This paper is set to analytically describe properties of the hyperbolic distribution. This law, along with the variance-gamma distribution, is one of the most popular normal mean–variance mixtures from the point of view of various applications. We have found closed form expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The obtained formulas use the values of the Humbert confluent hypergeometric and Whittaker special functions. The results are applied to the problem of European option pricing in the related Lévy model of financial market. The research demonstrates that the discussed normal mean–variance mixture is analytically tractable.
{"title":"On Properties of the Hyperbolic Distribution","authors":"Roman V. Ivanov","doi":"10.3390/math12182888","DOIUrl":"https://doi.org/10.3390/math12182888","url":null,"abstract":"This paper is set to analytically describe properties of the hyperbolic distribution. This law, along with the variance-gamma distribution, is one of the most popular normal mean–variance mixtures from the point of view of various applications. We have found closed form expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The obtained formulas use the values of the Humbert confluent hypergeometric and Whittaker special functions. The results are applied to the problem of European option pricing in the related Lévy model of financial market. The research demonstrates that the discussed normal mean–variance mixture is analytically tractable.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"2 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let C⊂Np be an integer polyhedral cone. An affine semigroup S⊂C is a C-semigroup if |C∖S|<+∞. This structure has always been studied using a monomial order. The main issue is that the choice of these orders is arbitrary. In the present work, we choose the order given by the semigroup itself, which is a more natural order. This allows us to generalise some of the definitions and results known from numerical semigroup theory to C-semigroups.
设 C⊂Np 是一个整数多面体圆锥。如果 |C∖S|<+∞，则仿射半群 S⊂C 是一个 C 半群。对这种结构的研究一直使用单项式阶。主要问题在于这些阶的选择是任意的。在本研究中，我们选择半群本身给出的阶，这是一种更自然的阶。这使我们能够将数字半群理论中的一些定义和结果推广到 C 半群。
{"title":"C-Semigroups and Their Induced Order","authors":"Daniel Marín-Aragón, Raquel Tapia-Ramos","doi":"10.3390/math12182889","DOIUrl":"https://doi.org/10.3390/math12182889","url":null,"abstract":"Let C⊂Np be an integer polyhedral cone. An affine semigroup S⊂C is a C-semigroup if |C∖S|<+∞. This structure has always been studied using a monomial order. The main issue is that the choice of these orders is arbitrary. In the present work, we choose the order given by the semigroup itself, which is a more natural order. This allows us to generalise some of the definitions and results known from numerical semigroup theory to C-semigroups.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"186 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem on the unbounded closed interval . Necessary and sufficient conditions for the existence and uniqueness of the solution are pointed out. Operator-valued moment problems and a scalar-valued moment problem are solved.
{"title":"Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation","authors":"Octav Olteanu","doi":"10.3390/math12182878","DOIUrl":"https://doi.org/10.3390/math12182878","url":null,"abstract":"We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem on the unbounded closed interval . Necessary and sufficient conditions for the existence and uniqueness of the solution are pointed out. Operator-valued moment problems and a scalar-valued moment problem are solved.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"46 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper explores fixed points for both contractive and non-contractive mappings in traditional b-metric spaces, preordered b-metric spaces, and random b-metric spaces. Our findings provide insights into the behavior of mappings under various constraints and extend our approach to include coincidence and common fixed-point theorems in these spaces. We present new examples and graphical representations for the first time, offering novel results and enhancing several related findings in the literature, while broadening the scope of earlier works of Ran and Reurings, Nieto and Rodríguez-López, Górnicki, and others.
