Mathematical modeling and optimization under uncertainty

数学模型产生的应用领域，如物流，供应链设计，分销和运输规划仍然具有挑战性，尽管最近的进展， 软件和硬件能力。除了尺寸大之外，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的和恒定的，可变性是固有的， 所有的实际问题，通常情况下，由确定性模型产生的解决方案， 在实施时是不可行的或远非最佳的。另一方面，优化 由于随机性，明确建模可变性的问题通常更具挑战性 和非线性。模型随机性的适当方法取决于 information available.在完全信息下，假设参数已知， 在规划范围内是恒定的，并且所产生的模型是确定性的。在风险之下， 随机建模假设参数遵循概率分布。鲁棒 最优化是一种新兴的框架，用于在不确定性下建模， 有资料可查。 本研究的重点是不确定性下的建模和优化。我们提出 模型和开发专门的优化工具来处理不确定性。幸运的是， 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Benders 分解两个应用领域将在这个框架下进行调查：灾害响应网络设计和航空公司业务规划。在这两个应用领域，随机 和鲁棒优化将被探索。我们的目标是提出稳健的设计， 优化利润或成本等传统目标，但也保持有效或允许平稳恢复 当设置改变时。 该建议提供了丰富的研究问题，适合严格的高素质人才培养。 总共有三个博士学位。四个MASC。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工作应用于灾害管理和航空公司运营规划的决策。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.