Scalar Method Research Articles

An effective approach for bi-objective multi-period touristic itinerary planning

Planning effective itineraries for tourists is a major problem that has been gaining attention over the last years. This paper proposes a new bi-objective integer linear programming model for this problem. Decisions include the choice of the best itinerary to be performed considering multi-period routing, time windows for the visited attractions and the choice of restaurants and hotels. The conflicting objectives considered are: (i) maximizing the level of service offered by the itinerary, and (ii) minimizing the total distance traveled. The problem resolution, even for small instances by exact methods, is limited. This motivated the proposition of a new customized MIP-heuristic based on decomposition, fix-and-optimize and MIP-start, to produce good-quality solutions with moderate computational effort. Tchebycheff’s scalarization method was coupled to this heuristic and multiple compromise solutions were obtained. Extensive results with problem instances of different sizes and characteristics showed a good performance of this approach, capable of producing effective solutions within short runtimes. The analysis of the solutions indicated a strong conflict between the objectives, allowing the user to quantify the losses and gains when one criterion is prioritized over the other. A brief sensitivity analysis of some model parameters revealed interesting managerial insights. Some examples include quantifying the negative impacts in terms of the level of service offered by concentrating hotels and restaurants in the center of tourist attractions, increasing visit and transfer times between attractions and reducing the planning horizon for the entire itinerary. These aspects validate the potential of using this MIP model and applying this MIP-heuristic in real situations.

Stability and error estimates of GPAV-based unconditionally energy-stable scheme for phase field crystal equation

In this article, we propose a linear, second-order, semi-discrete time stepping scheme for the phase field crystal equation based on generalized positive auxiliary variable (GPAV) approach. This scheme reduces the operation counts by half compared to the GPAV and the scalar auxiliary variable methods in previous works. We prove the unconditionally energy stability and provide bounds and error estimates for the field function. Numerical experiments are carried out to verify our theoretical results and demonstrate the robustness and accuracy of the proposed scheme.

Measurements of the azimuthal anisotropy of prompt and nonprompt charmonia in PbPb collisions at sNN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{s_{\ extrm{NN}}} $$\\end{document} = 5.02 TeV

The second-order (v2) and third-order (v3) Fourier coefficients describing the azimuthal anisotropy of prompt and nonprompt (from b-hadron decays) J/ψ, as well as prompt ψ(2S) mesons are measured in lead-lead collisions at a center-of-mass energy per nucleon pair of sNN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{s_{\ extrm{NN}}} $$\\end{document} = 5.02 TeV. The analysis uses a data set corresponding to an integrated luminosity of 1.61 nb−1 recorded with the CMS detector. The J/ψ and ψ(2S) mesons are reconstructed using their dimuon decay channel. The v2 and v3 coefficients are extracted using the scalar product method and studied as functions of meson transverse momentum and collision centrality. The measured v2 values for prompt J/ψ mesons are found to be larger than those for nonprompt J/ψ mesons. The prompt J/ψ v2 values at high pT are found to be underpredicted by a model incorporating only parton energy loss effects in a quark-gluon plasma medium. Prompt and nonprompt J/ψ meson v3 and prompt ψ(2S) v2 and v3 values are also reported for the first time, providing new information about heavy quark interactions in the hot and dense medium created in heavy ion collisions.

Automatic Soil Sampling Site Selection in Management Zones Using a Multi-Objective Optimization Algorithm

Precision agriculture hinges on accurate soil condition data obtained through soil testing across the field, which is a foundational step for subsequent processes. Soil testing is expensive, and reducing the number of samples is an important task. One viable approach is to divide the farm fields into homogenous management zones that require only one soil sample. As a result, these sample points must be the best representatives of the management zones and satisfy some other geospatial conditions, such as accessibility and being away from field borders and other test points. In this paper, we introduce an algorithmic method as a framework for automatically determining locations for test points using a constrained multi-objective optimization model. Our implementation of the proposed algorithmic framework is significantly faster than the conventional GIS process. While the conventional process typically takes several days with the involvement of GIS technicians, our framework takes only 14 s for a 200-hectare field to find optimal benchmark sites. To demonstrate our framework, we use time-varying Sentinel-2 satellite imagery to delineate management zones and a digital elevation model (DEM) to avoid steep regions. We define the objectives for a representative area of a management zone. Then, our algorithm optimizes the objectives using a scalarization method while avoiding constraints. We assess our method by testing it on five fields and showing that it generates optimal results. This method is fast, repeatable, and extendable.