本文探讨了传统二元空间、有序二元空间和随机二元空间中的收缩映射和非收缩映射的定点。我们的发现让我们深入了解了各种约束条件下的映射行为，并将我们的方法扩展到这些空间中的重合和共同定点定理。我们首次提出了新的示例和图形表示，提供了新的结果，并增强了文献中的一些相关发现，同时拓宽了 Ran 和 Reurings、Nieto 和 Rodríguez-López、Górnicki 等人早期著作的范围。
{"title":"Fixed-Point and Random Fixed-Point Theorems in Preordered Sets Equipped with a Distance Metric","authors":"Himanshu Baranwal, Ravindra Kishor Bisht, Arya Kumar Bedabrata Chand, Jen-Chih Yao","doi":"10.3390/math12182877","DOIUrl":"https://doi.org/10.3390/math12182877","url":null,"abstract":"This paper explores fixed points for both contractive and non-contractive mappings in traditional b-metric spaces, preordered b-metric spaces, and random b-metric spaces. Our findings provide insights into the behavior of mappings under various constraints and extend our approach to include coincidence and common fixed-point theorems in these spaces. We present new examples and graphical representations for the first time, offering novel results and enhancing several related findings in the literature, while broadening the scope of earlier works of Ran and Reurings, Nieto and Rodríguez-López, Górnicki, and others.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"9 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Over the past few years, there has been substantial interest and research activity surrounding the application of Convolutional Neural Networks (CNNs) for post-filtering in video coding. Most current research efforts have focused on using CNNs with various kernel sizes for post-filtering, primarily concentrating on High-Efficiency Video Coding/H.265 (HEVC) and Versatile Video Coding/H.266 (VVC). This narrow focus has limited the exploration and application of these techniques to other video coding standards such as AV1, developed by the Alliance for Open Media, which offers excellent compression efficiency, reducing bandwidth usage and improving video quality, making it highly attractive for modern streaming and media applications. This paper introduces a novel approach that extends beyond traditional CNN methods by integrating three different self-attention layers into the CNN framework. Applied to the AV1 codec, the proposed method significantly improves video quality by incorporating these distinct self-attention layers. This enhancement demonstrates the potential of self-attention mechanisms to revolutionize post-filtering techniques in video coding beyond the limitations of convolution-based methods. The experimental results show that the proposed network achieves an average BD-rate reduction of 10.40% for the Luma component and 19.22% and 16.52% for the Chroma components compared to the AV1 anchor. Visual quality assessments further validated the effectiveness of our approach, showcasing substantial artifact reduction and detail enhancement in videos.
{"title":"Multi-Type Self-Attention-Based Convolutional-Neural-Network Post-Filtering for AV1 Codec","authors":"Woowoen Gwun, Kiho Choi, Gwang Hoon Park","doi":"10.3390/math12182874","DOIUrl":"https://doi.org/10.3390/math12182874","url":null,"abstract":"Over the past few years, there has been substantial interest and research activity surrounding the application of Convolutional Neural Networks (CNNs) for post-filtering in video coding. Most current research efforts have focused on using CNNs with various kernel sizes for post-filtering, primarily concentrating on High-Efficiency Video Coding/H.265 (HEVC) and Versatile Video Coding/H.266 (VVC). This narrow focus has limited the exploration and application of these techniques to other video coding standards such as AV1, developed by the Alliance for Open Media, which offers excellent compression efficiency, reducing bandwidth usage and improving video quality, making it highly attractive for modern streaming and media applications. This paper introduces a novel approach that extends beyond traditional CNN methods by integrating three different self-attention layers into the CNN framework. Applied to the AV1 codec, the proposed method significantly improves video quality by incorporating these distinct self-attention layers. This enhancement demonstrates the potential of self-attention mechanisms to revolutionize post-filtering techniques in video coding beyond the limitations of convolution-based methods. The experimental results show that the proposed network achieves an average BD-rate reduction of 10.40% for the Luma component and 19.22% and 16.52% for the Chroma components compared to the AV1 anchor. Visual quality assessments further validated the effectiveness of our approach, showcasing substantial artifact reduction and detail enhancement in videos.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"15 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tanuj Joshi, Abhinav Gupta, Om Parkash, Ralph Kristoffer B. Gallegos, Nay Lin Oo, Gopal Krishan
The study establishes two mathematical formulations to predict the pressure drop in a solid–liquid slurry flowing through a straight pipe. Employing the Eulerian–Eulerian RNG k-ε model, the computational investigation uses water as the carrier fluid and glass beads as solid particles. The analysis spans various particle sizes (d50 = 75–175 μm), volumetric concentrations (Cvf = 10–50%), and velocities (Vm = 1–5 m/s). The first model, developed using the MATLAB curve-fitting tool, is complemented by a second empirical equation derived through non-polynomial mathematical formulation. Results from both models are validated against existing experimental and computational data, demonstrating accurate predictions for d50 = 75–175 µm particles within a Reynolds number range of 20,000 ≤ Re ≤ 320,000.
