论文速递 | Operations Research11月文章合集

编者按

在本文章中，我们梳理了运筹学顶刊Operations Research11月份发布的12篇文章的基本信息，旨在帮助读者快速洞察行业最新动态。

文章1

● 题目：Static Pricing for Multi-unit Prophet Inequalities

多单位先知不等式的静态定价

● 原文链接：https://doi.org/10.1287/opre.2023.0031

● 作者：Shuchi Chawla, Nikhil Devanur, Thodoris Lykouris

● 发布时间：2023.11.1

● 摘要：

We study a pricing problem where a seller has k identical copies of a product, buyers arrive sequentially, and the seller prices the items aiming to maximize social welfare. When k = 1, this is the so-called prophet inequality problem for which there is a simple pricing scheme achieving a competitive ratio of 1/2. On the other end of the spectrum, as k goes to infinity, the asymptotic performance of both static and adaptive pricing is well understood. We provide a static pricing scheme for the small-supply regime: where k is small but larger than one. Prior to our work, the best competitive ratio known for this setting was the 1/2 that follows from the single-unit prophet inequality. Our pricing scheme is easy to describe as well as practical; it is anonymous, nonadaptive, and order oblivious. We pick a single price that equalizes the expected fraction of items sold and the probability that the supply does not sell out before all customers are served; this price is then offered to each customer while supply lasts. This extends an approach introduced by Samuel-Cahn for the case of k = 1. This pricing scheme achieves a competitive ratio that increases gradually with the supply. Subsequent work shows that our pricing scheme is the optimal static pricing for every value of k.

我们研究的是这样一个定价问题：卖方有 k 份相同的产品，买方依次到达，卖方以社会福利最大化为目标对产品进行定价。当 k = 1 时，这就是所谓的先知不等式问题，有一个简单的定价方案可以达到 1/2 的竞争比率。而在另一端，当 k 变为无穷大时，静态定价和自适应定价的渐近性能都很好理解。我们为小供给机制提供了一种静态定价方案：k 较小但大于 1。在我们的研究之前，在这种情况下已知的最佳竞争比率是单个单位先知不等式得出的 1/2。我们的定价方案既易于描述，又切实可行；它是匿名的、非自适应的，而且无视订单。我们选择一个单一的价格，使预期售出的商品数量和供应量在满足所有客户需求之前不售罄的概率相等；然后在供应量持续的情况下，向每位客户提供这个价格。这扩展了Samuel-Cahn针对 k = 1 的情况提出的方法。这种定价方案实现了随供应量逐渐增加的竞争比率。随后的工作表明，针对每个k值，我们的定价方案都是最优静态定价方案。

文章2

● 题目：Cloud Cost Optimization: Model, Bounds, and Asymptotics

云成本优化：模型、边界和渐进

● 原文链接：https://doi.org/10.1287/opre.2022.0362

● 作者：Zihao Qu, Milind Dawande, Ganesh Janakiraman

● 发布时间：2023.11.2

● 摘要：

Motivated by the rapid growth of the cloud cost management and optimization (CCMO) industry to support the exploding cloud-computing market, we study an infinite-horizon, stochastic optimization problem from the viewpoint of a firm that employs cloud resources to process incoming orders (or jobs) over time. We model the following vital practical features of CCMO in our problem. There are several types of resources that differ in their costs and performance attributes (e.g., processor speed, memory, storage). For each type of resource, capacity can either be reserved over the long term at a discounted price or be used on demand at a relatively higher price. Orders of several types arrive stochastically through time; orders differ in their completion-time deadlines and in their resource-specific processing-time distributions. Moreover, the progress of an order can be observed periodically, and if required, the order can be moved from one resource type to another. Penalty costs are incurred for orders not completed by their deadlines. The firm’s goal is to minimize the long-run average expected cost per period, taking into account reserved-capacity costs, on-demand capacity costs, and order-delay costs. We derive a lower bound on the optimal cost by considering a set of decoupled problems, one for each order. The solutions of these problems are then used to construct a feasible policy for the original problem and derive an upper bound on that policy’s optimality gap. Importantly, we show that our policy is asymptotically optimal; when the demand rates of the orders are scaled by a factor θ>0, the policy’s optimality gap scales proportional to 1/√θ. We also report results of a comprehensive numerical study—on a test bed informed by capacity and pricing data from Amazon Web Services—to demonstrate the impressive performance of our policy.

