Optimization models and methods in Operations Management

我们研究和开发运营管理中决策问题的优化模型和方法，沿着五个主要研究方向。* (1)子模块和格是出现在离散数学、运筹学、人工智能、微观经济学和博弈论等领域的数学结构。我们研究了代数格和次模优化问题的结构，以及次模代价划分问题的算法发展。我们还研究了在手术持续时间不确定的情况下，结合离散凸性和随机优化的方法在医院中调度手术的计算特性 *（2）图和超图中的最小切割形成了这种分割问题的特殊情况，并应用于工程和离散优化的各个领域。 我们调查算法的后果显着的结构特性的全球最小切割，特别是随机算法和成功的概率和运行时间之间的权衡。 我们还研究了具有最小零件尺寸约束的子模块零件成本的划分问题。* (3)在划分、聚类和定位问题中，通常希望部分（簇、区域）具有“漂亮的形状”，或者至少“不太奇怪”。 我们计划研究这种形状约束的公式和性质，特别是在混合整数规划公式中，以及寻找满足这种约束的最大权重子集的“对偶”优化问题。(4)露天和地下矿山规划和排序：在合作研究项目中，我们研究露天矿山排序项目的解决方案，这些项目涉及高度结构化但庞大的整数规划模型（具有数千万个变量和约束）;以及使用连续空间（和连续时间）无限维优化模型作为这些巨大离散模型的替代方案。(5)调度方法和应用，特别是完成时间目标的加权和。 我们研究了连续松弛的计算复杂性，基于平均忙碌日期，对于非常一般的单阶段调度环境;以及包括维护活动的调度在内的问题，在此期间机器不可用于处理。这项研究计划预计将有助于基础知识，并通过其应用，加拿大经济和加拿大社会的几个部门，例如通过帮助减少加拿大医疗保健系统的等待时间。

We study and develop optimization models and methods for decision problems in Operations Management, along five main research directions.***(1) Submodularity and lattices are mathematical structures that arise in fields such as discrete mathematics, operations research, artificial intelligence, microeconomics and game theory. We study the structure of algebraic lattices and submodular optimization problems, and the development of algorithms for partitioning problems with submodular costs. We also investigate computational properties of methods combining discrete convexity and stochastic optimization for scheduling of surgeries in a hospital in the presence of uncertain operations durations***(2) Minimum cuts in graphs and hypergraphs form a special case of such partitioning problems, with applications to various areas of engineering and discrete optimization. We investigate algorithmic consequences of the remarkable structural properties of global minimum cuts, in particular randomized algorithms and the tradeoffs between probability of success and running time. We also study partitioning problems with submodular part costs subject to minimum part size constraints.***(3) In partitioning, clustering and location problems, it is often desirable that the parts (clusters, regions) have “nice shapes”, or at least are “not too odd”. We plan to investigate formulations and properties of such shape constraints, particularly within mixed-integer programming formulations, as well as “dual” optimization problems of finding maximum weight subsets satisfying such constraints.***(4) Surface and underground mine planning and sequencing: in collaborative research projects we study the solution of open pit mine sequencing projects that induce highly structured but huge integer programming models (with tens of millions of variables and constraints); and the use of continuous-space (and continuous-time) infinite-dimensional optimization models as alternatives to these huge discrete models.***(5) Scheduling methods and applications, in particular for weighted sums of completion times objectives. We study the computational complexity of a continuous relaxation, based on mean busy dates, for very general single-stage scheduling environments; and problems that include the scheduling of maintenance activities, during which the machines are not available for processing.******This research program is expected to contribute to fundamental knowledge and, through its applications, to several sectors of the Canadian economy and of the Canadian society, for example by helping to reduce waiting time in the Canadian health care system.**