Understanding the relationships between objectives in a multiobjective optimisation problem is important for developing tailored and efficient solving techniques. In particular, when tackling combinatorial optimisation problems with many objectives, that arise in real-world logistic scenarios, better support for the decision maker can be achieved through better understanding of the often complex fitness landscape. This paper makes a contribution in this direction by presenting a technique that allows a visualisation and analysis of the local and global relationships between objectives in optimisation problems with many objectives. The proposed technique uses four steps: First, the global pairwise relationships are analysed using the Kendall correlation method; then, the ranges of the values found on the given Pareto front are estimated and assessed; next, these ranges are used to plot a map using Gray code, similar to Karnaugh maps, that has the ability to highlight the trade-offs between multiple objectives; and finally, local relationships are identified using scatter plots. Experiments are presented for three combinatorial optimisation problems: multiobjective multidimensional knapsack problem, multiobjective nurse scheduling problem, and multiobjective vehicle routing problem with time windows . Results show that the proposed technique helps in the gaining of insights into the problem difficulty arising from the relationships between objectives.

Abstract-- V ertical T otal Electron Content (vTEC) is an iono-spheric characteristic used to derive the signal delay impo sed by the ionosphere on near-vertical trans-ionospheric link s. The major aim of this paper is to design a prediction model based o n the main factors that influence the variability of this param eter on a diurnal, seasonal and long-term time-scale. The model should be accurate and general (comprehensive) enough for efficiently approximating the high variations of vTEC. Howe ver, good approximation and generalization are conflicting obje ctives. For this reason a Genetic Programming (GP) with Multi-objec tive Evolutionary Algorithm based on Decomposition characteri stics (GP-MOEA/D) is designed and proposed for modeling vTEC over Cyprus. Experimental results show that the Multi-Objectiv e GPmodel, considering real vTEC measurements obtained over a period of 11 years, has produced a good approximation of the modeled parameter and can be implemented as a local model to account for the ionospheric imposed error in positioning . Particulary, the GP-MOEA/D approach performs better than a Single Objective Optimization GP, a GP with Non-dominated Sorting Genetic Algorithm-II (NSGA-II) characteristics a nd the previously proposed Neural Network-based approach in most cases. The ionosphere is defined as a region of the earth's upper atmosphere where sufficient ionisation can exist to affect t he propagation of radio waves. It ranges in height above the surface of the earth from approximately 50 km to 1000 km.

In the context of labor shortages and rising costs, construction robots are regarded as the key to revolutionizing traditional construction methods and improving efficiency and quality in the construction industry. In order to ensure that construction robots can perform tasks efficiently and accurately in complex construction environments, traditional single-objective trajectory optimization methods are difficult to meet the complex requirements of the changing construction environment. Therefore, we propose a multi-objective trajectory optimization for the robotic arm used in the curtain wall installation. First, we design a robotic arm for curtain wall installation, integrating serial, parallel, and folding arm elements, while considering its physical properties and motion characteristics. In addition, this paper proposes an NSGA-III-FO algorithm (NSGA-III with Focused Operator, NSGA-III-FO) that incorporates a focus operator screening mechanism to accelerate the convergence of the algorithm towards the Pareto front, thereby effectively balancing the multi-objective constraints of construction robots. The proposed algorithm is tested against NSGA-III, MOEA/D, and MSOPS-II in ten consecutive trials on the DTLZ3 and WFG3 test functions, showing significantly better convergence efficiency than the other algorithms. Finally, we conduct two sets of experiments on the designed robotic arm platform, which confirm the efficiency and practicality of the NSGA-III-FO algorithm in solving multi-objective trajectory planning problems for curtain wall installation tasks.

The evaluation of fairness models in Machine Learning involves complex challenges, such as defining appropriate metrics, balancing trade-offs between utility and fairness, and there are still gaps in this stage. This work presents a novel multi-objective evaluation framework that enables the analysis of utility-fairness trade-offs in Machine Learning systems. The framework was developed using criteria from Multi-Objective Optimization that collect comprehensive information regarding this complex evaluation task. The assessment of multiple Machine Learning systems is summarized, both quantitatively and qualitatively, in a straightforward manner through a radar chart and a measurement table encompassing various aspects such as convergence, system capacity, and diversity. The framework's compact representation of performance facilitates the comparative analysis of different Machine Learning strategies for decision-makers, in real-world applications, with single or multiple fairness requirements. The framework is model-agnostic and flexible to be adapted to any kind of Machine Learning systems, that is, black- or white-box, any kind and quantity of evaluation metrics, including multidimensional fairness criteria. The functionality and effectiveness of the proposed framework is shown with different simulations, and an empirical study conducted on a real-world dataset with various Machine Learning systems.

Path planning is essential for unmanned aerial vehicles (UAVs) as it determines the path that the UAV needs to follow to complete a task. This work addresses this problem by introducing a new algorithm called navigation variable-based multi-objective particle swarm optimization (NMOPSO). It first models path planning as an optimization problem via the definition of a set of objective functions that include optimality and safety requirements for UAV operation. The NMOPSO is then used to minimize those functions through Pareto optimal solutions. The algorithm features a new path representation based on navigation variables to include kinematic constraints and exploit the maneuverable characteristics of the UAV. It also includes an adaptive mutation mechanism to enhance the diversity of the swarm for better solutions. Comparisons with various algorithms have been carried out to benchmark the proposed approach. The results indicate that the NMOPSO performs better than not only other particle swarm optimization variants but also other state-of-the-art multi-objective and metaheuristic optimization algorithms. Experiments have also been conducted with real UAVs to confirm the validity of the approach for practical flights. The source code of the algorithm is available at https://github.com/ngandng/NMOPSO.