云成本管理与优化（CCMO）行业的快速发展为爆炸式增长的云计算市场提供了支持，受此激励，我们从一家公司的角度出发，研究了一个无限视距的随机优化问题，该公司使用云资源来处理随时间推移收到的订单（或作业）。我们在问题中模拟了 CCMO 的以下重要实际特征。有几种类型的资源，其成本和性能属性（如处理器速度、内存、存储）各不相同。对于每种类型的资源，既可以以折扣价长期保留容量，也可以以相对较高的价格按需使用。多种类型的订单会随着时间随机到达；订单的完成期限和特定资源的处理时间分布各不相同。此外，订单的进度可以定期观察，如有需要，订单可以从一种资源类型转移到另一种资源类型。未按期完成的订单会产生惩罚成本。公司的目标是在考虑到预留容量成本、按需容量成本和订单延迟成本的情况下，最大限度地降低每期的长期平均预期成本。我们通过考虑一系列解耦问题（每个订单一个问题），得出了最优成本的下限。然后，利用这些问题的解决方案为原始问题构建可行的策略，并推导出该策略最优性差距的上限。重要的是，我们证明了我们的策略是渐进最优的；当订单的需求率按系数 θ>0 缩放时，策略的最优性差距按 1/√θ 的比例缩放。我们还报告了以亚马逊网络服务的容量和定价数据为基础的测试平台的综合数值研究结果，以证明我们的策略具有优秀的性能。

文章3

● 题目：Treatment Planning for Victims with Heterogeneous Time Sensitivities in Mass Casualty Incidents

为大规模伤亡事件中时间敏感性不同的受害者制定救治计划

● 原文链接：https://doi.org/10.1287/mnsc.2020.03249

● 作者：Yunting Shi, Nan Liu, Guohua Wan

● 发布时间：2023.11.2

● 摘要：

The current emergency response guidelines suggest giving priority of treatment to those victims whose initial health conditions are more critical. Although this makes intuitive sense, it does not consider potential deterioration of less critical victims. Deterioration may lead to longer treatment time and irrecoverable health damage, but could be avoided if these victims were to receive care in time. Informed by a unique timestamps data set of surgeries carried out in a field hospital set up in response to a large-scale earthquake, we develop scheduling models to aid treatment planning for mass casualty incidents (MCIs). A distinguishing feature of our modeling framework is to simultaneously consider victim health deterioration and wait-dependent service times in making decisions. We identify conditions under which victims with a less critical initial condition have higher or lower priority than their counterparts in an optimal schedule—the priority order depends on victim deterioration trajectories and the resource (i.e., treatment time) availability. A counterfactual analysis based on our data shows that adopting our model would significantly reduce the surgical makespan and the total numbers of overdue and deteriorated victims compared with using the then-implemented treatment plan; dynamic adjustment of treatment plans (if a second batch of victims arrive) and care coordination among surgical teams could further improve operational efficiency and health outcomes. By demonstrating the value of adopting data-driven approaches in MCI response, our research holds strong potentials to improve emergency response and to inform its policy making.

目前的应急指南建议优先治疗那些最初健康状况较为危急的受害者。虽然这很直观，但却没有考虑到病情较轻的受害者可能出现的病情恶化。病情恶化可能会导致更长的治疗时间和无法挽回的健康损害，但如果这些受害者能及时得到救治，这些情况是可以避免的。根据在应对大规模地震的野战医院中开展手术的独特时间戳数据集，我们开发了调度模型，以帮助制定大规模伤亡事件（MCI）的治疗计划。我们的建模框架的一个显著特点是在决策过程中同时考虑了受害者的健康状况恶化和依赖于等待的服务时间。我们确定了在什么情况下，初始条件不那么危急的受害者在最佳时间安排中比同类受害者具有更高或更低的优先级--优先级顺序取决于受害者的恶化轨迹和资源（即治疗时间）的可用性。根据我们的数据进行的反事实分析表明，与采用当时实施的治疗计划相比，采用我们的模型将大大缩短手术时间，并减少逾期和病情恶化的受害者总数；动态调整治疗计划（如果第二批受害者到达）和手术团队之间的护理协调可进一步提高手术效率和医疗效果。通过证明在 MCI 响应中采用数据驱动方法的价值，我们的研究极有可能改善应急响应并为其政策制定提供依据。

文章4

● 题目：Characterizing Rational Transplant Program Response to Outcome-Based Regulation

描述移植项目对基于结果的监管的合理反应

● 原文链接：https://doi.org/10.1287/opre.2018.0721

● 作者：David Mildebrath, Taewoo Lee, Saumya Sinha, Andrew J. Schaefer, A. Osama Gaber

● 发布时间：2023.11.1

● 摘要：

Organ transplantation is an increasingly common therapy for many types of end-stage organ failure, including lungs, hearts, kidneys, and livers. The last 20 years have seen increased scrutiny of posttransplant outcomes in the United States to ensure the efficient utilization of the scarce organ supply. Under regulations by the Organ Procurement Transplantation Network (OPTN) and Centers for Medicare and Medicaid Services (CMS), the United States has seen a rise in risk-averse patient selection among transplant programs, resulting in decreased transplantation volume for some programs. Despite this observed decrease, the response of transplant programs to OPTN/CMS regulations remains poorly understood. In this work, we consider the perspective of a transplant program that seeks to simultaneously maximize transplant volume and control the risk of OPTN/CMS penalization. Using a chance-constrained mixed-integer programming model, we demonstrate that under certain conditions, it may be rational for a transplant program to curtail its transplant volume to avoid penalization under OPTN/CMS regulations. This finding, which confirms empirical results observed in the clinical literature, suggests that such regulations may be inherently unsuitable for use in incentivizing improved program performance. We also highlight other structural shortcomings of OPTN/CMS regulations that have not been observed previously in the literature.

器官移植是治疗肺、心脏、肾脏和肝脏等多种终末期器官衰竭的一种日益常见的疗法。在过去 20 年中，美国对器官移植后的治疗效果进行了越来越严格的审查，以确保有效利用稀缺的器官供应。根据器官获取移植网络（OPTN）和医疗保险与医疗补助服务中心（CMS）的规定，美国的移植项目在选择病人时越来越倾向于规避风险，导致一些项目的移植量减少。尽管观察到了这种减少，但人们对移植项目对 OPTN/CMS 规定的反应仍然知之甚少。在这项工作中，我们从移植项目的角度出发，考虑如何同时实现移植量最大化和控制 OPTN/CMS 惩罚风险。通过使用一个机会受限的混合整数规划模型，我们证明了在某些条件下，移植项目减少移植量以避免受到 OPTN/CMS 法规的处罚可能是合理的。这一发现与临床文献中观察到的经验结果相吻合，表明此类法规可能在本质上不适合用于激励项目绩效的提高。我们还强调了 OPTN/CMS 法规的其他结构性缺陷，这些缺陷在以前的文献中尚未发现。

文章5

● 题目：Fair and Efficient Online Allocations

公平高效的在线分配

● 原文链接：https://doi.org/10.1287/opre.2022.0332

● 作者：Gerdus Benadè, Aleksandr M. Kazachkov, Ariel D. Procaccia, Alexandros Psomas, David Zeng

● 发布时间：2023.11.3

● 摘要：

We study trade-offs between fairness and efficiency when allocating indivisible items online. We attempt to minimize envy, the extent to which any agent prefers another’s allocation to their own, while being Pareto efficient. We provide matching lower and upper bounds against a sequence of progressively weaker adversaries. Against worst-case adversaries, we find a sharp trade-off; no allocation algorithm can simultaneously provide both nontrivial fairness and nontrivial efficiency guarantees. In a slightly weaker adversary regime where item values are drawn from (potentially correlated) distributions, it is possible to achieve the best of both worlds. We give an algorithm that is Pareto efficient ex post and either envy free up to one good or envy free with high probability. Neither guarantee can be improved, even in isolation. En route, we give a constructive proof for a structural result of independent interest. Specifically, there always exists a Pareto-efficient fractional allocation that is strongly envy free with respect to pairs of agents with substantially different utilities while allocating identical bundles to agents with identical utilities (up to multiplicative factors).

