Mathematical modeling and optimization under uncertainty

尽管最近在软件和硬件能力方面取得了进步，但从物流、供应链设计、分销和运输规划等应用领域产生的数学模型仍然具有挑战性。除了大尺寸之外，设计参数****的可变性导致建模和解决方案方法的显著复杂性。虽然通常假设参数是已知且恒定的，但可变性是所有实际问题所固有的，而且通常情况下，由确定性模型生成的解决方案在实现时被证明是不可行的或远非最佳的。另一方面，由于随机性和非线性，明确建模可变性的优化问题通常更具挑战性。建模随机性的适当方法取决于可用信息的数量。在完全信息条件下，假设规划范围内的参数是已知的，且*为常数，得到的模型是确定性的。在风险下，随机建模假设参数服从概率分布。鲁棒优化是一种新兴的框架，用于不确定情况下没有可用的分布信息的建模。****本研究的重点是不确定条件下的建模与优化。我们提出模型并开发专门的优化工具来处理不确定性。幸运的是，这种复杂的问题往往表现出特殊的结构，这使得它们适合于分解方法，如拉格朗日松弛/列生成、dantzigg - wolfe分解和Benders分解。在这个框架下，将研究两个应用领域：灾难响应网络设计和航空公司运营规划。在这两个应用领域，将探讨随机*和鲁棒优化。目标是提出稳健的设计，以优化利润或成本等经典目标，同时在设置改变时保持有效或允许平滑恢复。***提案提供了丰富的研究问题，适合严格的高素质人才培养。总共有3名博士和4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的优化软件。学生将把他们的研究成果应用于灾害管理和航空公司运营计划的决策。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，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.  **