{"title":"Mathematical Formulations for Predicting Pressure Drop in Solid–Liquid Slurry Flow through a Straight Pipe Using Computational Modeling","authors":"Tanuj Joshi, Abhinav Gupta, Om Parkash, Ralph Kristoffer B. Gallegos, Nay Lin Oo, Gopal Krishan","doi":"10.3390/math12182879","DOIUrl":"https://doi.org/10.3390/math12182879","url":null,"abstract":"The study establishes two mathematical formulations to predict the pressure drop in a solid–liquid slurry flowing through a straight pipe. Employing the Eulerian–Eulerian RNG k-ε model, the computational investigation uses water as the carrier fluid and glass beads as solid particles. The analysis spans various particle sizes (d50 = 75–175 μm), volumetric concentrations (Cvf = 10–50%), and velocities (Vm = 1–5 m/s). The first model, developed using the MATLAB curve-fitting tool, is complemented by a second empirical equation derived through non-polynomial mathematical formulation. Results from both models are validated against existing experimental and computational data, demonstrating accurate predictions for d50 = 75–175 µm particles within a Reynolds number range of 20,000 ≤ Re ≤ 320,000.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"24 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Signifiable computability aims to separate what is theoretically computable from what is computable through performable processes on computers with finite amounts of memory. Real numbers and sequences thereof, data types, and instances are treated as finite texts, and memory limitations are made explicit through a requirement that the texts be stored in the available memory on the devices that manipulate them. In Part I of our investigation, we define the concepts of signification and reference of real numbers. We extend signification to number tuples, data types, and data instances and show that data structures representable as tuples of discretely finite numbers are signifiable. From the signification of real tuples, we proceed to the constructive signification of multidimensional matrices and show that any data structure representable as a multidimensional matrix of discretely finite numbers is signifiable.
{"title":"On Signifiable Computability: Part I: Signification of Real Numbers, Sequences, and Types","authors":"Vladimir A. Kulyukin","doi":"10.3390/math12182881","DOIUrl":"https://doi.org/10.3390/math12182881","url":null,"abstract":"Signifiable computability aims to separate what is theoretically computable from what is computable through performable processes on computers with finite amounts of memory. Real numbers and sequences thereof, data types, and instances are treated as finite texts, and memory limitations are made explicit through a requirement that the texts be stored in the available memory on the devices that manipulate them. In Part I of our investigation, we define the concepts of signification and reference of real numbers. We extend signification to number tuples, data types, and data instances and show that data structures representable as tuples of discretely finite numbers are signifiable. From the signification of real tuples, we proceed to the constructive signification of multidimensional matrices and show that any data structure representable as a multidimensional matrix of discretely finite numbers is signifiable.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hon Yiu So, Man Ho Ling, Narayanaswamy Balakrishnan
One-shot devices are products that can only be used once. Typical one-shot devices include airbags, fire extinguishers, inflatable life vests, ammo, and handheld flares. Most of them are life-saving products and should be highly reliable in an emergency. Quality control of those productions and predicting their reliabilities over time is critically important. To assess the reliability of the products, manufacturers usually test them in controlled conditions rather than user conditions. We may rely on public datasets that reflect their reliability in actual use, but the datasets often come with missing observations. The experimenter may lose information on covariate readings due to human errors. Traditional missing-data-handling methods may not work well in handling one-shot device data as they only contain their survival statuses. In this research, we propose Multiple Imputation with Unsupervised Learning (MIUL) to impute the missing data using Hierarchical Clustering, k-prototype, and density-based spatial clustering of applications with noise (DBSCAN). Our simulation study shows that MIUL algorithms have superior performance. We also illustrate the method using datasets from the Crash Report Sampling System (CRSS) of the National Highway Traffic Safety Administration (NHTSA).