An efficient linear and unconditionally stable numerical scheme for the phase field sintering model

In this article, the phase field sintering model, which is composed of a Cahn–Hilliard type equation and several Allen–Cahn type equations, has been considered. On the scalar auxiliary variable framework, we propose a theoretically efficient and stable method for solid-state sintering. In order to overcome the nonlinear issues, we define a stabilized scalar auxiliary variable method and reformulate the phase field sintering model. An efficient and accurate numerical scheme is investigated to solve our model. The scheme consist of several decoupled diffusion equations at every time step. Therefore, our scheme is easy to implement. Then we prove the numerical discrete energy is unconditionally stable. Several numerical simulations in two- and three-dimensional spaces are presented to demonstrate the robustness of our method.

Joint Multi-Objective Optimization for Radio Access Network Slicing Using Multi-Agent Deep Reinforcement Learning

Radio access network (RAN) slices can provide various customized services for next-generation wireless networks. Thus, multiple performance metrics of different types of RAN slices need to be jointly optimized. However, existing efforts in multi-objective optimization problem (MOOP) for RAN slicing are only in the scalar form, which is difficult to achieve simultaneous optimization. In this paper, we consider a non-scalar MOOP for RAN slicing with three types of slices, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , the high-bandwidth slice, the low-delay slice, and the wide-coverage slice over the same underlying physical network. We jointly optimize the throughput, the transmission delay, and the coverage area by user-oriented dynamic virtual base stations (vBSs)' deployment, and sub-channel and power allocation. An improved multi-agent deep deterministic policy gradient (IMADDPG) algorithm, having the characteristics of centralized training and distributed execution, is proposed to solve the above non-deterministic polynomial-time hard (NP-hard) problem. The rank voting method is introduced in the inference process to obtain near-Pareto optimal solutions. Simulation results verify that the proposed scheme can ensure better performance than the traditional scalar utility method and other benchmark algorithms. The proposed scheme has the advantage of flexibly approaching any point of the Pareto boundary, while the traditional scalar method only subjectively approaches one of the Pareto optimal solutions. Furthermore, our proposal strikes a compelling tradeoff among three types of RAN slices due to the non-dominance between Pareto optimal solutions.

Performance of passive scalar method in heat transfer simulation of a two-dimensional droplet focusing on the parasitic temperatures

Performance of passive scalar method in heat transfer simulation of a two-dimensional droplet focusing on the parasitic temperatures

Scattering direction sampling methods for polarized Monte Carlo simulation of oceanic lidar.

Monte Carlo techniques have been widely applied in polarized light simulation. Based on different preconditions, there are two main types of sampling strategies for scattering direction: one is the scalar sampling method; the others are polarized sampling approaches, including the one- and two-point rejection methods. The polarized simulation of oceanic lidar involves a variety of mediums, and an efficient scattering sampling method is the basis for the coupling simulation of the atmosphere and ocean. To determine the optimal scattering sampling method for oceanic lidar simulation, we developed a polarized Monte Carlo model and simulated Mie scattering, Rayleigh scattering, and Petzold average-particle scattering experiments. This simulation model has been validated by comparison with Ramella-Roman's program [Opt. Express13, 4420 (2005)OPEXFF1094-408710.1364/OPEX.13.004420], with differences in reflectance and transmittance Stokes less than 1% in Mie scattering. The simulation results show these scattering sampling methods differ in runtime, scattering angle distributions, and reflectance and transmittance Stokes. Considering the current simulation accuracy of oceanic lidar, the differences in reflectance and transmittance Stokes are acceptable; thus, the runtime becomes the main evaluation factor. The one-point rejection method and scalar sampling method are preferable for the oceanic lidar polarized simulation. Under complex atmosphere-ocean coupling systems, scalar sampling methods may be a better choice since the calculation process of the sampling is independent of the incident Stokes vector.

Residual useful life prognosis of equipment based on modified hidden semi-Markov model with a co-evolutional optimization method

Health diagnosis and prognosis of equipment is considered crucial for condition-based maintenance. This paper presents a prognosis health and monitoring scheme framework based on the modified hidden semi-Markov model (HSMM). First, a novel scalarization method for raw signals is developed based on the Weibull distribution with a short-time function window. Then, the conglutination coefficient and deterioration kernel are introduced into HSMM to simulate the inherent deterioration process of equipment. A co-evolutional algorithm based on genetic algorithm and salp swarm algorithm is proposed to estimate the parameters of the modified HSMM. A model base is established based on various sub-models of different faults for the health prognosis and diagnosis of equipment. The residual useful life can be calculated based on the historical whole useful life (WUL) data and the real-time status of equipment. Finally, the feasibility and effectiveness of the proposed model is verified on the turbofan engine dataset. The proposed scheme can provide a novel solution for the health prognosis of the non-vibration signal.