我们研究了在线分配不可分割物品时公平与效率之间的权衡。我们试图在保证帕累托效率的同时，最大限度地减少妒忌，即任何代理在多大程度上偏好他人的分配而非自己的分配。我们针对一系列逐渐变弱的对手提供了匹配的下限和上限。针对最坏情况下的对手，我们发现了一个尖锐的权衡；没有一种分配算法能同时提供非对等的公平性和非对等的效率保证。在对手稍弱的情况下，即项目值来自（可能相关的）分布，则有可能实现两全其美。我们给出了一种事后帕累托效率算法，这种算法要么在一种物品上是无嫉妒的，要么在高概率上是无嫉妒的。这两种保证都无法改进，甚至无法单独改进。在此过程中，我们给出了一个具有独立意义的结构性结果的构造性证明。具体地说，总是存在一种帕累托效率的部分分配，这种分配对于效用大不相同的代理人对来说是强烈无嫉妒的，同时对效用相同的代理人分配相同的捆绑物（不超过乘法因子）。

文章6

● 题目：Assortment Optimization Under the Multinomial Logit Model with Utility-Based Rank Cutoffs

基于效用等级截止值的多项式对数模型下的分类优化

● 原文链接：https://doi.org/10.1287/opre.2021.0060

● 作者：Yicheng Bai, Jacob Feldman, Huseyin Topaloglu, Laura Wagner

● 发布时间：2023.11.7

● 摘要：

We study assortment optimization problems under a natural variant of the multinomial logit model where the customers are willing to focus only on a certain number of products that provide the largest utilities. In particular, each customer has a rank cutoff, characterizing the number of products that she will focus on during the course of her choice process. Given that we offer a certain assortment of products, the choice process of a customer with rank cutoff k proceeds as follows. The customer associates random utilities with all of the products as well as the no-purchase option. The customer ignores all alternatives whose utilities are not within the k largest utilities. Among the remaining alternatives, the customer chooses the available alternative that provides the largest utility. Under the assumption that the utilities follow Gumbel distributions with the same scale parameter, we provide a recursion to compute the choice probabilities. Considering the assortment optimization problem to find the revenue-maximizing assortment of products to offer, we show that the problem is NP-hard and give a polynomial time approximation scheme. Because the customers ignore the products below their rank cutoffs in our variant of the multinomial logit model, intuitively speaking, our variant captures choosier choice behavior than the standard multinomial logit model. Accordingly, we show that the revenue-maximizing assortment under our variant includes the revenue-maximizing assortment under the standard multinomial logit model, so choosier behavior leads to larger assortments offered to maximize the expected revenue. We conduct computational experiments on both synthetic and real data sets to demonstrate that incorporating rank cutoffs can yield better predictions of customer choices and yield more profitable assortment recommendations.

我们研究的是多项式对数模型(MNL)自然变体下的分类优化问题，在这种模型中，顾客只愿意关注一定数量的能提供最大效用的产品。具体来说，每个客户都有一个等级截止值，它描述了客户在选择过程中会关注的产品数量。鉴于我们提供的产品种类繁多，具有等级截止值 k 的顾客的选择过程如下。客户会将随机效用与所有产品以及不购买选项联系起来。顾客会忽略所有效用不在 k 个最大效用范围内的备选方案。在剩下的备选方案中，客户选择能提供最大效用的备选方案。假设效用遵循具有相同标度参数的 Gumbel 分布，我们提供了计算选择概率的递推方法。考虑到分类优化问题是为了找到收入最大化的产品种类，我们证明了该问题是 NPhard的 ，并给出了一个多项式时间近似方案。由于在我们的MNL变体中，顾客会忽略低于其等级分界线的产品，因此直观地说，我们的变体比标准MNL模型更能捕捉到顾客的选择行为。因此，我们的变式表明，在我们的变式中，收益最大化的组合包括标准MNL模型中收益最大化的组合，因此，选择行为会导致提供更大的组合，从而使预期收益最大化。我们在合成数据集和真实数据集上进行了计算实验，证明纳入等级截止值可以更好地预测客户的选择，并提供更有利可图的组合推荐。

