Composite Rule Research Articles

FRKVF: High-accuracy motion interpolation for polishing operations using fourth-order Runge-Kutta and velocity flexibility planning

Traditional interpolation algorithms often fail to meet the precision requirements of ultra-precision machining when applied to super-precision polishing machines. Moreover, the processing of fused silica glass optical components is frequently threatened by mechanical impacts due to their fragility. In order to address this issue, this paper proposes a high-precision motion interpolation method based on fourth-order Runge-Kutta and velocity-flexible planning. This method is designed for open-architecture small multi-axis optical polishing machines to polish quartz glass. The algorithm initially employs composite Simpson's rule to calculate the lengths of sub-paths within the polishing trajectory. Based on these length values, flexible velocity planning is executed to ensure the smooth continuity of velocity, acceleration, and jerk during motion interpolation. This reduces the risk of mechanical impacts that could damage the components during the machining process. The introduction of the adaptive fourth-order Runge-Kutta method significantly enhances the parameter point calculation accuracy of NURBS curves. The incorporation of adaptive principles also maintains a higher processing speed, thereby greatly improving processing efficiency. This method comprehensively addresses both the precision of curve interpolation and execution efficiency. Finally, experimental validation is conducted on an open-architecture small multi-axis optical polishing machine. The proposed method based on FRKVF not only mitigates mechanical impacts resulting from discontinuous acceleration, thereby ensuring the machining quality of optical components, but also satisfies the high-precision requirements for processing fused silica optical elements.

Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods

The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (M<i>T_3</i>) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.

Out-of-order execution enabled deep reinforcement learning for dynamic additive manufacturing scheduling

Additive Manufacturing (AM) has revolutionized the production landscape by enabling on-demand customized manufacturing. However, the efficient management of dynamic AM orders poses significant challenges for production planning and scheduling. This paper addresses the dynamic scheduling problem considering batch processing, random order arrival and machine eligibility constraints, aiming to minimize total tardiness in a parallel non-identical AM machine environment. To tackle this problem, we propose the out-of-order enabled dueling deep Q network (O3-DDQN) approach. In the proposed approach, the problem is formulated as a Markov decision process (MDP). Three-dimensional features, encompassing dynamic orders, AM machines, and delays, are extracted using a ‘look around’ method to represent the production status at a rescheduling point. Additionally, five novel composite scheduling rules based on the out-of-order principle are introduced for selection when an AM machine completes processing or a new order arrives. Moreover, we design a reward function that is strongly correlated with the objective to evaluate the agent’s chosen action. Experimental results demonstrate the superiority of the O3-DDQN approach over single scheduling rules, randomly selected rules, and the classic DQN method. The average improvement rate of performance reaches 13.09% compared to composite scheduling rules and random rules. Additionally, the O3-DDQN outperforms the classic DQN agent with a 6.54% improvement rate. The O3-DDQN algorithm improves scheduling in dynamic AM environments, enhancing productivity and on-time delivery. This research contributes to advancing AM production and offers insights into efficient resource allocation.

Integrated scheduling of multi-objective lot-streaming hybrid flowshop with AGV based on deep reinforcement learning

The conventional approach to shop floor scheduling often overlooks handling time, either neglecting it or assimilating it into processing time. This results in inadequate information sharing, hindering operational synergy and impeding overall optimisation. Addressing this issue, this study focuses on the three-stage lot-streaming hybrid flowshop scheduling problem with automated guided vehicles (LSHFSP-AGV). It introduces a multi-objective scheduling model to enhance collaboration between machine production and AGV distribution, three objectives of minimum maximum completion time, minimum total machine idle time, and shortest AGV transportation distance are set. An improved multi-objective double-depth Q learning algorithm (NSGA2-MDDQN) based on NSGA-II is proposed to solve the problem. Taguchi experiments were conducted to determine the optimal parameter combination among alternative parameter combinations, and extensive numerical experiments involving 27 instances demonstrated the superiority of NSGA2-MDDQN over combinatorial scheduling rules, MDDQN, and NSGA-II, proving the superiority of the algorithm. The experimental results show that on the objective of minimising makespan, NSGA2-MDDQN reduces by an average of 23.17% compared to the composite scheduling rule, NSGA-II and MDDQN. And achieving an average reduction of 43.78% in machine idle time, and 9.12% in AGV transport distance.

The Unification of Evolutionary Dynamics through the Bayesian Decay Factor in a Game on a Graph.

