Computers & Operations Research最新文献

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Characteristics-based Estimation of distribution algorithm for the steelmaking-refining-continuous casting scheduling problem in the real-world steel plants 炼钢-精炼-连铸调度问题的基于特征估计的分布算法
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-21 DOI: 10.1016/j.cor.2025.107107
Long Zhang, Xi Hu, XiaoMing Wu
{"title":"Characteristics-based Estimation of distribution algorithm for the steelmaking-refining-continuous casting scheduling problem in the real-world steel plants","authors":"Long Zhang,&nbsp;Xi Hu,&nbsp;XiaoMing Wu","doi":"10.1016/j.cor.2025.107107","DOIUrl":"10.1016/j.cor.2025.107107","url":null,"abstract":"<div><div>As a key production process in the steel industry, excellent scheduling of Steelmaking-refining-Continuous Casting (SCC) manufacturing process can improve production efficiency, shorten the steel production cycle, and reduce the production cost for steel enterprises. This paper presents a Characteristics-based Estimation of Distribution Algorithm (CEDA) for the SCC scheduling problem in the real-world steel plants. Considering the processing characteristics of the continuous casting machine, a novel caster-based encoding scheme and an improved decoding scheme are proposed. Also, a distance concept is introduced to mitigate the impact of similar individuals on the probability model, and an importance-based probability model updating mechanism is designed to increase the impact of excellent individual on the probability model. Furthermore, an individual sampling scheme with enhanced probability is constructed to ensure continuous processing of the continuous casting machine as much as possible. Finally, this paper designs a limited insertion operation in the local search to address the exploitation of the proposed algorithm. Extensive numerical simulations demonstrate that the proposed CEDA for the SCC scheduling process is more efficient than some state-of-the-art algorithms in the literature.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"181 ","pages":"Article 107107"},"PeriodicalIF":4.1,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dynamic inventory control and pricing strategies for perishable products considering both profit and waste 考虑利润与浪费的易腐产品动态库存控制与定价策略
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-19 DOI: 10.1016/j.cor.2025.107103
Melda Hasiloglu-Ciftciler , Onur Kaya
{"title":"Dynamic inventory control and pricing strategies for perishable products considering both profit and waste","authors":"Melda Hasiloglu-Ciftciler ,&nbsp;Onur Kaya","doi":"10.1016/j.cor.2025.107103","DOIUrl":"10.1016/j.cor.2025.107103","url":null,"abstract":"<div><div>With the increasing sustainability considerations throughout the world, there is an increasing interest in the effective management of perishable products both in the industry and the academia. There is a need to control the inventories, as well as the prices of perishable products in order to increase the profits while minimizing the waste. In this study, we focus on a retailer who sells old and new perishable food products, enabling demand shifts between products based on their prices and consumer behaviors. A bi-objective dynamic programming model is developed to optimize the discounted price, sale price, and order quantity of perishable food products in order to maximize the retailer’s profit and minimize food waste. We develop four static and dynamic pricing policies commonly practiced and quantify the advantages of dynamic pricing and price differentiation between old and new products in terms of both profit and waste. Our findings reveal that significant benefits can be obtained when the order quantity and the old product’s sale price decisions are given in a dynamic manner by considering the available inventory at hand. Additionally, this research analyzes the results of various weight combinations for profit and waste in the objective function. The findings highlight the significance of waste and sustainability concerns, underline the tradeoff between profit and waste and provide insights to companies to achieve improvements in their system results.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"181 ","pages":"Article 107103"},"PeriodicalIF":4.1,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143854650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Deep reinforcement learning-based resource allocation method for multi-satellite scheduling 基于深度强化学习的多卫星调度资源分配方法
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-17 DOI: 10.1016/j.cor.2025.107088
Xiaoyu Chen , Tian Tian , Guangming Dai , Maocai Wang , Zhiming Song , Lining Xing
{"title":"Deep reinforcement learning-based resource allocation method for multi-satellite scheduling","authors":"Xiaoyu Chen ,&nbsp;Tian Tian ,&nbsp;Guangming Dai ,&nbsp;Maocai Wang ,&nbsp;Zhiming Song ,&nbsp;Lining Xing","doi":"10.1016/j.cor.2025.107088","DOIUrl":"10.1016/j.cor.2025.107088","url":null,"abstract":"<div><div>Agile Earth observation satellites (AEOSs) scheduling represents a complex domain within combinatorial optimization, crucial for the regular operations and mission success of in-orbit satellites. In order to timelessly tackle the allocation of complex resources and corresponding time windows, a satellite resource adaptive allocation method, named SRADA-DRL, is proposed in this paper. By combining deep reinforcement learning (DRL) with rule-based heuristics, the SRADA-DRL is designed to optimize the allocation of satellite resources in dynamic environments. Concerning maximizing the total rewards of allocated missions, a mathematical model and a corresponding Markov decision model are constructed within the scheduling process. After analyzing the spatial–temporal distribution features of all resources and missions, the time-dependent missions are first decomposed into meta-missions corresponding to satellite resources, and a meta-mission is then selected to generate an allocation sequence in each stage. On this basis, the execution times for all missions are assigned in the single-satellite scheduling process. In which, the DRL updates the gradient information contingent upon the rewards garnered from the allocation sequence. In addition, the classical scheduling scenarios of varying scales are also conducted. Experimental results demonstrate the effectiveness and efficiency of the proposed SRADA-DRL method in addressing the AEOSs scheduling.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"181 ","pages":"Article 107088"},"PeriodicalIF":4.1,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143854652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Energy-efficient single-machine scheduling with group processing features under time-of-use electricity tariffs 分时电价下具有组加工特性的节能单机调度
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-17 DOI: 10.1016/j.cor.2025.107100
Shuaipeng Yuan, Bailin Wang, Yihan Pei, Tieke Li
{"title":"Energy-efficient single-machine scheduling with group processing features under time-of-use electricity tariffs","authors":"Shuaipeng Yuan,&nbsp;Bailin Wang,&nbsp;Yihan Pei,&nbsp;Tieke Li","doi":"10.1016/j.cor.2025.107100","DOIUrl":"10.1016/j.cor.2025.107100","url":null,"abstract":"<div><div>This work studies a novel single machine scheduling problem with group-processing features under time-of-use tariffs, which is derived from the realistic hot milling process in modern steel manufacturing industry. The objective is to minimize the total energy cost while adhering to a bounded maximum completion time. We first propose two mixed integer linear programming (MILP) models: a time-indexed MILP and a period-based MILP. Next, we analyze the problem’s properties and design a block-based dynamic programming algorithm. To solve instances of practical size, an improved iterative greedy algorithm is introduced. In the algorithm, a problem-specific heuristic is presented to construct an initial solution. Both block-based and job-based disruption and reconstruction strategies, along with six local search operators, are designed to direct the algorithm towards promising regions. Moreover, a deep search strategy based on a 0–1 programming model is developed to optimize the sequence of jobs within each price interval. Computational results indicate that: (i) the efficiency of the period-based MILP is superior to the time-indexed MILP; (ii) the dynamic programming algorithm exhibits higher performance in solving some small-scale instances compared to the period-based MILP; and (iii) the proposed algorithm is highly effective for both small- and large- scale instances, which can provide effective support for the production management of enterprises.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"181 ","pages":"Article 107100"},"PeriodicalIF":4.1,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143851584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Adjusted distributionally robust bounds on expected loss functions 期望损失函数的调整分布鲁棒界
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-16 DOI: 10.1016/j.cor.2025.107081
Yasemin Merzifonluoğlu , Joseph Geunes
{"title":"Adjusted distributionally robust bounds on expected loss functions","authors":"Yasemin Merzifonluoğlu ,&nbsp;Joseph Geunes","doi":"10.1016/j.cor.2025.107081","DOIUrl":"10.1016/j.cor.2025.107081","url":null,"abstract":"<div><div>Optimization problems in operations and finance often include a cost that is proportional to the expected amount by which a random variable exceeds some fixed quantity, known as the expected loss function. Representation of this function often leads to computational challenges, depending on the distribution of the random variable of interest. Moreover, in practice, a decision maker may possess limited information about this probability distribution, such as the mean and variance, but not the exact form of the associated probability density or distribution function. In such cases, a distributionally robust (DR) optimization approach seeks to minimize the maximum expected cost among all possible distributions that are consistent with the available information. Past research has recognized the overly conservative nature of this approach because it accounts for worst-case probability distributions that almost surely do not arise in practice. Motivated by this, we propose a DR approach that accounts for the worst-case performance with respect to a broad class of common continuous probability distributions, while producing solutions that are less conservative (and, therefore, less expensive, on average) than those produced by existing DR approaches in the literature. The methods we propose also permit approximation of the expected loss function for probability distributions under which exact representation of the function is difficult or impossible. Finally, we draw a connection between Scarf-type bounds from the literature, and mean-MAD (mean absolute deviation) bounds when MAD information is available in addition to variance.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"181 ","pages":"Article 107081"},"PeriodicalIF":4.1,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143847779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Capacitated profitable tour problem with cross-docking 交叉对接的有能力盈利旅游问题
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-12 DOI: 10.1016/j.cor.2025.107077
Pengfei He , Wenchong Chen , Qinghua Wu , Fengjun Xiao
{"title":"Capacitated profitable tour problem with cross-docking","authors":"Pengfei He ,&nbsp;Wenchong Chen ,&nbsp;Qinghua Wu ,&nbsp;Fengjun Xiao","doi":"10.1016/j.cor.2025.107077","DOIUrl":"10.1016/j.cor.2025.107077","url":null,"abstract":"<div><div>This paper addresses a real-world transportation problem arising from Industrial Internet platforms, where logistics companies selectively respond to requests for shipping products from manufacturers to customers. We formulate the problem as the capacitated profitable tour problem with cross-docking (CPTPC), which involves not only the selection of requests based on profit, but also the planning of vehicle routes with respect to capacitated constraints. The CPTPC, a generalization of the profitable tour problem and the vehicle routing problem with cross-docking, presents significant computational complexity. In this paper, we propose an effective hybrid genetic algorithm (HGA) tailored to address the problem. The algorithm integrates a dedicated two-level edge assembly crossover operator to generate promising offspring solutions. Additionally, it incorporates a streamlined technique-driven local search approach to improve each solution. Empirical evaluations showcase the robust performance of the algorithm on benchmark instances, and experimental analyses provide insights into the key search components inherent in the proposed algorithm. In addition, we conduct a case study to assess the practical utility of our HGA in improving the operational efficiency and profitability of logistics companies.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"181 ","pages":"Article 107077"},"PeriodicalIF":4.1,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143838951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A matheuristic for complex pricing problems: An application to rentable resources 复杂定价问题的数学方法:可租用资源的应用
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-09 DOI: 10.1016/j.cor.2025.107083
Kristina Bayer, Robert Klein
{"title":"A matheuristic for complex pricing problems: An application to rentable resources","authors":"Kristina Bayer,&nbsp;Robert Klein","doi":"10.1016/j.cor.2025.107083","DOIUrl":"10.1016/j.cor.2025.107083","url":null,"abstract":"<div><div>We consider the problem of a service provider who offers resources (such as equipment or accommodation) for rent that are substitutable and renewable over time. The provider aims to set static, yet time-dependent prices to maximize revenue while adhering to business-specific pricing rules. As customers arrive consecutively, they base their rental decisions on their willingness to pay, the prices set by the provider, and resource availability, leading to dynamic substitution.</div><div>To solve the problem, we propose a mixed-integer linear program (MIP) for a given stream of customers and different price constraints. The problem is difficult to solve because it requires modeling customers’ choices and resource availability over the course of time and also includes many prices that are closely intertwined by price constraints, constituting a complex pricing system. When developing heuristics, applying construction or improvement approaches becomes difficult because knowing all prices is necessary to evaluate a customer stream. Therefore, we develop a matheuristic based on the destroy/repair paradigm. To regain feasibility in the repair step, our approach uses a sub-MIP, which can easily consider different price constraints. As a benchmark, we also implement an enumeration-based approach.</div><div>We conduct a comprehensive computational study that covers 198 different instance classes of realistic size considering various price constraints. The study findings indicate that the new MIP-based heuristic outperforms the enumeration-based approach and a standard solver when applied to the initial MIP formulation.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"181 ","pages":"Article 107083"},"PeriodicalIF":4.1,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143854651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Improved algorithm for minimizing total late work on a proportionate flow shop and extensions to job rejection and generalized due dates 改进了比例流水车间中最小化总延迟工作的算法,并扩展到工作拒绝和广义截止日期
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-07 DOI: 10.1016/j.cor.2025.107046
Baruch Mor , Xin-Na Geng
{"title":"Improved algorithm for minimizing total late work on a proportionate flow shop and extensions to job rejection and generalized due dates","authors":"Baruch Mor ,&nbsp;Xin-Na Geng","doi":"10.1016/j.cor.2025.107046","DOIUrl":"10.1016/j.cor.2025.107046","url":null,"abstract":"<div><div>Gerstl et al (2019) studied the problem of minimizing the total late work (TLW) on an <span><math><mi>m</mi></math></span>-machine proportionate flow shop. They solved the case where the total late work refers to the last operation of the job (i.e., the operation performed on the last machine of the flow shop). As the problem is known to be NP-hard, the authors proved two crucial properties of an optimal schedule and introduced a pseudo-polynomial dynamic programming (DP) algorithm. In this research, we revisit the same problem and present enhanced algorithms by the factor of <span><math><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo></mrow></math></span>, where <span><math><mi>n</mi></math></span> is the number of jobs and <span><math><mi>m</mi></math></span> is the number of machines. Furthermore, based on the improved algorithm, we extend the fundamental problem to consider optional job rejection. We focus on minimizing the TLW subject to an upper bound on the total rejection cost and introduce DP algorithms. Next, we address the problem of minimizing the TLW with generalized due dates, with an upper bound on the permitted rejection cost, and likewise introduce DP algorithms. We conducted an extensive numerical study to evaluate the efficiency of all DP algorithms.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"181 ","pages":"Article 107046"},"PeriodicalIF":4.1,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Deep reinforcement learning for solving efficient and energy-saving flexible job shop scheduling problem with multi-AGV 基于深度强化学习的多agv柔性作业车间高效节能调度问题
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-04 DOI: 10.1016/j.cor.2025.107087
Weiyao Cheng , Chaoyong Zhang , Leilei Meng , Biao Zhang , Kaizhou Gao , Hongyan Sang
{"title":"Deep reinforcement learning for solving efficient and energy-saving flexible job shop scheduling problem with multi-AGV","authors":"Weiyao Cheng ,&nbsp;Chaoyong Zhang ,&nbsp;Leilei Meng ,&nbsp;Biao Zhang ,&nbsp;Kaizhou Gao ,&nbsp;Hongyan Sang","doi":"10.1016/j.cor.2025.107087","DOIUrl":"10.1016/j.cor.2025.107087","url":null,"abstract":"<div><div>The flexible job shop scheduling problem with multi-automatic guided vehicles (FJSP-AGV) exists widely in the industrial field. To enhance production efficiency and conserve energy, efficient and energy-saving FJSP-AGV is studied. Three optimization tasks: optimizing the makespan, optimizing the total energy consumption (TEC), and simultaneously optimizing the makespan and TEC are solved. To address these challenges, a deep reinforcement learning (DRL) framework is developed. Specifically, in the Markov decision process of the scheduling agent, twelve features are extracted from the shop floor, and sixteen composite scheduling rules are used as the action space. Based on the three optimization tasks, two single-reward functions and a weighted comprehensive reward function are presented. Additionally, the deep Q-network algorithm is used to train the scheduling agent. Comprehensive experiments are conducted on 98 test instances to evaluate the performance of the proposed method. The experiment results demonstrate its effectiveness compared to composite scheduling rules, exact methods, <em>meta</em>-heuristic methods, and other DRL methods.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"181 ","pages":"Article 107087"},"PeriodicalIF":4.1,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143815601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Integrated electric ground vehicle and drone with blockchain-driven approach for routing delivery 集成电动地面车辆和无人机,采用区块链驱动的路线交付方法
IF 4.1 2区 工程技术
Computers & Operations Research Pub Date : 2025-04-02 DOI: 10.1016/j.cor.2025.107057
Ramdhan Nugraha , Adriansyah Dwi Rendragraha , Soo Young Shin
{"title":"Integrated electric ground vehicle and drone with blockchain-driven approach for routing delivery","authors":"Ramdhan Nugraha ,&nbsp;Adriansyah Dwi Rendragraha ,&nbsp;Soo Young Shin","doi":"10.1016/j.cor.2025.107057","DOIUrl":"10.1016/j.cor.2025.107057","url":null,"abstract":"<div><div>This study explores the integration of electric ground vehicles (EGVs) and drones within the logistics sector to address environmental and operational challenges in transportation. By introducing a novel multi-cooperative EV routing problem with flexible drones (MCEVRPFD), the research leverages both technologies to enhance delivery efficiency and extend the operational range of EGVs. Additionally, the study highlights the potential of blockchain technology to ensure efficiency, security, and transparency in supply chain management, enhancing operational reliability and supporting sustainability. The findings suggest that the combined use of EGVs, drones, and blockchain can revolutionize logistics by offering more sustainable, efficient, and transparent solutions, thereby increasing consumer trust and satisfaction.</div></div>","PeriodicalId":10542,"journal":{"name":"Computers & Operations Research","volume":"180 ","pages":"Article 107057"},"PeriodicalIF":4.1,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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