Decision Making Under Uncertainty Using Mathematical Optimization: A Survey

Quantitative optimization techniques have proven to be a powerful tool in tackling complicated decision-making problems in various application domains. Scheduling workforces, pricing airline tickets, managing production systems and designing engineering projects are just a small sample of the myriad industrial applications that have benefited from optimization techniques such as linear, nonlinear and integer programming, just to name a few. Deterministic optimization is a welldeveloped field that, while perhaps not fully mature, has definitely passed out of adolescence. However, despite the many successes, there are drawbacks to deterministic optimization techniques. A prominent underlying assumption is frequently violated in practice, namely, that problem data are known exactly. For example, in production systems, forecasts of future customer demand are routinely input into deterministic production optimization models. While this approximate approach can sometimes lead to an acceptable solution, in many cases practitioners resort to optimization techniques that account for uncertainty in the problem data. In this paper, we survey various techniques for optimization under uncertainty. Outline: In Section II, we consider optimization techniques when problem data uncertainty is characterized by probabilistic distributions. In particular, we consider chance-constrained programming, recourse-based stochastic programming and dynamic programming. In Section III, we consider approaches that are appropriate when there are not distributions available to characterize uncertain problem parameters; more specifically, we survey robust and online optimization in this section.