In agent-based simulations, synthetic populations of agents are commonly used to represent the structure, behaviour, and interactions of individuals. However, generating a synthetic population that accurately reflects real population statistics is a challenging task, particularly when performed at scale. In this paper, we propose a multi objective combinatorial optimisation technique for large scale population synthesis. We demonstrate the effectiveness of our approach by generating a synthetic population for selected regions and validating it on contingency tables from real population data. Our approach supports complex hierarchical structures between individuals and households, is scalable to large populations and achieves minimal contigency table reconstruction error. Hence, it provides a useful tool for policymakers and researchers for simulating the dynamics of complex populations.

The Multi-Objective Vehicle Routing Problem (MOVRP) is a complex optimization problem in the transportation and logistics industry. This paper proposes a novel approach to the MOVRP that aims to create routes that consider drivers' and operators' decisions and preferences. We evaluate two approaches to address this objective: visually attractive route planning and data mining of historical driver behavior to plan similar routes. Using a real-world dataset provided by Amazon, we demonstrate that data mining of historical patterns is more effective than visual attractiveness metrics found in the literature. Furthermore, we propose a bi-objective problem to balance the similarity of routes to historical routes and minimize routing costs. We propose a two-stage GRASP algorithm with heuristic box splitting to solve this problem. The proposed algorithm aims to approximate the Pareto front and to present routes that cover a wide range of the objective function space. The results demonstrate that our approach can generate a small number of non-dominated solutions per instance, which can help decision-makers to identify trade-offs between routing costs and drivers' preferences. Our approach has the potential to enhance the last-mile delivery operations of logistics companies by balancing these conflicting objectives.

Large Language Models (LLMs) have garnered considerable attention owing to their remarkable capabilities, leading to an increasing number of companies offering LLMs as services. Different LLMs achieve different performance at different costs. A challenge for users lies in choosing the LLMs that best fit their needs, balancing cost and performance. In this paper, we propose a framework for addressing the cost-effective query allocation problem for LLMs. Given a set of input queries and candidate LLMs, our framework, named OptLLM, provides users with a range of optimal solutions to choose from, aligning with their budget constraints and performance preferences, including options for maximizing accuracy and minimizing cost. OptLLM predicts the performance of candidate LLMs on each query using a multi-label classification model with uncertainty estimation and then iteratively generates a set of non-dominated solutions by destructing and reconstructing the current solution. To evaluate the effectiveness of OptLLM, we conduct extensive experiments on various types of tasks, including text classification, question answering, sentiment analysis, reasoning, and log parsing. Our experimental results demonstrate that OptLLM substantially reduces costs by 2.40% to 49.18% while achieving the same accuracy as the best LLM. Compared to other multi-objective optimization algorithms, OptLLM improves accuracy by 2.94% to 69.05% at the same cost or saves costs by 8.79% and 95.87% while maintaining the highest attainable accuracy.

Surrogate-assisted evolutionary algorithms have been widely developed to solve complex and computationally expensive multi-objective optimization problems in recent years. However, when dealing with high-dimensional optimization problems, the performance of these surrogate-assisted multi-objective evolutionary algorithms deteriorate drastically. In this work, a novel Classifier-assisted rank-based learning and Local Model based multi-objective Evolutionary Algorithm (CLMEA) is proposed for high-dimensional expensive multi-objective optimization problems. The proposed algorithm consists of three parts: classifier-assisted rank-based learning, hypervolume-based non-dominated search, and local search in the relatively sparse objective space. Specifically, a probabilistic neural network is built as classifier to divide the offspring into a number of ranks. The offspring in different ranks uses rank-based learning strategy to generate more promising and informative candidates for real function evaluations. Then, radial basis function networks are built as surrogates to approximate the objective functions. After searching non-dominated solutions assisted by the surrogate model, the candidates with higher hypervolume improvement are selected for real evaluations. Subsequently, in order to maintain the diversity of solutions, the most uncertain sample point from the non-dominated solutions measured by the crowding distance is selected as the guided parent to further infill in the uncertain region of the front. The experimental results of benchmark problems and a real-world application on geothermal reservoir heat extraction optimization demonstrate that the proposed algorithm shows superior performance compared with the state-of-the-art surrogate-assisted multi-objective evolutionary algorithms. The source code for this work is available at https://github.com/JellyChen7/CLMEA.

Multi-objective optimisation problems involve finding solutions with varying trade-offs between multiple and often conflicting objectives. Ising machines are physical devices that aim to find the absolute or approximate ground states of an Ising model. To apply Ising machines to multi-objective problems, a weighted sum objective function is used to convert multi-objective into single-objective problems. However, deriving scalarisation weights that archives evenly distributed solutions across the Pareto front is not trivial. Previous work has shown that adaptive weights based on dichotomic search, and one based on averages of previously explored weights can explore the Pareto front quicker than uniformly generated weights. However, these adaptive methods have only been applied to bi-objective problems in the past. In this work, we extend the adaptive method based on averages in two ways: (i)~we extend the adaptive method of deriving scalarisation weights for problems with two or more objectives, and (ii)~we use an alternative measure of distance to improve performance. We compare the proposed method with existing ones and show that it leads to the best performance on multi-objective Unconstrained Binary Quadratic Programming (mUBQP) instances with 3 and 4 objectives and that it is competitive with the best one for instances with 2 objectives.