Modeling Uncertainty in Optimization Problems

This expository article discusses approaches for modeling optimization problems that involve uncertainty. The emphasis of the paper is on motivation and intuition rather than technical completeness. By virtue of its length, the paper is inevitably incomplete, and the reader is advised to look at other papers in this encyclopedia to explore the richness of modeling approaches that this paper can only hint at. How to model real decision problems using optimization models depends on the exact form of the problem being faced, and so it is difficult to give a general description of modeling techniques without some context in which to view the techniques. For this reason, we have chosen to illustrate the modeling approaches we discuss by placing them in the context of optimization of some models of electricity generation. Optimization problems that involve uncertainty fall into several broad classes that admit different modeling approaches. A good starting point is to consider any system with random effects that can be simulated. As an example consider a simple unit commitment problem in which the owner of a thermal generation unit and a wind farm must decide ahead of time whether to start up their thermal unit to meet a known future demand. If the wind blows enough then there will be sufficient power to meet the load, and so the cost incurred by the startup will be unnecessary. On the other hand, if the wind does not blow, and the unit is not started then the owner will incur some penalty of not supplying the demand. The decision in this case is a simple binary choice, and our model should enable the decision maker to make the correct choice. The natural question to ask at this point is “how do we determine the correct choice?” Since there are only two alternatives, one can investigate all the implications of making each choice and compare them. We suppose here, and throughout the rest of the paper that the best choice would be clear if there were no uncertainty in the model. In other words, if we