We unify evolutionary dynamics on graphs in strategic uncertainty through a decaying Bayesian update. Our analysis focuses on the Price theorem of selection, which governs replicator(-mutator) dynamics, based on a stratified interaction mechanism and a composite strategy update rule. Our findings suggest that the replication of a certain mutation in a strategy, leading to a shift from competition to cooperation in a well-mixed population, is equivalent to the replication of a strategy in a Bayesian-structured population without any mutation. Likewise, the replication of a strategy in a Bayesian-structured population with a certain mutation, resulting in a move from competition to cooperation, is equivalent to the replication of a strategy in a well-mixed population without any mutation. This equivalence holds when the transition rate from competition to cooperation is equal to the relative strength of selection acting on either competition or cooperation in relation to the selection differential between cooperators and competitors. Our research allows us to identify situations where cooperation is more likely, irrespective of the specific payoff levels. This approach provides new perspectives into the intended purpose of Price's equation, which was initially not designed for this type of analysis.

Deep reinforcement learning for dynamic distributed job shop scheduling problem with transfers

Dynamic events and transportation constraints would significantly affect the full utilization of resources and the reduction of production costs in distributed job shops. Therefore, in this paper, a deep reinforcement learning algorithm (DRL)-based real-time scheduling method is developed to minimize the mean tardiness of the dynamic distributed job shop scheduling problem with transfers (DDJSPT) considering random job arrivals. Firstly, the proposed DDJSPT is modeled as a Markov decision process (MDP). Then, ten problem-oriented state features covering four aspects of factories, machines, jobs, and operations are elaborately extracted from the dynamic distributed job shop. After that, eleven composite rules considering the uniqueness of DDJSPT are constructed as a pool of actions to intelligently prioritize unfinished jobs and allocate the selected job to an appropriate factory. Moreover, a justified reward function adapted from the objective is designed for better convergence of DRLs. Subsequently, five DRLs are employed to address the DDJSPT, encompassing deep Q-network (DQN), double DQN (DDQN), dueling DQN (DlDQN), trust region policy optimization (TRPO), and proximal policy optimization (PPO). Finally, grounded in numerical comparison experiments under 243 production configurations of the DDJSPT, the effectiveness and generalization of DRL-based scheduling methods are credibly verified and confirmed.

Low-latency Virtual Network function Scheduling Algorithm Based on Deep Reinforcement Learning

This paper addresses the problem of mapping, scheduling, and routing of virtual network functions (VNF) on a service function chain (SFC) that is sensitive to latency in a virtual network. A scheduling algorithm for VNF is proposed, which aims to minimize the SFC rejection rate while taking into account VNF mapping, scheduling, and traffic routing during the scheduling process. To achieve this goal, a Markov decision process (MDP)-based VNF scheduling model is established that guarantees SFC resource requirements are met. The model uses the D3QN (Dueling Double DQN) algorithm based on composite rules to select the SFC at each scheduling time point, and selects virtual nodes and routes using a routing optimization algorithm to minimize the SFC rejection rate. We compare our algorithm with the single rule, DQN and genetic algorithm, and the simulation results show that the proposed algorithm can reduce the rejection rate of SFC by approximately 8% compared to genetic algorithms.

Dynamic Intelligent Scheduling in Low-Carbon Heterogeneous Distributed Flexible Job Shops with Job Insertions and Transfers.

With the rapid development of economic globalization and green manufacturing, traditional flexible job shop scheduling has evolved into the low-carbon heterogeneous distributed flexible job shop scheduling problem (LHDFJSP). Additionally, modern smart manufacturing processes encounter complex and diverse contingencies, necessitating the ability to address dynamic events in real-world production activities. To date, there are limited studies that comprehensively address the intricate factors associated with the LHDFJSP, including workshop heterogeneity, job insertions and transfers, and considerations of low-carbon objectives. This paper establishes a multi-objective mathematical model with the goal of minimizing the total weighted tardiness and total energy consumption. To effectively solve this problem, diverse composite scheduling rules are formulated, alongside the application of a deep reinforcement learning (DRL) framework, i.e., Rainbow deep-Q network (Rainbow DQN), to learn the optimal scheduling strategy at each decision point in a dynamic environment. To verify the effectiveness of the proposed method, this paper extends the standard dataset to adapt to the LHDFJSP. Evaluation results confirm the generalization and robustness of the presented Rainbow DQN-based method.

On the integral solution of hyperbolic Kepler’s equation

In a recent paper of Philcox, Goodman and Slepian, the solution of the elliptic Kepler’s equation is given as a quotient of two contour integrals along a Jordan curve that contains in its interior the unique real solution of the elliptic Kepler’s equation and does not include other complex zeroes. In this paper, we show that a similar explicit integral solution can be given for the hyperbolic Kepler’s equation. With this purpose, we carry out a study of the complex zeros of the hyperbolic Kepler’s equation in order to define suitable Jordan contours in the integrals. Even more, we show that appropriate elliptic Jordan contours can be defined for such integrals, which reduces the computing time. Moreover, using the ideas behind the fast Fourier transform (FFT) algorithm, these integrals can be approximated by the composite trapezoidal rule which gives an algorithm with spectral convergence as a function of the number of nodes. The results of some numerical experiments are presented to show that this implementation is a reliable and very accurate algorithm for solving the hyperbolic Kepler’s equation.

Simulation-Based Approach for Multiproject Scheduling Based on Composite Priority Rules

Simulation-Based Approach for Multiproject Scheduling Based on Composite Priority Rules

A feedback learning-based selection hyper-heuristic for distributed heterogeneous hybrid blocking flow-shop scheduling problem with flexible assembly and setup time

Distributed manufacturing is increasingly common due to economic globalization. It has important practical significance to optimize global supply chains by considering cooperative scheduling of distributed production and flexible assembly. This paper studies a distributed heterogeneous hybrid blocking flow-shop scheduling problem with flexible assembly and setup time (DHHBFSP-FAST). The objective is to minimize makespan of all products. To tackle such problem, a mixed-integer linear programming model (MILP) is presented to formulate it. Then, a feedback learning-based selection hyper-heuristic (FLS-HH) is proposed, which contains high-level control strategies and low-level heuristics. For the high-level control strategies, the transition information between each pair of low-level heuristics is collected, then a feedback learning-based selection method is presented based on such transition information, which can automatically select the appropriate low-level heuristics. For the low-level heuristics, to assign jobs/products to machines, a composite dispatch rule is proposed. To generate an initial domain solution with high quality, a constructive heuristic based on double evaluation indexes is developed. The critical jobs and products are analyzed to avoid invalid searching. Based on such analysis, several low-level heuristics are presented to search the domain solution space, which combine six problem-specific perturbation heuristics and two local searches to balances exploration and exploitation. Comprehensive numerical experiments are carried out. The effectiveness of the proposed MILP and the special designs of FLS-HH are verified. To verify the effectiveness of FLS-HH, we compare it with 10 existing high-performing approaches. The comparison results show that FLS-HH significantly outperforms its competitors in solving DHHBFSP-FAST.

Composite Fractional Trapezoidal Rule with Romberg Integration

Composite Fractional Trapezoidal Rule with Romberg Integration

Modified inertial subgradient extragradient algorithms for generalized equilibria systems with constraints of variational inequalities and fixed points

<abstract><p>In this research, we studied modified inertial composite subgradient extragradient implicit rules for finding solutions of a system of generalized equilibrium problems with a common fixed-point problem and pseudomonotone variational inequality constraints. The suggested methods consisted of an inertial iterative algorithm, a hybrid deepest-descent technique, and a subgradient extragradient method. We proved that the constructed algorithms converge to a solution of the considered problem, which also solved some hierarchical variational inequality.</p></abstract>

A Composite Decision Rule of CD8+ T-cell Density in Tumor Biopsies Predicts Efficacy in Early-stage, Immunotherapy Trials.

To examine whether CD8+ T-cell numbers in paired tumor biopsies in early-stage clinical trials can be used as an early indicator of clinical benefit for cancer immunotherapies. Paraffin sections of tumor biopsies were stained immunohistochemically for CD8+ T cells, which were digitally enumerated. The tumor biopsies were from cancer patients in early-phase trials testing novel immunotherapeutic agents. Paired biopsies taken before the start of treatment and on-treatment were compared. A total of 155 patients were used as the training set and an additional 221 patients were used as the validation set. Using the Cox proportional hazard model, a ≥0.9- increase in fold change (FC) on a ln scale in CD8+ T cells (corresponding to a 2.5-fold increase on the linear scale), from baseline, demonstrated a greater association with prolonged progression-free survival and allowed improved differentiation between groups above and below the threshold. Similarly, a ≥6.2 threshold in geometric mean of the on-treatment density (OTD) of T cells, which approximately corresponds to 500 cells/mm2, correlated with longer PFS. The combination of both criteria (FC and OTD) provided the best discrimination between clinically nonactive and active compounds. We propose that a composite score of CD8+ T-cell density in paired biopsies taken before and on-treatment may be a new biomarker to inform on clinical outcomes in early immunotherapy clinical trials.

A monotone numerical integration method for mean–variance portfolio optimization under jump-diffusion models