文章7

● 题目：Preface to Special Issue on Computational Advances in Short-Term Power System Operations

特刊序言：Computational Advances in Short-Term Power System Operations

● 原文链接：https://doi.org/10.1287/opre.intro.v71.n6

● 作者：Ross Baldick , Steven Low , Richard O’Neill, Daniel Ralph , Golbon Zakeri

● 发布时间：2023.11.20

文章8

● 题目：Reshaping National Organ Allocation Policy

调整国家器官分配政策

● 原文链接：https://doi.org/10.1287/opre.2022.0035

● 作者：Theodore Papalexopoulos, James Alcorn, Dimitris Bertsimas, Rebecca Goff, Darren Stewart, Nikolaos Trichakis

● 发布时间：2023.11.20

● 摘要：

The Organ Procurement & Transplantation Network (OPTN) initiated in 2018 a major overhaul of all U.S. deceased-donor organ allocation policies, aiming to gradually migrate them to a so-called continuous distribution model, with the goal of creating an allocation system that is more efficient, more equitable, and more inclusive. Development of policies within this model, however, represents a major challenge because multiple efficiency and fairness objectives need to be delicately balanced. We introduce a novel analytical framework that leverages machine learning, simulation, and optimization to illuminate policy tradeoffs and enable dynamic exploration of the efficient frontier of policy options. In collaboration with the OPTN, we applied the framework to design a new national allocation policy for lungs. Since March 9, 2023, all deceased-donor lungs in the United States have been allocated according to this policy that we helped design, projected to reduce waitlist mortality by approximately 20% compared with current policy based on simulations. We discuss how we extended our collaboration with the OPTN to the redesign of kidney, pancreas, heart, and liver allocation and how our framework can be applied to other application domains, such as school choice or public housing allocation systems.

器官获取与移植网络（OPTN）于2018年启动了对美国所有已故供体器官分配政策的重大改革，旨在将其逐步迁移至所谓的连续分配模式，目标是建立一个更高效、更公平、更具包容性的分配系统。然而，在这种模式下制定政策是一项重大挑战，因为需要微妙地平衡效率和公平的多重目标。我们引入了一个新颖的分析框架，利用机器学习、模拟和优化来阐明政策权衡，并对政策选项的有效前沿进行动态探索。我们与 OPTN 合作，应用该框架设计了新的肺部国家分配政策。自 2023 年 3 月 9 日起，美国所有已故捐献者的肺都将根据我们帮助设计的这一政策进行分配，根据模拟预测，与现行政策相比，等待者死亡率将降低约 20%。我们讨论了如何将与 OPTN 的合作扩展到肾脏、胰腺、心脏和肝脏分配的重新设计，以及我们的框架如何应用于其他应用领域，如学校选择或公共住房分配系统。

文章9

● 题目：A Unifying Framework for the Capacitated Vehicle Routing Problem Under Risk and Ambiguity

风险和模糊性条件下有容量限制的车辆路径规划问题(CVRP)的统一框架

● 原文链接：https://doi.org/10.1287/opre.2021.0669

● 作者：Shubhechyya Ghosal, Chin Pang Ho, Wolfram Wiesemann

● 发布时间：2023.11.22

● 摘要：

We propose a generic model for the capacitated vehicle routing problem (CVRP) under demand uncertainty. By combining risk measures, satisficing measures, or disutility functions with complete or partial characterizations of the probability distribution governing the demands, our formulation bridges the popular but often independently studied paradigms of stochastic programming and distributionally robust optimization. We characterize when an uncertainty-affected CVRP is (not) amenable to a solution via a popular branch-and-cut scheme, and we elucidate how this solvability relates to the interplay between the employed decision criterion and the available description of the uncertainty. Our framework offers a unified treatment of several CVRP variants from the recent literature, such as formulations that optimize the requirements violation or the essential riskiness indices, and it, at the same time, allows us to study new problem variants, such as formulations that optimize the worst case expected disutility over Wasserstein or ϕ-divergence ambiguity sets. All of our formulations can be solved by the same branch-and-cut algorithm with only minimal adaptations, which makes them attractive for practical implementations.

