Operations Research Perspectives最新文献_第2页

Unified tail assignment and maintenance task scheduling: A decision support framework for improved efficiency and stability 统一的机尾分配和维护任务调度：提高效率和稳定性的决策支持框架

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-11-19 DOI: 10.1016/j.orp.2025.100363

Luigi Pescio, Marta Ribeiro, Bruno F. Santos

Flight and maintenance scheduling pose conflicting objectives: while maintenance is vital for ensuring aircraft airworthiness, it comes at the cost of taking aircraft out of operation. In current operations, airlines manually handle tail assignment and maintenance task scheduling separately, missing an opportunity to strike a better balance. This division leads to wasted maintenance resources, restricted fleet availability for schedule flexibility, inconsistent planning, and neglect of schedule resilience. This study presents a novel approach that integrates tail assignment and maintenance scheduling into a unified decision-support framework. An integer program, tailored to meet airline-specific requirements and constraints, is combined with an innovative time-space network (TSN). The TSN incorporates two distinct spaces for maintenance and network activities. The primary objective is to generate feasible plans that increase schedule efficiency (i.e., no cancellations, high fleet availability, high fleet health, and optimal use of maintenance resources) and schedule stability (i.e., limited number of late arrival disruptions during operations) the day before operation. Additionally, this framework addresses overlooked aspects in the literature: it treats maintenance tasks as variable interval activities based on aircraft-specific needs, departing from the traditional fixed interval approach. The performance of the framework is tested with real-data provided by a major European single hub-to-spoke airline, with a heterogeneous fleet of over 50 wide-body aircraft. Historical data from arrival delays is used to create robust buffers that mitigate delay propagation. A 17% reduction in maintenance time was achieved compared to the airline’s current plans, resulting in a 10% increase in fleet availability on the day of operations. This improvement is attributed to higher labour and task interval utilization, indicating the framework’s superior efficiency in scheduling maintenance tasks. Lastly, the framework produced plans more resilient to arrival delays, reducing the number of disruptions and delay propagation over 40%. This framework can be used as a decision-support tool for airlines, enabling the creation of schedules that are both robust against delays and optimized for fleet utilization.

飞行和维修计划带来了相互冲突的目标：虽然维修对确保飞机适航至关重要，但它的代价是让飞机停止运行。在目前的操作中，航空公司手动处理机尾分配和维修任务调度，失去了一个更好地平衡的机会。这种划分导致浪费维护资源，限制机队对时间表灵活性的可用性，不一致的计划，以及忽视时间表弹性。本研究提出了一种将机尾分配和维修计划整合到统一决策支持框架中的新方法。为满足航空公司的特定要求和限制而量身定制的整数方案与创新的时空网络（TSN）相结合。TSN包含两个不同的空间，用于维护和网络活动。主要目标是生成可行的计划，以提高运行前一天的调度效率（即无取消、高机队可用性、高机队健康状况和维护资源的最佳使用）和调度稳定性（即运行期间延迟到达中断的数量有限）。此外，该框架解决了文献中被忽视的方面：它将维护任务视为基于飞机特定需求的可变间隔活动，与传统的固定间隔方法不同。该框架的性能通过欧洲一家大型单一枢纽到辐航空公司提供的真实数据进行了测试，该航空公司拥有50多架宽体飞机的异构机队。来自到达延迟的历史数据用于创建鲁棒缓冲器，以减轻延迟传播。与航空公司目前的计划相比，维修时间减少了17%，从而使运营当天的机队可用性增加了10%。这种改进归因于更高的劳动力和任务间隔利用率，表明该框架在调度维护任务方面具有更高的效率。最后，该框架制定的计划对到达延迟更具弹性，将中断和延迟传播的数量减少了40%以上。该框架可以用作航空公司的决策支持工具，使其能够创建既能抵御延误又能优化机队利用率的时间表。

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引用次数: 0

Reinforcement learning for solving the pricing problem in column generation for routing 用于解决路由列生成中定价问题的强化学习

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-11-12 DOI: 10.1016/j.orp.2025.100364