Stability of the approximate solution sets for set optimization problems with the perturbations of feasible set and objective mapping

The aim of this paper is to study stability of the sets of l-minimal approximate solutions and weak l-minimal approximate solutions for set optimization problems with respect to the perturbations of feasible sets and objective mappings. We introduce a new metric between two set-valued mappings by utilizing a Hausdorff-type distance proposed by Han [A Hausdorff-type distance, the Clarke generalized directional derivative and applications in set optimization problems. Appl Anal. 2022;101:1243–1260]. The new metric between two set-valued mappings allows us to discuss set optimization problems with respect to the perturbation of objective mappings. Then, we establish semicontinuity and Lipschitz continuity of l-minimal approximate solution mapping and weak l-minimal approximate solution mapping to parametric set optimization problems by using the scalarization method and a density result. Finally, our main results are applied to stability of the approximate solution sets for vector optimization problems.

A Generalized Scalarization Method for Evolutionary Multi-Objective Optimization

The decomposition-based multi-objective evolutionary algorithm (MOEA/D) transforms a multi-objective optimization problem (MOP) into a set of single-objective subproblems for collaborative optimization. Mismatches between subproblems and solutions can lead to severe performance degradation of MOEA/D. Most existing mismatch coping strategies only work when the L∞ scalarization is used. A mismatch coping strategy that can use any Lp scalarization, even when facing MOPs with non-convex Pareto fronts, is of great significance for MOEA/D. This paper uses the global replacement (GR) as the backbone. We analyze how GR can no longer avoid mismatches when L∞ is replaced by another Lp with p ∈ [1, ∞), and find that the Lp-based (1 ≤ p &lt; ∞) subproblems having inconsistently large preference regions. When p is set to a small value, some middle subproblems have very small preference regions so that their direction vectors cannot pass through their corresponding preference regions. Therefore, we propose a generalized Lp (GLp) scalarization to ensure that the subproblem’s direction vector passes through its preference region. Our theoretical analysis shows that GR can always avoid mismatches when using the GLp scalarization for any p ≥ 1. The experimental studies on various MOPs conform to the theoretical analysis.

Evaluation of the Trade-Off between Ground Delays and Intersecting Departures under Various Pilot Acceptance Rate Scenarios

The increasing demand for air traffic at airports necessitates the efficient utilization of ground facilities such as runways and taxiways. Intersecting departures, in which one or more aircraft take off from intersecting points on the runway, is a commonly used approach to increase runway capacity and reduce ground delays and taxi times, as well as noise and air pollution. However, the procedure carries potential risks such as runway incursion and excursion. This creates a trade-off between minimizing the number of intersecting departures and minimizing ground delays. In practice, the decision to perform an intersecting departure is ultimately up to the pilot, resulting in uncertainty in the acceptance rate of these types of takeoffs. In this study, a departure sequencing model was developed for a single-runway airport that considers intersecting departures and various pilot acceptance rate scenarios. The primary objective of the model is to minimize total ground delay, including taxi delays, runway holds, and conflict holds. The secondary objective is to minimize the number of intersecting departures by directing the most operationally critical aircraft to the intersection takeoff. The epsilon constraint method—a multi-objective scalarization method—was used to reveal the trade-offs between the objective functions. The results of the model were compared with a traditional scenario that only allows take offs from the beginning of the runway. As a result, average delay savings ranged from 17.1% to 31.5% in various acceptance rate scenarios, as well as average taxi time savings ranging from 4.9% to 8.4% compared with the traditional scenario.

A weighted metric scalarization approach for multiobjective BOHB hyperparameter optimization in LSTM model for sentiment analysis

In recent years, hyperparameter tuning has become increasingly essential, especially for neural network models, where models with good performance generally require a long time to train. In contrast, fast-trained models often result in poor model performance. Thus, it is essential to determine the optimal hyperparameter configuration for a neural network model that can produce good model performance and fast training time. This paper presents a multiobjective optimization for Long-Short Term Memory (LSTM) model using Bayesian optimization-Hyperband (BOHB) with a weighted metric scalarization method. The objective functions used are the F1-score (Macro) and model training time, where both of the objective functions are considered equally important. The data used in this paper are the SmSA and EmoT datasets taken from the IndoNLU benchmark. The results are optimized model using multiobjective BOHB optimization with weighted metric scalarization method produced higher F1-score and faster model training time in both datasets compared to the baseline model and optimized model using single objective BOHB.

