Sequential Decision Making under System-inherent Uncertainty: Mathematical Optimization Methods for Time-dynamic Applications

在数学优化中，通常假设问题的数据是完全已知的。然而，在实践中，这一假设往往得不到满足，因为数据是随着时间的推移而变得已知的，需要在不确定的情况下进行迭代决策。 时间动态优化问题的应用出现在不同的层次上：例如，在战略供应链计划中，数据是按季度提供的，而在操作机器调度中，新订单必须按分钟计算。统一的因素是时间进程中的不确定性。目前还没有系统的、跨学科的研究方法。有几种研究方法是依赖于问题的：在在线优化中，没有关于未来事件的知识，决策是这样的，即使在最坏的情况下，解决方案也不会离最优解太远，可以追溯计算。在随机规划中，人们假设一组未来的情景沿着概率，并在预期结果的意义上做出决定。在鲁棒优化中，所有场景都保证了可行性，这就是为什么优化的自由度受到限制。目前研究的主要不足在于对时间和不确定性因素的处理不一致。在线优化存在最坏情况导向，随机规划结果基于未知的随机假设，鲁棒优化不是针对多阶段问题设计的。一般性的扩展，例如对未来数据价值的分析，才刚刚开始其科学发展。在供应链管理中，计划和控制工具的灵活实现是不存在的，因此，我们的目标包括在一个统一的框架中处理不确定性的不同方法的整合，以促进一个合适的解决方案方法（在线优化算法，随机规划，或鲁棒优化）的上下文相关的选择。这包括分布式分析方法，可以评估和比较算法的质量和数据的价值。然后使用敏感性分析来检查不同方法在不同情况下的行为，即，不同类型的不确定性。基于生产和物流的典型问题，我们检查的方法在实践中的适用性。此外，我们打算引导规划和控制工具走向自适应功能逻辑。实践中可获得的数据越来越多，例如，从全球定位系统或RFID芯片，表明一个全面的方法论的理解是必要的，以便能够决定信息的价值和应用适当的优化方法。这一事实得到了正在进行的工业计划的支持，例如德国的工业4.0或美国的工业互联网。

In mathematical optimization it is often assumed that a problem's data is known entirely. In practice however this assumption is often not met because data is made known over the course of time requiring iterative decision making under uncertainty. Applications for time-dynamic optimization problems arise on different levels: For instance, in strategic supply chain planning data is made available quarterly, whereas in operational machine scheduling new orders have to be accounted for on a minute-basis. The unifying element are uncertainties in the course of time. There is no systematic, interdisciplinary approach.Several approaches are pursued in research problem-dependently: In online optimization there is no knowledge on future events and decision are made such that even in the worst case the solution is not too far away from the optimal solution that can be computed in retrospective. In stochastic programming one assumes a set of future scenarios along with probabilities and one decides in the sense of the expected outcome. In robust optimization feasibility is guaranteed for all scenarios which is why the degrees of freedom for optimization are restricted. The main deficiency of current research comprises the inconsistent handling of the factors time and uncertainty. Online optimization suffers from the worst case orientation, stochastic programming results are based on unknown stochastic assumptions, robust optimization is not specifically designed to multi-stage problems. General extensions, such as an analysis of the value of future data, is just in the beginning of its scientific evolution. Flexible implementations within planning and control tools in supply chain management do not exist.Therefore, our goal consists in the consolidation of the different approaches dealing with uncertainty in a unified framework facilitating a context-dependent selection of a suitable solution methodology (algorithm from online optimization, stochastic programming, orrobust optimization). This comprises distributional methods of analysis which allow to assess and compare the quality of algorithms and the value of data. Sensitivity analysis is used then in order to check the behavior of different methods under varying circumstances, i.e., under different types of uncertainty. Based on exemplary Problems from production and logistics, we check the applicability of the methods in practice. Furthermore, we intend to lead planning and control tools towards an adaptive functional logic. The increasing amounts of data available in practice, e.g., from GPS or RFID chips, indicate that a comprehensive methological understanding is necessary in order to be able to decide on the value of information and to apply suitable optimization methods. This fact is supported by on-going industrial initiatives such as Industry 4.0 in Germany or the Industrial Internet in the US.