Abdo Abouelrous , Laurens Bliek , Adriana F. Gabor , Yaoxin Wu , Yingqian Zhang

In this paper, we address the problem of Column Generation (CG) for routing problems using Reinforcement Learning (RL). Specifically, we use a RL model based on the attention-mechanism architecture to find the columns with most negative reduced cost in the Pricing Problem (PP). Unlike previous Machine Learning (ML) applications for CG, our model deploys an end-to-end mechanism that independently solves the pricing problem without the help of any heuristic. We consider a variant of Vehicle Routing Problem (VRP) as a case study for our method. Through a series of experiments comparing our approach with a Dynamic Programming (DP)-based heuristic for solving the PP, we demonstrate that the proposed method obtains solutions for the linear relaxation up to a reasonable objective gap and significantly faster than the DP-based heuristic for the PP.

在本文中，我们使用强化学习（RL）解决了路由问题的列生成（CG）问题。具体来说，我们使用基于注意力机制架构的RL模型来寻找定价问题（PP）中负降低成本最多的列。与以前的CG机器学习（ML）应用程序不同，我们的模型部署了一个端到端机制，可以独立解决定价问题，而无需任何启发式的帮助。我们考虑了车辆路径问题（VRP）的一个变体作为我们方法的案例研究。通过与基于动态规划（DP）的启发式方法求解PP的一系列实验比较，我们证明了所提出的方法在合理的客观间隙内获得线性松弛的解，并且比基于DP的启发式方法求解PP的速度快得多。

引用次数: 0

A Probabilistic and adaptive strategy for the newsvendor problem with periodic demand 具有周期性需求的报贩问题的概率自适应策略

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-11-12 DOI: 10.1016/j.orp.2025.100365

Hui Yu , Yu Gong , Xiaoli Yan

The newsvendor problem with periodic demand (PFNV) is a common and significant challenge in practice, where traditional methods such as optimization, statistical analysis, and artificial intelligence often struggle to balance effectiveness and operability. We propose the Probability-based Adaptive Strategy (PAS) for the PFNV problem, which formulates decisions through the dual reference points and probabilities. The decision-making process comprises four steps that simulate human behavior based on bounded rationality. The design of reference points is data-driven, using either a linear method or a multi-armed bandit (MAB), while probability calculation is guided by an optimization objective that reflects human regret psychology. The final decision is made through either a random sampling (RS) or an expectation construction (EC) scheme. Experiments with both simulated and real-world data show that PAS effectively captures periodic trends in both stable and volatile datasets. The PAS combining classification, MAB, and the EC scheme performs better in average cost in most cases, while other variants exhibit different characteristics under varying conditions. Compared with several benchmarks, PAS demonstrates potential for cost optimization in certain scenarios while maintaining both operability and interpretability.

具有周期性需求的报贩问题（PFNV）在实践中是一个常见而重大的挑战，传统的方法如优化、统计分析和人工智能往往难以平衡有效性和可操作性。针对PFNV问题，提出了基于概率的自适应策略（Probability-based Adaptive Strategy， PAS），该策略通过双重参考点和概率来制定决策。决策过程包括四个步骤，模拟基于有限理性的人类行为。参考点的设计是数据驱动的，采用线性法或多臂强盗法（MAB），而概率计算则以反映人类后悔心理的优化目标为指导。通过随机抽样（RS）或期望构造（EC）方案做出最终决定。模拟和真实数据的实验表明，PAS有效地捕获了稳定和不稳定数据集的周期性趋势。结合分类、MAB和EC方案的PAS在大多数情况下的平均成本表现较好，而其他变体在不同条件下表现出不同的特征。与几个基准相比，PAS显示了在某些情况下成本优化的潜力，同时保持了可操作性和可解释性。

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引用次数: 0

Customer order scheduling in a permutation flow shop environment 排列流车间环境中的客户订单调度

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-11-03 DOI: 10.1016/j.orp.2025.100362

Julius Hoffmann , Janis S. Neufeld , Udo Buscher

Various recent scheduling literature has studied the so called customer order scheduling problem. In this class of scheduling problems, there are multiple customer orders, and each of them consists of several jobs. The order finishes and is ready to be shipped when the last job of the order finishes. In this paper, we consider the customer order scheduling problem in a permutation flow shop environment with

m

machines. There are

n

orders and each order has

o

jobs. The objective is to minimize the total completion time of the orders. We present multiple problem properties and a MINLP formulation of the problem. Furthermore, four heuristics which follow the Iterated Greedy Algorithm are developed. In a computational experiment, we evaluated the four heuristics on their practicability. They showed good results in short calculation time when compared with the MINLP solution from a solver. Afterwards, we compared the four heuristics with each other for different problem sizes.

最近的各种调度文献都研究了所谓的客户订单调度问题。在这类调度问题中，有多个客户订单，每个订单由几个作业组成。当订单的最后一个作业完成时，订单完成并准备发货。本文研究了有m台机器的置换流车间环境下的客户订单调度问题。有n个订单，每个订单有0个工作。目标是最小化订单的总完成时间。我们提出了该问题的多个性质和一个MINLP公式。在此基础上，提出了迭代贪心算法的四种启发式算法。在计算实验中，我们评估了四种启发式的实用性。与求解器的MINLP解决方案相比，它们在较短的计算时间内显示出良好的结果。之后，我们对不同问题规模的四种启发式进行了比较。

引用次数: 0

Inventory prepositioning of relief material under the Joint Government-Enterprise Storage mode 政企联储模式下的救灾物资库存预配置

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-11-01 DOI: 10.1016/j.orp.2025.100361

Li Zhang, Jianqin Zhou, Xufeng Yang

To ensure the timely supply of relief materials at a low cost, many countries have adopted the Joint Government and Enterprises Storage (JGES) mode to prepositioning relief materials, where some enterprises replace the government in stockpiling emergency supplies for disasters. A critical problem faced by the enterprise is how to manage its inventory considering its daily business demand and the possible emergency demand. The government also wants to know the performance of the mode and how to subsidize the enterprise. To address these questions, we first consider the single-period problem and formulate it as a newsvendor-type model. We obtain the optimal conditions and analyze the impacts of some parameters on the optimal policy. Furthermore, we consider the multi-period case and the government’s optimal subsidy for the enterprise. For the former, we show that the optimal inventory policy is still the base-stock policy if the fixed ordering cost is zero, and is the

(s, S)

policy if the cost is positive. The government’s subsidy to the firm increases first and then decreases as the occurrence probability of the emergency increases. Finally, we conduct numerical experiments to compare the performance of the mode with that of the Separate Government-Enterprise Storage (SGES) mode, to demonstrate its advantages and the impacts of some parameters on its performance.

为了保证救灾物资的及时、低成本供应，许多国家都采取了政府与企业联合储备（JGES）的方式来预置救灾物资，由一些企业代替政府储备救灾应急物资。考虑到企业的日常业务需求和可能出现的紧急需求，如何对库存进行管理是企业面临的一个关键问题。政府也想知道这种模式的效果以及如何对企业进行补贴。为了解决这些问题，我们首先考虑单周期问题，并将其表述为一个报贩类型的模型。得到了最优条件，并分析了一些参数对最优策略的影响。此外，我们还考虑了多时期的情况和政府对企业的最优补贴。对于前者，我们证明了当固定订购成本为零时，最优库存策略仍然是基本库存策略；当固定订购成本为正时，最优库存策略是（s, s）策略。随着突发事件发生概率的增加，政府对企业的补贴先增加后减少。最后，通过数值实验对该模式与政企分离存储模式的性能进行了比较，论证了该模式的优点以及一些参数对其性能的影响。

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引用次数: 0

A meta-analysis of set partitioning/set covering based matheuristics for vehicle routing problems 基于集划分/集覆盖的车辆路径问题数学元分析

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-10-29 DOI: 10.1016/j.orp.2025.100357

Alejandro Arenas-Vasco , Daniela Alcázar , Juan G. Villegas

The Vehicle Routing Problem (VRP) is a cornerstone of operations research with broad real-world relevance. As variants increase in complexity, researchers increasingly adopt metaheuristics and matheuristics to find high-quality solutions. We focus on set partitioning (SP) and set covering (SC) formulations as enhancement mechanisms in matheuristics for VRPs. These methods exploit previously generated routes by decomposing and recombining solutions, either as a post-optimization step or iteratively. We conducted a meta-analysis of 30 implementations, selected from 54 eligible studies identified via systematic review of Web of Science and Scopus, complemented by backward snowballing. Using a random-effects model, we quantified the effect of SP/SC enhancements on solution quality. On average, SP/SC improved solutions by 0.51% (95% CI: 0.41%–0.61%). Although the numerical gains are modest, they are consistent and significant, highlighting the practical value of these classical formulations in hybrid heuristic frameworks.

车辆路径问题（VRP）是运筹学的基础，具有广泛的现实意义。随着变量复杂性的增加，研究人员越来越多地采用元启发式和数学来寻找高质量的解决方案。我们关注集划分（SP）和集覆盖（SC）公式作为vrp数学中的增强机制。这些方法通过分解和重组解决方案来利用先前生成的路线，要么作为后优化步骤，要么迭代。我们对从Web of Science和Scopus系统综述中选出的54项符合条件的研究进行了30项实施的荟萃分析，并辅以反向滚雪球法。使用随机效应模型，我们量化了SP/SC增强对溶液质量的影响。SP/SC平均改善溶液0.51% （95% CI: 0.41%-0.61%）。虽然数值上的收益是适度的，但它们是一致的和重要的，突出了这些经典公式在混合启发式框架中的实用价值。

引用次数: 0

Multi commodity network design problem with minimum flow constraints 具有最小流量约束的多商品网络设计问题

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-10-17 DOI: 10.1016/j.orp.2025.100359

Luuk van Rijthoven

This research presents a fast heuristic method for solving large-scale real-life Stock Rebalancing problems with minimum transfer constraints on the arcs, as well as a maximum supply and demand limitation on the nodes, which can be considered as a variation of the multi-commodity network design (MCND) problems. The proposed Rank-based Greedy Heuristic with Swapping (RGHS) ranks all feasible flow combinations according to a profit criteria. Then, the algorithm greedily considers the combinations until demand and supply constraints are met, followed by a flow swapping mechanism to further improve the solution. Furthermore, the RGHS is extended to a Reduced Set Hybrid Model (RSHM) that combines the heuristic approach with a commercial solver on the reduced solution space. The proposed methods are evaluated against the published Modified Greedy (MG) algorithm that showed good results on benchmark instances with a significantly improved computation time compared to other state-of-the-art methods. This study contributes by proposing a fast algorithm tailored for the real-life instances on considerably larger instances compared to existing literature, and introduces the concept of minimum transfer restrictions in contrast to the more common maximum capacities. This paper reports the results on various large-scale real-life instances and larger simulated instances and shows the scalability and solution quality compared to existing methods.

本文提出了一种快速的启发式方法，用于求解具有最小转移约束和节点最大供需限制的大规模现实库存再平衡问题，该方法可视为多商品网络设计（MCND）问题的一种变化。提出了基于排序的贪心交换启发式算法（RGHS），该算法根据盈利标准对所有可行的流组合进行排序。然后，算法贪婪地考虑组合，直到满足需求和供给约束，然后通过流交换机制进一步改进解。进一步，将RGHS扩展为一种将启发式方法与简化解空间上的商业求解器相结合的简化集混合模型（RSHM）。针对已发表的Modified Greedy （MG）算法对所提出的方法进行了评估，该算法在基准实例上显示出良好的结果，与其他最先进的方法相比，计算时间显着提高。与现有文献相比，本研究提出了一种针对更大实例的现实实例量身定制的快速算法，并引入了最小传输限制的概念，而不是更常见的最大容量。本文报告了在各种大规模现实实例和更大的模拟实例上的结果，并与现有方法相比显示了可扩展性和解决方案的质量。

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引用次数: 0

Recursive delivery multiple flying sidekicks traveling salesman problem: An enlightenment of the Covid-19 pandemic 递归配送多飞伴旅行商问题：新冠肺炎疫情的启示

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-10-16 DOI: 10.1016/j.orp.2025.100360