一次性设备是只能使用一次的产品。典型的一次性装置包括安全气囊、灭火器、充气救生衣、弹药和手持信号弹。它们大多是救生产品，在紧急情况下应具有高度可靠性。对这些产品进行质量控制并预测其在一段时间内的可靠性至关重要。为了评估产品的可靠性，制造商通常会在受控条件下而非用户条件下对产品进行测试。我们可以依靠公共数据集来反映产品在实际使用中的可靠性，但这些数据集往往会丢失观测数据。实验人员可能会因为人为失误而丢失协变量读数信息。传统的缺失数据处理方法可能无法很好地处理一次性设备数据，因为这些数据只包含其存活状态。在这项研究中，我们提出了 "无监督学习多重估算"（Multiple Imputation with Unsupervised Learning，MIUL）方法，利用层次聚类、k-原型和基于密度的带噪声应用空间聚类（DBSCAN）来估算缺失数据。我们的模拟研究表明，MIUL 算法性能优越。我们还使用美国国家公路交通安全管理局（NHTSA）的碰撞报告采样系统（CRSS）数据集对该方法进行了说明。
{"title":"Imputing Missing Data in One-Shot Devices Using Unsupervised Learning Approach","authors":"Hon Yiu So, Man Ho Ling, Narayanaswamy Balakrishnan","doi":"10.3390/math12182884","DOIUrl":"https://doi.org/10.3390/math12182884","url":null,"abstract":"One-shot devices are products that can only be used once. Typical one-shot devices include airbags, fire extinguishers, inflatable life vests, ammo, and handheld flares. Most of them are life-saving products and should be highly reliable in an emergency. Quality control of those productions and predicting their reliabilities over time is critically important. To assess the reliability of the products, manufacturers usually test them in controlled conditions rather than user conditions. We may rely on public datasets that reflect their reliability in actual use, but the datasets often come with missing observations. The experimenter may lose information on covariate readings due to human errors. Traditional missing-data-handling methods may not work well in handling one-shot device data as they only contain their survival statuses. In this research, we propose Multiple Imputation with Unsupervised Learning (MIUL) to impute the missing data using Hierarchical Clustering, k-prototype, and density-based spatial clustering of applications with noise (DBSCAN). Our simulation study shows that MIUL algorithms have superior performance. We also illustrate the method using datasets from the Crash Report Sampling System (CRSS) of the National Highway Traffic Safety Administration (NHTSA).","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"116 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Xiu Shu, Feng Huang, Zhaobing Qiu, Xinming Zhang, Di Yuan
The limited availability of thermal infrared (TIR) training samples leads to suboptimal target representation by convolutional feature extraction networks, which adversely impacts the accuracy of TIR target tracking methods. To address this issue, we propose an unsupervised cross-domain model (UCDT) for TIR tracking. Our approach leverages labeled training samples from the RGB domain (source domain) to train a general feature extraction network. We then employ a cross-domain model to adapt this network for effective target feature extraction in the TIR domain (target domain). This cross-domain strategy addresses the challenge of limited TIR training samples effectively. Additionally, we utilize an unsupervised learning technique to generate pseudo-labels for unlabeled training samples in the source domain, which helps overcome the limitations imposed by the scarcity of annotated training data. Extensive experiments demonstrate that our UCDT tracking method outperforms existing tracking approaches on the PTB-TIR and LSOTB-TIR benchmarks.
热红外（TIR）训练样本的有限性导致卷积特征提取网络的目标表示不理想，从而对 TIR 目标跟踪方法的准确性产生不利影响。为解决这一问题，我们提出了一种用于 TIR 跟踪的无监督跨域模型 (UCDT)。我们的方法利用 RGB 域（源域）的标记训练样本来训练通用特征提取网络。然后，我们采用跨域模型来调整该网络，以便在 TIR 域（目标域）中有效提取目标特征。这种跨域策略有效地解决了 TIR 训练样本有限的难题。此外，我们还利用无监督学习技术为源域中未标注的训练样本生成伪标签，这有助于克服标注训练数据稀缺所带来的限制。大量实验证明，在 PTB-TIR 和 LSOTB-TIR 基准上，我们的 UCDT 跟踪方法优于现有的跟踪方法。
{"title":"Learning Unsupervised Cross-Domain Model for TIR Target Tracking","authors":"Xiu Shu, Feng Huang, Zhaobing Qiu, Xinming Zhang, Di Yuan","doi":"10.3390/math12182882","DOIUrl":"https://doi.org/10.3390/math12182882","url":null,"abstract":"The limited availability of thermal infrared (TIR) training samples leads to suboptimal target representation by convolutional feature extraction networks, which adversely impacts the accuracy of TIR target tracking methods. To address this issue, we propose an unsupervised cross-domain model (UCDT) for TIR tracking. Our approach leverages labeled training samples from the RGB domain (source domain) to train a general feature extraction network. We then employ a cross-domain model to adapt this network for effective target feature extraction in the TIR domain (target domain). This cross-domain strategy addresses the challenge of limited TIR training samples effectively. Additionally, we utilize an unsupervised learning technique to generate pseudo-labels for unlabeled training samples in the source domain, which helps overcome the limitations imposed by the scarcity of annotated training data. Extensive experiments demonstrate that our UCDT tracking method outperforms existing tracking approaches on the PTB-TIR and LSOTB-TIR benchmarks.