Mathematical modeling and optimization under uncertainty

从物流、供应链设计、配送和运输计划等应用领域产生的数学模型仍然具有挑战性，尽管软件和硬件能力最近有所进步。除了大尺寸外，设计参数*的可变性导致建模和解决方案方法都非常复杂。虽然人们普遍认为参数是已知的和恒定的，但可变性是所有实际问题的固有特征，而且通常情况下，确定性模型产生的解决方案在实施时*是不可行的，或者远远不是最佳的。另一方面，由于随机性和非线性，显式建模可变性的优化问题通常更具挑战性。建立随机性模型的适当方法取决于可获得的信息量。在完全信息的情况下，假设参数在规划范围内是已知的且*恒定的，并且所得到的模型是确定性的。在风险情况下，*随机建模假设参数服从概率分布。稳健优化是一种在没有分布信息的不确定情况下进行建模的新兴框架。*本研究的重点是不确定性下的建模和优化。我们提出了*模型，并开发了专门的优化工具来处理不确定性。幸运的是，这样的复杂问题往往呈现出特殊的结构，这使得它们适合于分解方法，如拉格朗日松弛/列生成、Dantzig-Wolfe分解和Bders*分解。在这一框架下将研究两个应用领域：灾难应对网络设计和航空公司运营规划。在这两个应用领域，都将探索随机优化和稳健优化。我们的目标是提出稳健的设计，既能优化利润或成本等经典目标，又能在环境发生变化时保持有效或平稳复苏。*该提案提供了丰富的研究问题，适合严格的、高素质的人才培训。总共有三名博士和四名硕士。学生将有机会进行理论和应用研究。他们将获得定量决策的分析和建模技能，以及开发算法和使用复杂优化软件的技能。学生将把他们的研究成果应用于灾难管理和航空公司运营规划的决策中。这两个领域都非常适用，并为学生提供在学术界、航空业、政府、医疗援助机构等工作的资格。**

Mathematical models arising from application areas such as logistics, supply chain design, distribution, and transportation planning remain challenging despite the recent advances in*software and hardware capabilities. In addition to large size, variability in design parameters****leads to significant complexity both in modeling and in solution methodologies. While it*is common to assume that parameters are known and constant, variability is inherent to*all practical problems and often times, solutions generated by deterministic models turn out*to be infeasible or far from optimal when implemented. On the other hand, optimization*problems that model variability explicitly are generally more challenging due to stochasticity*and nonlinearity. The appropriate approach to model randomness depends on the amount of*information available. Under perfect information, parameters are assumed to be known and*constant during the planning horizon and the resulting models are deterministic. Under risk,*stochastic modeling assumes that parameters follow a probability distribution. Robust*optimization is an emerging framework for modeling under uncertainty where no distribution*information is available.****The focus of this research is on modeling and optimization under uncertainty. We propose*models and develop specialized optimization tools to handle uncertainty. Fortunately such*complex problems tend to exhibit special structure, which makes them suitable for decomposition methodologies such as Lagrangian relaxation/column generation, Dantzig-Wolfe decomposition, and Benders*decomposition. Two application areas will be investigated under this framework: disaster response network design and airline operations planning. In both application areas, stochastic*and robust optimization will be explored. The goal is to come up with robust designs that*optimize classical objectives like profit or cost but also remain valid or allow smooth recovery*when settings change.***The proposal provides rich research questions suitable for rigorous highly qualified personnel training.  In total, three Ph.D. and four MASc. students will have the opportunity to carry out both theoretical and applied research.  They will gain skills in analysis and modeling of quantitative decision making as well as in developing algorithms and using sophisticated optimization software. The students will apply their research work in decision making for disaster management and airline operations planning. Both areas are highly applicable and provide students the qualifications to work in academia, the airline industry, government,  aid agencies, etc.  **