Fatemeh Jamshidian, Saeed Yaghoubi, Mohammad Sadeghi

The last-mile delivery of COVID-19 vaccines required a structured timeframe, particularly for double-dose vaccines, where follow-up deliveries had to occur after vaccine type-specific intervals. This recurring service pattern inspired the development of a novel recursive delivery concept—where services must be repeated based on prior deliveries and elapsed time. We define recursive delivery as a multi-period mechanism that dynamically establishes whether, when, and how each customer should be revisited, built upon both service history and time-based constraints. To capture such recursive service patterns, this paper introduces a coordinated truck-drone delivery system, where either vehicle may recursively revisit previously serviced locations depending on the type of service—single or recursive— provided in earlier periods. To formalize this concept, we present a new variant of the Traveling Salesman Problem, termed the Recursive delivery multiple Flying Sidekicks Traveling Salesman Problem

(R - m F S T S P)

. This model extends traditional

T S P

by incorporating dynamic, service-type-dependent revisit scheduling. The

R - m F S T S P

has wide applicability in various domains requiring structured, time-sensitive service repetition, such as maintenance, health services, and supply replenishment. We formulate the

R - m F S T S P

as a mixed-integer linear programming model aimed at minimizing total transportation costs. Given the computational limitations of exact solvers for large-scale instances, a tailored metaheuristic algorithm has been developed to address the structural characteristics of the proposed

R - m F S T S P

. Its performance has been benchmarked against results from a relevant study, demonstrating competitive outcomes. Furthermore, a lower bound is provided to evaluate the quality of the obtained solutions.

COVID-19疫苗的最后一英里交付需要一个结构化的时间框架，特别是对于双剂量疫苗，后续交付必须在疫苗特定类型间隔之后进行。这种循环服务模式激发了一种新的递归交付概念的开发，即必须根据先前的交付和经过的时间来重复服务。我们将递归交付定义为一种多周期机制，它基于服务历史和基于时间的约束，动态地确定是否、何时以及如何重新访问每个客户。为了捕获这种递归服务模式，本文引入了一种协调的卡车-无人机交付系统，其中任何一辆车都可以根据早期提供的服务类型递归地重新访问先前服务的位置——单次或递归。为了形式化这个概念，我们提出了旅行商问题的一个新变体，称为递归交付多飞伙伴旅行商问题（R - mFSTSP）。该模型通过合并动态的、依赖于服务类型的重访调度来扩展传统的TSP。R - mFSTSP广泛适用于各种需要结构化、时间敏感的服务重复的领域，如维护、卫生服务和补给。我们将R - mfstspa表述为一个混合整数线性规划模型，旨在使总运输成本最小化。考虑到大规模实例的精确解算器的计算限制，我们开发了一种定制的元启发式算法来解决所提出的R - mFSTSP的结构特征。它的表现以一项相关研究的结果为基准，展示了具有竞争力的结果。并给出了一个下界来评价所得到的解的质量。

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引用次数: 0

A two-period model of counterterrorism with terrorist sponsoring and the role of hatred 恐怖分子支持和仇恨作用下的两期反恐模式

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-10-15 DOI: 10.1016/j.orp.2025.100358

Kjell Hausken

A two-period counterterrorism model is developed. One special case without terrorist sponsoring enables the government to degrade the terrorist’s resources substantially, especially with low unit counterterrorism cost or high asset valuations. If the terrorist in a second special case is sponsored proportionally to its resources, the government increases its counterterrorism in both periods, even at a higher unit cost. Degrading the terrorist’s resources becomes more challenging. If the terrorist in a third special case is sponsored proportionally to the government’s counterterrorism, due to developing hatred, and proportionally to its resources, counterterrorism is high in both periods, even at a high unit cost. Two contrasts with the two other special cases are that the terrorist’s resources are U shaped in both the government’s unit counterterrorism cost and their asset valuations. Both low unit counterterrorism cost and high asset valuations cause high counterterrorism and substantial terrorist sponsoring due to hatred. The government’s counterterrorism is inverse V shaped in the proportionality parameter for how counterterrorism causes hatred and terrorist sponsoring. An intermediate parameter causes maximum counterterrorism where the government strikes a balance between degrading the terrorist’s resources while constraining terrorist sponsoring due to hatred. The three special cases illustrate the government’s counterterrorism dilemmas.

建立了一个两期反恐模型。一个没有恐怖分子资助的特殊情况使政府能够大幅降低恐怖分子的资源，特别是在单位反恐成本低或资产估值高的情况下。如果在第二个特殊情况下，恐怖分子得到的资助与其资源成比例，政府就会在两个时期都加大反恐力度，即使单位成本更高。削弱恐怖分子的资源变得更具挑战性。如果在第三个特殊情况下，恐怖分子得到的资助与政府的反恐行动成比例，由于仇恨的发展，与政府的资源成比例，反恐在两个时期都是高的，即使单位成本很高。与其他两种特殊情况的两个对比是，恐怖分子的资源在政府的单位反恐成本和资产估值中都是U形的。低单位反恐成本和高资产估值导致高反恐和大量仇恨恐怖赞助。在反恐怖主义如何导致仇恨和恐怖主义赞助的比例参数中，政府的反恐怖主义是反V形的。中间参数导致最大程度的反恐怖主义，即政府在减少恐怖分子的资源和限制因仇恨而资助恐怖分子之间取得平衡。这三个特殊案例说明了政府的反恐困境。

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引用次数: 0

Cloud seeding optimization under uncertainty: A Markov chain approach in a two-stage fuzzy-stochastic framework 不确定性下的播云优化：两阶段模糊随机框架下的马尔可夫链方法

IF 3.7 4区管理学 Q2 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Perspectives

Pub Date : 2025-10-10 DOI: 10.1016/j.orp.2025.100356

Mohammad Sadeghi, Saeed Yaghoubi

The occurrence of sequential droughts and various forms of water shortages globally underscores the urgent need for sustainable water management solutions. In this context, cloud seeding has gained attention for its potential to enhance precipitation, yet its effectiveness is often uncertain due to complex cloud microphysics and atmospheric conditions. Acknowledging the inherent uncertainty in this endeavor, in this study, we employ a two-stage stochastic framework, integrating strategic decisions (facility location and network design) and operational realizations (seeding planning according to storm trajectories). Additionally, our model also considers fuzzy nature of seeding parameters. Above all, we develop a Markov chain procedure to mathematically model the prediction of expected increase in precipitation across cloud seeding decision-making processes. The integration of these stochastic methods into existing deterministic models from the literature results in a multi-objective Mixed-Integer Linear Programming (MILP) model designed to maximize rain probability and coverage while minimizing system-wide costs. To enhance the scalability and efficiency of the model, valid inequalities are developed to reduce the domain of binary variables. Additionally, a Lagrangian relaxation technique is proposed, yielding exact optimal solutions within reasonable timeframes and facilitating the handling of continuous space instances. Finally, a real-world case study in Iran demonstrates significant enhancements in precipitation predictions, with the Markov chain procedure showing an average 55 % increase in expected rain probability based on optimized seeding decisions. Scenario-based stochastic programming yields an 11.7 % value of stochastic solution and 16.5 % expected value of perfect information for cloud seeding initiatives.

全球接连发生的干旱和各种形式的水资源短缺突出表明迫切需要可持续的水管理解决办法。在这种情况下，人工降雨因其增强降水的潜力而受到关注，但由于复杂的云微物理和大气条件，其有效性往往不确定。考虑到这一努力中固有的不确定性，在本研究中，我们采用了一个两阶段的随机框架，整合了战略决策（设施选址和网络设计）和业务实现（根据风暴轨迹进行播种规划）。此外，我们的模型还考虑了种子参数的模糊性。最重要的是，我们开发了一个马尔可夫链过程来数学模拟在云播决策过程中预期降水增加的预测。将这些随机方法整合到现有的确定性模型中，得到一个多目标混合整数线性规划（MILP）模型，该模型旨在最大化降雨概率和覆盖范围，同时最小化系统范围的成本。为了提高模型的可扩展性和效率，提出了有效的不等式来减少二元变量的定义域。此外，提出了一种拉格朗日松弛技术，在合理的时间范围内得到精确的最优解，并便于处理连续空间实例。最后，伊朗的一个实际案例研究表明，降水预测的显著增强，马尔可夫链程序显示，基于优化的播种决策，预期降雨概率平均增加55%。基于场景的随机规划得到的播云方案的随机解值为11.7%，完美信息期望值为16.5%。

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引用次数: 0