The Value of Stochastic Modeling in Two-Stage Stochastic Programs with Cost Uncertainty

Although stochastic programming is probably the most effective frameworks for handling decision problems that involve uncertain variables, it is always a costly task to formulate the stochastic model that accurately embodies our knowledge of these variables. In practice, this might require one to collect a large amount of observations, to consult with experts of the specialized field of practice, or to make simplifying assumptions about the underlying system. When none of these options seem feasible, a common heuristic has been to simply seek the solution of a version of the problem where each uncertain variable takes on its expected value (otherwise known as the solution of the mean value problem). In this paper, we show that when 1) the stochastic program takes the form of a two-stage mixed-integer stochastic linear programs, and 2) the uncertainty is limited to the objective function, the solution of the mean value problem is in fact robust with respect to the selection of a stochastic model. We also propose tractable methods that will bound the actual value of stochastic modeling: i.e., how much improvement can be achieved by investing more efforts in the resolution of the stochastic model. Our framework is applied to an airline fleet composition problem. In the three cases that are considered, our results indicate that resolving the stochastic model can not lead to more than a 7% improvement of expected profits thus providing arguments against the need to develop these more sophisticated models.