Ordinal optimization through multi-objective reformulation

We analyze combinatorial optimization problems with ordinal, i.e., non-additive, objective functions that assign categories (like good, medium and bad) rather than cost coefficients to the elements of feasible solutions. We review different optimality concepts for ordinal optimization problems and discuss their similarities and differences. We then focus on two prevalent optimality concepts that are shown to be equivalent. Our main focus lies on the investigation of a bijective linear transformation that transforms ordinal optimization problems to associated standard multi-objective optimization problems with binary cost coefficients. Since this transformation preserves all properties of the underlying problem, problem-specific solution methods remain applicable. A prominent example is dynamic programming and Bellman’s principle of optimality, that can be applied, e.g., to ordinal shortest path and ordinal knapsack problems. We investigate the interrelation between scalarization techniques and methods based on the hypervolume indicator when applied to the ordinal and the transformed problem, respectively. Furthermore, we extend our results to multi-objective optimization problems that combine ordinal and real-valued objective functions.

Analysis of the weighted Tchebycheff weight set decomposition for multiobjective discrete optimization problems

Scalarization is a common technique to transform a multiobjective optimization problem into a scalar-valued optimization problem. This article deals with the weighted Tchebycheff scalarization applied to multiobjective discrete optimization problems. This scalarization consists of minimizing the weighted maximum distance of the image of a feasible solution to some desirable reference point. By choosing a suitable weight, any Pareto optimal image can be obtained. In this article, we provide a comprehensive theory of this set of eligible weights. In particular, we analyze the polyhedral and combinatorial structure of the set of all weights yielding the same Pareto optimal solution as well as the decomposition of the weight set as a whole. The structural insights are linked to properties of the set of Pareto optimal solutions, thus providing a profound understanding of the weighted Tchebycheff scalarization method and, as a consequence, also of all methods for multiobjective optimization problems using this scalarization as a building block.

Multi-Objective Approach for User Association to Improve Load Balancing and Blockage in Millimeter Wave Cellular Networks

One of the leading enabling technologies of 5G wireless networks is to use the mm-Wave spectrum band. Despite its large and wide frequency bandwidth, the obtained data rate can be diminished due to link blockage in this frequency band. We formulate a bi-objective optimization problem to optimize user association in cellular networks with mm-Wave enabled base stations. The two objectives to minimize are maximum base station utility and blockage score. We simulate three different scalarization methods to turn a bi-objective vector into a scalar. Since the combinatorial bi-objective problem is NP-Hard, we conduct Lagrangian dual analysis on all of the scalarization methods. Solving the dual problem decreases the time complexity of the solver algorithm, but the solution has a distance from the optimal point created by solving the primal. We also solve the primal optimization problem with a single objective optimization tool. Compared to the time complexity of the primal problem of scalarization methods, the time complexities of solutions to the dual problems are lower. The results show that our solution to the bi-objective optimization problem has a better outcome in terms of the number of link blockage and the maximum base station utility compared to optimizing each objective alone.

CFD-based geometrical shape optimization of a packed-bed reactor combining multi-objective and adjoint system methods

This paper presents the development of a geometric shape optimization methodology based on the so-called ”Hadamard boundary variation” method for performing very general domain deformations, and the related concept of domain differentiation. The resulting method is used to determine the optimal configuration of a two-dimensional packed-bed reactor that simultaneously optimizes its conversion rate and fluid energy dissipation, and where a homogeneous first-order reaction or a catalytic surface reaction takes place. The considered multi-objective optimization problem is subjected to four constraints: the process model constraints consisting of the Navier–Stokes, continuity and mass balance equations, an iso-volume and two manufacturing constraints. The approach to solve the problem is based on the linear scalarization method which converts the multi-objective problem into a single objective problem. The adjoint system method is used to compute the gradient of the performance indices and constraints. Since the indices are conflicting, the solution of the problem is a set of solutions, called Pareto front. Each optimal solution is evaluated using multi-attribute utility theory (MAUT) to determine the best optimal shape of the reactor. Finally, the resulting shape is fabricated using a 3D printing technique and will be experimentally validated.