我们为需求不确定情况下的CVRP问题提出了一个通用模型。通过将风险度量、满足度量或无用性功能与需求概率分布的完整或部分特征相结合，我们的表述为随机规划和分布鲁棒性优化这两个流行但通常独立研究的范例架起了桥梁。我们描述了受不确定性影响的 CVRP 何时可以（不可以）通过流行的分支切割法求解，并阐明了这种可求解性与所采用的决策标准和对不确定性的可用描述之间的相互作用的关系。我们的框架统一处理了近期文献中的几种 CVRP 变体，如优化违反要求或基本风险指数的公式，同时还允许我们研究新的问题变体，如优化 Wasserstein 或 ϕ -divergence模糊集上最坏情况预期效用的公式。我们的所有公式都可以用相同的分支切割法求解（只需做极少的调整），这使得它们在实际应用中很有吸引力。

文章10

● 题目：Screening with Limited Information: A Dual Perspective

在信息有限的情况下进行筛查：对偶视角

● 原文链接：https://doi.org/10.1287/opre.2022.0016

● 作者：Zhi Chen, Zhenyu Hu, Ruiqin Wang

● 发布时间：2023.11.23

● 摘要：

Consider a seller seeking a selling mechanism to maximize the worst-case revenue obtained from a buyer whose valuation distribution lies in a certain ambiguity set. Such a mechanism design problem with one product and one buyer is known as the screening problem. For a generic convex ambiguity set, we show via the minimax theorem that strong duality holds between the problem of finding the optimal robust mechanism and a minimax pricing problem where the adversary first chooses a worst-case distribution, and then the seller decides the best posted price mechanism. This implies that the extra value of optimizing over more sophisticated mechanisms amounts exactly to the value of eliminating distributional ambiguity under a posted price mechanism. The duality result also connects prior literature that separately studies the primal (robust screening) and problems related to the dual (e.g., robust pricing, buyer-optimal pricing, and personalized pricing). We further analytically solve the minimax pricing problem (as well as the robust pricing problem) for several important ambiguity sets, such as the ones with mean and various dispersion measures, and with the Wasserstein metric, and we provide a unified geometric intuition behind our approach. The solutions are then used to construct the optimal robust mechanism and to compare with the solutions to the robust pricing problem. We also establish the uniqueness of the worst-case distribution for some cases.

考虑卖方寻求一种销售机制，以最大限度地提高从买方那里获得的最坏情况收益，而买方的估值分布位于某个模糊集合中。这种只有一种产品和一个买方的机制设计问题被称为筛查问题。对于一般的凸模糊集，我们通过minimax theorem证明，寻找最优稳健机制问题与minimax定价问题之间存在强对偶性，在minimax定价问题中，对手首先选择最坏情况分布，然后卖方决定最佳发布价格机制。这意味着，在更复杂的机制上进行优化的额外价值，恰好等于在公布价格机制下消除分布模糊性的价值。这一对偶性结果还连接了之前分别研究原始问题（稳健筛查）和与对偶性相关问题（如稳健定价、买方最优定价和个性化定价）的文献。我们进一步分析求解了几个重要模糊集的minimax定价问题（以及稳健定价问题），如具有均值和各种离散度量的模糊集，以及具有 Wasserstein 度量的模糊集。然后，我们利用这些解来构建最优稳健机制，并与稳健定价问题的解进行比较。我们还确定了某些情况下最坏情况分布的唯一性。

