不确定连续设施选址新鲁棒方法研究

Facility location is one of the popular research topics in operation research. Many factors in real-life facility location are uncertain. This fact has made it essential to study the facility location under uncertainty. When no probability distribution information of the random factors is known, the existing researches on this topic mostly focus on robust facility location, i.e., minimizing the cost function or regret function of the location system under the worst-case scenario. The worst-case scenario is perhaps a small probability event, and thus the classical robust location methods are possible to be overly conservative. Different from the classical ones, two new robust methods are proposed for continuous facility location under uncertainty in this proposal: distributionally robust location method and stochastic-variational-inequality-based (SVI-based) robust location method. For the former one, the moment constraints for the random factors are first estimated based on historical or empirical data and then a probability distribution set, whose elements satisfy the moment constraints, is constructed. The distributionally robust location method is proposed for locating facilities over the choice of a distribution in this distribution set. For the latter one, some appropriate residual functions are constructed for the equivalent stochastic variational inequality of the uncertain location problem. Through the optimization of the residual functions, the SVI-based robust location method is proposed for siting new facilities under uncertainty. Based on the two new methods and their combination, three robust location models will be built and then studied from theoretical analysis, algorithm design and application research in this proposal. Due to the consideration of all scenarios as well as the variance of system performance in different scenarios, rather than only emphasizing the worst-case scenario, the new robust location methods will be more practical than the classical ones and furthermore provide significant theoretical and algorithmic basis for continuous facility location under uncertainty.

设施选址是运筹学研究热点之一。由于实际选址中很多因素具有不确定性，研究不确定选址问题就变得尤为重要。当随机因素的概率分布信息未知时，已有的研究工作主要集中于鲁棒选址：最小化选址系统在最坏情形下的费用或缺憾函数。最坏情形也许是小概率事件，这导致传统鲁棒选址方法可能过于保守。与传统方法不同，本项目拟对不确定连续设施选址提出两种新鲁棒方法：分布鲁棒选址方法和基于随机变分不等式的鲁棒选址方法。前者将根据历史或经验数据确定随机因素的矩约束，并在满足矩约束的概率分布集上进行鲁棒选址；后者将对等价的随机变分不等式构造合适的残量函数，并通过优化残量函数进行鲁棒选址。根据并融合两种新方法，项目拟构建三种鲁棒选址模型并对其进行理论分析、算法设计与应用研究。由于考虑随机因素的所有情形以及不同情形下系统表现的差异性，新方法将比仅关注最坏情形的传统方法更具实用性，并为不确定连续设施选址研究提供重要的理论与算法基础。

设施选址是运筹学研究热点之一。由于实际选址问题中存在大量的随机因素，不确定选址问题的研究就变得尤为重要。当随机因素的概率分布信息未知时，已有的研究工作主要集中于鲁棒选址：最小化选址系统在最坏情形（worst-case scenario）下的费用函数或缺憾函数。鲁棒选址方法由于仅关注最坏情形（小概率事件）常常得到过保守的决策。本项目对不确定连续设施选址提出了两种新鲁棒方法：分布鲁棒选址方法和基于随机变分不等式的鲁棒选址方法。分布鲁棒选址方法根据历史数据或经验数据确定随机因素的矩约束构造概率分布集，并在该分布集上进行最差分布（worst-case distribution）下的鲁棒选址。基于随机变分不等式的鲁棒选址则是对等价的随机变分不等式构造合适的残量函数，通过优化残量函数的期望进行鲁棒选址。在两种鲁棒选址方法的基础上，项目还将两者融合设计了基于随机变分不等式的分布鲁棒选址方法，分析了其理论性质并设计算法。除此之外，本项目还研究了其他一些选址模型。本项目顺利完成了预期的研究目标，取得了如下几个方面的研究成果：.1. 对有实际应用背景的不确定选址模型，构造了满足矩约束的模糊集，对其中的参数选择及参数可信度进行了分析，建立了不确定选址的分布鲁棒模型，分析了其理论性质，设计了快速求解算法，并通过数值实验进行了验证；.2. 对有实际应用背景的连续设施选址问题，构造了凸的残量函数，建立了基于随机变分不等式的鲁棒模型，分析了模型解的存在性，设计了样本平均近似方法并证明了其收敛性，并通过数值实验对理论和算法进行了验证；.3. 对2中设计的残量函数在分布鲁棒优化框架下建立了基于随机变分不等式的分布鲁棒模型，对模型进行了理论分析和算法设计，并通过数值实验对模型和算法进行验证；.4. 已发表标注本项目资助的论文8篇，包括6篇在SCI检索期刊发表，2篇在核心期刊发表；.5. 本项目期间招收博士研究生2名，硕士研究生3名，形成了稳定的科研团队。