real case of public investment demonstrates the effectiveness of the approach in collaborative contexts that used natural language. Full article
(This article belongs to the Special Issue Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition))
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29 pages, 666 KB  
Article
Hestenes–Stiefel-Type Conjugate Direction Algorithm for Interval-Valued Multiobjective Optimization Problems
by Rupesh Krishna Pandey, Balendu Bhooshan Upadhyay, Subham Poddar and Ioan Stancu-Minasian
Algorithms 2025, 18(7), 381; https://doi.org/10.3390/a18070381 - 23 Jun 2025
Cited by 2 | Viewed by 683
Abstract
This article investigates a class of interval-valued multiobjective optimization problems (IVMOPs). We define the Hestenes–Stiefel (HS)-type direction for the objective function of IVMOPs and establish that it has a descent property at noncritical points. An Armijo-like line search is employed to determine an [...] Read more.
This article investigates a class of interval-valued multiobjective optimization problems (IVMOPs). We define the Hestenes–Stiefel (HS)-type direction for the objective function of IVMOPs and establish that it has a descent property at noncritical points. An Armijo-like line search is employed to determine an appropriate step size. We present an HS-type conjugate direction algorithm for IVMOPs and establish the convergence of the sequence generated by the algorithm. We deduce that the proposed algorithm exhibits a linear order of convergence under appropriate assumptions. Moreover, we investigate the worst-case complexity of the sequence generated by the proposed algorithm. Furthermore, we furnish several numerical examples, including a large-scale IVMOP, to demonstrate the effectiveness of our proposed algorithm and solve them by employing MATLAB. To the best of our knowledge, the HS-type conjugate direction method has not yet been explored for the class of IVMOPs. Full article
(This article belongs to the Special Issue Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition))
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18 pages, 984 KB  
Article
A Linear Regression Prediction-Based Dynamic Multi-Objective Evolutionary Algorithm with Correlations of Pareto Front Points
by Junxia Ma, Yongxuan Sang, Yaoli Xu and Bo Wang
Algorithms 2025, 18(6), 372; https://doi.org/10.3390/a18060372 - 19 Jun 2025
Viewed by 646
Abstract
The Dynamic Multi-objective Optimization Problem (DMOP) is one of the common problem types in academia and industry. The Dynamic Multi-Objective Evolutionary Algorithm (DMOEA) is an effective way for solving DMOPs. Despite the existence of many research works proposing a variety of DMOEAs, the [...] Read more.
The Dynamic Multi-objective Optimization Problem (DMOP) is one of the common problem types in academia and industry. The Dynamic Multi-Objective Evolutionary Algorithm (DMOEA) is an effective way for solving DMOPs. Despite the existence of many research works proposing a variety of DMOEAs, the demand for efficient solutions to DMOPs in drastically changing scenarios is still not well met. To this end, this paper is oriented towards DMOEA and innovatively proposes to explore the correlation between different points of the optimal frontier (PF) to improve the accuracy of predicting new PFs for new environments, which is the first attempt, to our best knowledge. Specifically, when the DMOP environment changes, this paper first constructs a spatio-temporal correlation model between various key points of the PF based on the linear regression algorithm; then, based on the constructed model, predicts a new location for each key point in the new environment; subsequently, constructs a sub-population by introducing the Gaussian noise into the predicted location to improve the generalization ability; and then, utilizes the idea of NSGA-II-B to construct another sub-population to further improve the population diversity; finally, combining the previous two sub-populations, re-initializing a new population to adapt to the new environment through a random replacement strategy. The proposed method was evaluated by experiments on the CEC 2018 test suite, and the experimental results show that the proposed method can obtain the optimal MIGD value on six DMOPs and the optimal MHVD value on five DMOPs, compared with six recent research results. Full article
(This article belongs to the Special Issue Multi-Objective and Multi-Level Optimization: Algorithms and Applications (2nd Edition))
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