So far, in terms of concrete procedures, we have considered only decision trees, which are well suited to cope with discrete decisions, when uncertainty can be represented by a finite set of scenarios. More generally, we would like to solve a problem like where S is a subset of , and the expectation can be taken with respect… Continue reading TWO-STAGE STOCHASTIC PROGRAMMING MODELS
Month: February 2023
Quantile-based risk measures: value at risk
Given the limitations of standard deviation and variance as risk measures, alternative ones have been proposed. To be specific, we will refer once more to a financial investment problem, where risk is related to portfolio loss. The most widely known such measure is value at risk [VaR; not to be confused with variance (Var)]. The VaR concept… Continue reading Quantile-based risk measures: value at risk
Mean–risk optimization
If asked about our utility function, we would hardly be able to give a sensible answer. However, in real life, we do trade off expectations against risk; to do so, we need a way to measure risk. DEFINITION 13.1 A risk measure is a function ρ(X), mapping a random variable X into the set of nonnegative real numbers . In… Continue reading Mean–risk optimization
A conceptual tool: the utility function
The idea that most decision makers are risk-averse is intuitively clear, but what does risk aversion really mean? A theoretical answer, commonly put forward in economic theory, can be found by assuming that decision makers order uncertain outcomes by a utility function rather than by straightforward expected monetary values. To introduce the concept, let us consider simple… Continue reading A conceptual tool: the utility function
RISK AVERSION AND RISK MEASURES
So far, when dealing with a decision problem under risk, we have used expected profit or expected cost as the criterion of choice. We did so, e.g., for the newsvendor problem,2 as well as for the decision trees of the previous section. But does this actually make sense? The following examples show that this need not… Continue reading RISK AVERSION AND RISK MEASURES
Expected value of perfect information
Decision trees are a very simple tool for framing decision problems with a discrete set of alternatives and a discrete representation of uncertainty. We may start moving to more complicated cases by expressing a decision problem under risk in a more general way: It is important to understand what problem (13.1) represents: This is a here-and-now decision,… Continue reading Expected value of perfect information
DECISION TREES
Decision trees are a natural way to describe decision problems under risk, involving a sequence of decisions among a finite set of alternative options and a set of discrete scenarios, modeling uncertain outcomes that follow our decisions. Actually, we have already dealt with decision trees informally in earlier examples.1 Now we should treat this formalism more… Continue reading DECISION TREES
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… Continue reading Introduction
The impact of model formulation
We have seen that commercial branch and bound procedures compute bounds by LP-based (continuous) relaxations. Given a MILP problem where S denotes its feasible set, the continuous relaxation is obtained by relaxing the integrality constraints, which yields the relaxed feasible set and the relaxed problem: If we could find the convex hull of S, which is a polyhedron, the… Continue reading The impact of model formulation
LP-based branch and bound method
When dealing with a MILP, the simplex method cannot guarantee an integer solution; more often than not, we obtain a fractional solution. There is an even more troubling fact, which is essentially due to the lack of convexity. In fact, if some magical oracle hands over the optimal solution of a continuous LP, we know… Continue reading LP-based branch and bound method