Stochastic modeling concepts and applications

Stochastic modeling is a mathematical technique of decision making where some of the data that are incorporated into the objective or constraints of the mathematical functions are uncertain. Mathematically, uncertainty is normally characterized through a probability distribution on the parameters. Theoretically, the uncertainty can be defined rigorously, but in practice it can range in detail from a few scenarios (possible outcomes of the data) to specific and precise joint probability distributions [1]. The outcomes are generally described in terms of elements w of a set W. W can be, for example, the set of possible demands over the next few months. When some of the data are random, then solutions and the optimal objective value to the optimization problem are themselves random, then stochastic modeling is used.

Deterministic optimization problems are formulated with known parameters, but real world problems almost invariably include some unknown parameters [2]. When the parameters are known only within certain bounds, one approach to tackling such problems is called robust optimization. Here the goal is to find a solution, which is feasible for all such data and optimal in some sense [3]. Stochastic programming models are similar in style but take advantage of the fact that probability distributions governing the data are known or can be estimated. The goal here is to find some policy that is feasible for all (or almost all) the possible data instances and maximizes the expectation of some function of the decisions and the random variables. More generally, such models are formulated, solved analytically or numerically, and analyzed in order to provide useful information to a decision-maker.

The most widely applied and studied stochastic programming models are two-stage linear programs [4]. Here the decision maker takes some action in the first stage, after which a random event occurs affecting the outcome of the first-stage decision. A recourse decision can then be made in the second stage that compensates for any bad effects that might have been experienced, as a result of the first-stage decision. The optimal policy from such a model is a single first-stage policy and a collection of recourse decisions (a decision rule) defining which second-stage action should be taken in response to each random outcome [5].