Energy-aware JPEG image compression: A multi-objective approach

Customer satisfaction is crucially affected by energy consumption in mobile devices. One of the most energy-consuming parts of an application is images. This paper, first, investigates that there is a correlation between energy consumption and image quality as well as image file size. Therefore, these two can be considered as a proxy for energy consumption. In the next step, we focused on proposing a multi-objective strategy to enhance image quality and reduce image file size based on the quantisation table (QT) in JPEG image compression. To this end, we have used two general multi-objective approaches: scalarisation and Pareto-based. In this paper, we embed our strategy into five scalarisation algorithms, including energy-aware multi-objective genetic algorithm (EnMOGA), energy-aware multi-objective particle swarm optimisation (EnMOPSO), energy-aware multi-objective differential evolution (EnMODE), energy-aware multi-objective evolutionary strategy (EnMOES), and energy-aware multi-objective pattern search (EnMOPS). Also, two Pareto-based methods, including a non-dominated sorting genetic algorithm (NSGA-II) and a reference-point-based NSGA-II (NSGA-III) are used for the embedding scheme, and two Pareto-based algorithms, EnNSGAII and EnNSGAIII, are presented. With our proposed scalarisation method, user’s preferences can be set before starting the optimisation process and the algorithm generates only one solution based on the preference, while our Pareto-based approaches generate a set of solutions so that a user can select one of the preferred solutions after the optimisation process.Experimental studies show that the performance of the baseline algorithm is improved by embedding the proposed strategy into metaheuristic algorithms. In particular, EnMOGA, EnMOPS, and EnNSGA-II can perform competitively, among others. From the results, the baseline algorithm in all cases and in comparison to all algorithms yields the worst results. Among the scalarisation methods, EnMOGA and EnMOPS can achieve the first rank in 6 and 7 out of 13 cases and the second rank in 7 and 5 cases in terms of objective function. Also, EnMOES achieved the fifth or worst rank among the scalarisation algorithms. Regarding the Pareto-based algorithms, the table shows that EnNSGAII outperforms EnNSGAIII in 10 out of 13 cases in terms of hyper-volume measure, while it fails in 3 cases. Furthermore, we statistically verify the proposed algorithm’s effectiveness based on the Wilcoxon-signed rank test. Finally, a sensitivity analysis of the parameters is provided. The source code for reproducing the results is available in: https://github.com/SeyedJalaleddinMousavirad/MultiobjectiveJPEGImageCompression.

A Spherical Volume-Rendering Method of Ocean Scalar Data Based on Adaptive Ray Casting

There are some limitations in traditional ocean scalar field visualization methods, such as inaccurate expression and low efficiency in the three-dimensional digital Earth environment. This paper presents a spherical volume-rendering method based on adaptive ray casting to express ocean scalar field. Specifically, the minimum bounding volume based on spherical mosaic is constructed as the proxy geometry, and the depth texture of the seabed terrain is applied to determine the position of sampling points in the spatial interpolation process, which realizes the fusion of ocean scalar field and seabed terrain. Then, we propose an adaptive sampling step algorithm according to the heterogeneous depth distribution and data change rate of the ocean scalar field dataset to improve the efficiency of the ray-casting algorithm. In addition, this paper proposes a nonlinear color-mapping enhancement scheme based on the skewness characteristics of the datasets to optimize the expression effect of volume rendering, and the transparency transfer function is designed to realize volume rendering and local feature structure extraction of ocean scalar field data in the study area.

Fast Error Calibration for Three-Axis SQUID Magnetometers Based on Full-Space Rotational and Frequency-Controllable Magnetic Field

In order to overcome the limitations of the traditional error calibration method for three-axis superconducting quantum interference device (SQUID) magnetometers, this article presents a fast error calibration method for three-axis SQUID magnetometers. In the proposed method, a full-space rotational and frequency-controllable magnetic field with constant total field intensity can be generated by a triaxial Helmholtz coil. Meanwhile, the nonorthogonal error, zero offset error, and proportional coefficient error of the three-axis SQUID magnetometers, which measure the relative value rather than the absolute value, can be accurately solved by an error modeling and ellipsoid fitting theory. Compared with the traditional scalar calibration method, this method replaces the rotation of Dewar and three-axis SQUID magnetometers with a rotating magnetic field, thus avoiding the interference and influence of motion. Compared with the traditional vector calibration method, this method does not need to ensure the alignment between the three-axis SQUID magnetometers and triaxial Helmholtz coil. Moreover, the interference of low-frequency magnetic field can be suppressed by shortening the period of rotating magnetic field. Simulation and experiments prove the effectiveness of this method. After calibration by the proposed method, the root-mean-square error is reduced to 0.414 from 138.411 nT in the electromagnetic shielding room. Repetitive and comparative experiments with the traditional method in field also prove the superiority and stability of the proposed method.