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"16 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Single-machine due-window assignment scheduling with delivery times and variable processing times is investigated, where the variable processing time of a job means that the processing time is a function of its position in a sequence and its resource allocation. Currently, there are multiple optimization objectives for the due-window assignment problem, and there is a small amount of research on optimization problems where the window starting time, the rejected cost and the optimal scheduling are jointly required. The goal of this paper is to minimize the weighed sum of scheduling cost, resource consumption cost and outsourcing measure under the optional job outsourcing (rejection). Under two resource allocation models (i.e., linear and convex resource allocation models), the scheduling cost is the weighted sum of the number of early–tardy jobs, earliness–tardiness penalties and due-window starting time and size, where the weights are positional-dependent. The main contributions of this paper include the study and data simulation of single-machine scheduling with learning effects, delivery times and outsourcing cost. For the weighed sum of scheduling cost, resource consumption cost and outsourcing measure, we prove the polynomial solvability of the problem. Under the common and slack due-window assignments, through the theoretical analysis of the optimal solution, we reveal that four problems can be solved in O(n6) time, where n is the number of jobs.
研究了具有交货时间和可变处理时间的单机到期窗口分配调度，其中作业的可变处理时间意味着处理时间是其在序列中的位置和资源分配的函数。目前，到期窗口分配问题有多个优化目标，而对窗口起始时间、拒绝成本和优化调度共同要求的优化问题的研究较少。本文的目标是在可选工作外包（拒绝）的情况下，使调度成本、资源消耗成本和外包措施的权重之和最小化。在两种资源分配模型（即线性资源分配模型和凸资源分配模型）下，调度成本是早迟到作业数量、早迟到惩罚和到期窗口开始时间及大小的加权和，其中权重与位置有关。本文的主要贡献包括对具有学习效应、交货时间和外包成本的单机调度进行了研究和数据模拟。对于调度成本、资源消耗成本和外包成本的权重和，我们证明了问题的多项式可解性。在普通和松弛的到期窗口分配下，通过最优解的理论分析，我们发现四个问题可以在 O(n6) 时间内求解，其中 n 为作业数。
{"title":"Study on Single-Machine Common/Slack Due-Window Assignment Scheduling with Delivery Times, Variable Processing Times and Outsourcing","authors":"Bing Bai, Cai-Min Wei, Hong-Yu He, Ji-Bo Wang","doi":"10.3390/math12182883","DOIUrl":"https://doi.org/10.3390/math12182883","url":null,"abstract":"Single-machine due-window assignment scheduling with delivery times and variable processing times is investigated, where the variable processing time of a job means that the processing time is a function of its position in a sequence and its resource allocation. Currently, there are multiple optimization objectives for the due-window assignment problem, and there is a small amount of research on optimization problems where the window starting time, the rejected cost and the optimal scheduling are jointly required. The goal of this paper is to minimize the weighed sum of scheduling cost, resource consumption cost and outsourcing measure under the optional job outsourcing (rejection). Under two resource allocation models (i.e., linear and convex resource allocation models), the scheduling cost is the weighted sum of the number of early–tardy jobs, earliness–tardiness penalties and due-window starting time and size, where the weights are positional-dependent. The main contributions of this paper include the study and data simulation of single-machine scheduling with learning effects, delivery times and outsourcing cost. For the weighed sum of scheduling cost, resource consumption cost and outsourcing measure, we prove the polynomial solvability of the problem. Under the common and slack due-window assignments, through the theoretical analysis of the optimal solution, we reveal that four problems can be solved in O(n6) time, where n is the number of jobs.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"5 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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