Introduction

This represents a synthesis of what we have become acquainted with so far. Decision making under uncertainty is a quite challenging topic, merging probability theory and statistics with optimization modeling. This mix may result in quite demanding mathematics, which we will avoid by focusing on fundamental concepts and a few illustrative toy examples to clarify them.

One preliminary question that we should address is: Which kind of uncertainty should we consider? We take a rather standard view, i.e., that uncertainty may be represented by the classical tools of probability and statistics. In fact, this is not to be taken for granted, as there are quite different kinds of uncertainty. Compare the roll of a die against the production decision for a brand-new and truly innovative product. In the first case we do not know which number will be drawn, and betting on it means making a risky decision. However, we have no doubt about the rules of the game. In other words, we have a well-defined probability distribution of a random variable, and we just do not know in advance its realization. In the second case, we do not even know the probability distribution, which will be more subjective than fact-based. In extreme cases, even the very use of probabilities is questionable. It has been proposed to distinguish between decision making under risk and decision making under uncertainty. Strictly speaking, what we deal here with is decision making under risk, as we assume a known probability distribution. True uncertainty is a more elusive concept, possibly involving beliefs, rather than frequentist concepts. We will consider issues related to subjective probability. Here we introduce the fundamental concepts of risk aversion and the way we may account for it when making decisions. We also outline alternative frameworks, addressing additional issues like robustness, disappointment, and regret.

The formalization of decision trees, which is the subject of Section 13.1. Then, we consider the attitude toward risk. Much theory concerning random variables revolves around expected values; we have considered, for instance, the maximization of expected profit in newsvendor problems. However, such an approach may lead to unreasonable solutions for some decision problems, and this motivates the need to represent risk aversion. In Section 13.2 we introduce concepts related to utility theory and risk measures. Then, we start considering the extension of optimization models, namely, linear programming models, to decision making under risk. In Section 13.3 we consider two-stage stochastic linear programming, which is extended to the multistage case in Section 13.4. Stochastic linear programs can be quite challenging to solve, but we will not consider specific solution methods that have been proposed; what is really important is to consider a couple of basic examples, in order to understand the value of this modeling framework and the qualitative difference between solutions obtained when uncertainty is disregarded and those obtained by considering a set of alternative scenarios.