文章11

● 题目：Stochastic Liquidity as a Proxy for Nonlinear Price Impact

作为非线性价格影响替代物的随机流动性

● 原文链接：https://doi.org/10.1287/opre.2022.0627

● 作者：Johannes Muhle-Karbe, Zexin Wang, Kevin Webster

● 发布时间：2023.11.28

● 摘要：

Optimal execution and trading algorithms rely on price impact models, such as the propagator model, to quantify trading costs. Empirically, price impact is concave in trade sizes, leading to nonlinear models for which optimization problems are intractable, and even qualitative properties, such as price manipulation, are poorly understood. However, we show that in the diffusion limit of small and frequent orders, the nonlinear model converges to a tractable linear model. In this high-frequency limit, a stochastic liquidity parameter approximates the original impact function’s nonlinearity. We illustrate the approximation’s practical performance using limit order data.

最优执行和交易算法依赖于价格影响模型（如传播者模型）来量化交易成本。从经验上看，价格影响随交易规模呈凹形，导致非线性模型的优化问题难以解决，甚至连定性属性（如价格操纵）也不甚了解。然而，我们的研究表明，在小订单和频繁订单的扩散极限中，非线性模型会收敛到一个可行的线性模型。在这一高频极限中，随机流动性参数近似于原始影响函数的非线性。我们使用限价订单数据说明了该近似值的实际性能。

文章12

● 题目：On Proportionally Consistent Solutions to the Divorced-Parents Problem

关于离婚父母问题的比例一致的解决方案

● 原文链接：https://doi.org/10.1287/opre.2022.0470

● 作者：Ward Romeijnders, Nicky D. Van Foreest, Jacob Wijngaard

● 发布时间：2023.11.28

● 摘要：

When Dutch parents divorce, Dutch law dictates that the parental contributions to cover the financial needs of the children have to be proportionally consistent. This rule is clear when parents only have common children. However, cases can be considerably more complicated, for example, when parents have financial responsibilities to children from previous marriages. We show that, mathematically, this settlement problem can be modeled as a bipartite rationing problem for which a unique global proportionally proportional solution exists. Moreover, we develop two efficient algorithms for obtaining this proportionally proportional solution, and we show numerically that both algorithms are considerably faster than standard convex optimization techniques. The first algorithm is a novel tailor-made fixed-point iteration algorithm (FPA), whereas the second algorithm only iteratively applies simple lawsuits involving a single child and its parents. The inspiration for this latter algorithm comes from our main convergence proof in which we show that iteratively applying settlements on smaller subnetworks eventually leads to the same settlement on the network as a whole. This has significant societal importance because, in practice, lawsuits are often only held between two or a few parents. Moreover, our iterative algorithm is easy to understand, also by parents, legal counselors, and judges, which is crucial for its acceptance in practice. Finally, as the method provides a unique solution to any dispute, it removes the legal inequality perceived by parents. Consequently, it may considerably reduce the workload of courts because parents and lawyers can compute the proportionally proportional parental contributions before bringing their case to court.

当荷兰父母离婚时，荷兰法律规定，父母为满足子女的经济需求所做的贡献必须按比例一致。在父母只有共同子女的情况下，这一规则是明确的。然而，当父母对前次婚姻的子女负有经济责任时，情况就会复杂得多。我们的研究表明，从数学上讲，这个结算问题可以建模为一个双边配给问题，其中存在一个唯一的全局按比例配给解。此外，我们还开发了两种高效算法来获取这种按比例分配的解，并通过数值计算表明，这两种算法都比标准的凸优化技术快得多。第一种算法是一种新颖的定制不动点迭代算法（FPA），而第二种算法只迭代应用涉及单个子代及其父代的简单诉讼。后一种算法的灵感来源于我们的主要收敛性证明，我们在该证明中表明，在较小的子网络上迭代应用和解最终会导致在整个网络上应用相同的和解。这具有重要的社会意义，因为在实践中，诉讼往往只在两个或几个父母之间发生。此外，我们的迭代算法通俗易懂，父母、法律顾问和法官都很容易理解，这对其在实践中被接受至关重要。最后，由于该方法为任何争议提供了唯一的解决方案，因此消除了父母所认为的法律不平等。因此，它可以大大减少法院的工作量，因为父母和律师可以在将案件提交法院之前计算出父母按比例分担的费用。