We develop a highly efficient, easy-to-implement, and strictly monotone numerical integration method for Mean-Variance (MV) portfolio optimization. This method proves very efficient in realistic contexts, which involve jump-diffusion dynamics of the underlying controlled processes, discrete rebalancing, and the application of investment constraints, namely no-bankruptcy and leverage.A crucial element of the MV portfolio optimization formulation over each rebalancing interval is a convolution integral, which involves a conditional density of the logarithm of the amount invested in the risky asset. Using a known closed-form expression for the Fourier transform of this conditional density, we derive an infinite series representation for the conditional density where each term is strictly positive and explicitly computable. As a result, the convolution integral can be readily approximated through a monotone integration scheme, such as a composite quadrature rule typically available in most programming languages. The computational complexity of our proposed method is an order of magnitude lower than that of existing monotone finite difference methods. To further enhance efficiency, we propose an implementation of this monotone integration scheme via Fast Fourier Transforms, exploiting the Toeplitz matrix structure.The proposed monotone numerical integration scheme is proven to be both ℓ∞-stable and pointwise consistent, and we rigorously establish its pointwise convergence to the unique solution of the MV portfolio optimization problem. We also intuitively demonstrate that, as the rebalancing time interval approaches zero, the proposed scheme converges to a continuously observed impulse control formulation for MV optimization expressed as a Hamilton–Jacobi–Bellman Quasi-Variational Inequality. Numerical results show remarkable agreement with benchmark solutions obtained through finite differences and Monte Carlo simulation, underscoring the effectiveness of our approach.

Online semi-supervised learning of composite event rules by combining structure and mass-based predicate similarity

Symbolic event recognition systems detect event occurrences using first-order logic rules. Although existing online structure learning approaches ease the discovery of such rules in noisy data streams, they assume the existence of fully labelled training data. Splice is a recent online graph-based approach that estimates the labels of unlabelled data and makes it possible to learn such rules from semi-supervised training sequences of logical interpretations. However, Splice labelling depends significantly on the metric used to compute the distances of unlabelled examples to their labelled counterparts. Moreover, there is no guarantee about the quality of the labelling found in the local graphs that are built as the data stream in. In this paper, we propose a new online learning method, which includes an enhanced hybrid measure that combines an optimised structural distance, and a data-driven one. The former is guided by feature selection targeted to kNN classification, while the latter is a mass-based dissimilarity. Additionally, the enhanced Splice method, improves the graph construction process, by storing a synopsis of the past, in order to achieve more informed labelling on the local graphs. We evaluate our approach by learning Event Calculus theories for the tasks of human activity recognition, maritime monitoring, and fleet management. The evaluation suggests that our approach outperforms its predecessor, in terms of inferring the missing labels and improving the predictive accuracy of the underlying structure learning system.

A Class of Quadrature Rules for Complex Cauchy Principal Value Integrals

This article is fully devoted to the numerical approximation of Cauchy-type integrals in the complex plane. A class of degree eight quadrature rules is formulated from a family of Gauss-type two-point rules based on the method of extrapolation. The basic rules are developed and then composite rules are constructed from the basic rules. The Error bounds for each rule are determined. The validity of the method has been demonstrated by the provision of numerical experiments and their results.

An equidistributed grid-based second-order scheme for a singularly perturbed Fredholm integro-differential equation with an interior layer

In this study, we introduce and evaluate the effectiveness of the central finite difference coupled with a composite trapezoidal rule on adapted meshes in solving a second-order singularly perturbed Fredholm integro-differential equation with a discontinuous source term. A grid equidistribution technique was employed to address the challenge posed by the perturbation parameter in classical numerical methods. Our approach demonstrated superior precision compared to existing schemes in the literature and achieved second-order uniform convergence.

A novel Romberg integration method for neutrosophic valued functions

This study proposes a numerical integration technique to determine the approximate integral of the neutrosophic valued function. Newton Cot’s method with a positive coefficient has been used for neutrosophic integration, and then the suggested technique is used to find the approximate value of the integral. This method is called the Romberg integration method. In this method, the composite trapezoidal rule is applied initially, and Richardson’s extrapolation technique is used next to improve the solution. Richardson’s extrapolation is the basis of Romberg’s integration. Formulas with high-order truncation errors are created using Richardson extrapolation, which employs an averaging method. Richardson’s extrapolation helps the proposed method to achieve high computational efficiency. The convergence of the numerical approximation is provided in terms of theorems. A comparative analysis of the proposed method with other existing methods is demonstrated to show the efficiency and reliability of the proposed method.

A variable step‐size extrapolated Crank–Nicolson method for option pricing under stochastic volatility model with jump

This paper studies the numerical solution of the stochastic volatility model with jumps under European options. This model can be transformed into a partial integro‐differential equation (PIDE) with spatial mixed derivative terms. Due to the nonsmoothness of the initial function, the variable step‐size extrapolated Crank–Nicolson (CN) method, which explicitly discretizes the jump term and implicitly the rest, is proposed to solve this model. The finite difference method is used to discretize the spatial differential operator, the composite trapezoidal rule to calculate the jump integral, and then a linear system with a nine‐diagonal coefficient matrix is obtained, which is easy to solve. The stability of the variable step‐size extrapolated CN method is then proved. Based on realistic regularity assumptions on the data, the consistency error and global error bounds of the variable step‐size extrapolated CN method are derived in norm. Compared with the variable step‐size IMEX BDF2 method and the variable step‐size IMEX MP method, the numerical results show the efficiency of the proposed variable step‐size extrapolated CN method.