Mathematical modeling and optimization under uncertainty

尽管最近在软件和硬件能力方面取得了进展，但物流、供应链设计、配送和运输规划等应用领域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。虽然它 * 是常见的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，往往时候，确定性模型产生的解决方案变成 * 是不可行的或远离最佳实施时。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用的信息量。在完全信息条件下，假设规划期内参数已知且为常数，得到的模型是确定性的。在风险下，* 随机建模假设参数遵循概率分布。鲁棒 * 优化是一种新兴的框架，用于在不确定性下建模，其中没有分布 * 信息可用。本研究的重点是不确定性下的建模和优化。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，这样的复杂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，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 * 和强大的优化将被探索。我们的目标是提出稳健的设计，* 优化经典目标，如利润或成本，但也保持有效或允许平稳恢复 * 当设置更改。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.  **