chance-constrained / scenario-based optimization

For optimization problems LP or MIP, if the input variables have a probability distribution, then objective function is likely following a probability distribution as well. Sometimes, if the variables' probability distributions are normal / Gaussian, then the combined distribution (e.g. the objective function) follows Normal distribution as well. Sometimes, the combined distribution is not Normal.

The question is usually optimizing for a goal, with a confidence level of e.g. 95% that all constraints are met.

Using advanced solvers like Gurobi, it can solve the problem if all following Normal distribution, because a normal distribution can be defined using Quadratic and Power constraints. If it is not Normal distribution, a solution is to sample N scenarios of the distribution, and the put in a constraint that at least 95% of the scenarios must meet the constraints.

Here are the detailed solutions: