Author: haroonkhan

The best way to introduce multistage stochastic models is a simple asset–liability management (ALM) model.24 We have an initial wealth W0, that should be properly invested in such a way to meet a liability L at the end of the planning horizon H. If possible, we would like to own a terminal wealth WH larger than L; however, we should account properly for…

Multistage stochastic programming formulations arise naturally as a generalization of two-stage models. At each stage, we gather new information and we make decisions accordingly, taking into account immediate costs and expected future recourse cost. The resulting decision process may be summarized as follows:23 From the point of view of time period t = 0, the decisions x1, …, xH are…

Mean–risk formulations are based on the idea of trading off expected profit (or return) against a risk measure. Classical mean–variance portfolio optimization relies on an analytical representation of variance, which leads to an easy convex quadratic programming problem. This need not be the case if we choose another risk measure. Value at risk is easy…

In Section 13.1.1 we defined EVPI, which is not only a way to price perfect information, but also a measure of the impact of uncertainty. If EVPI is low, uncertainty is not that relevant in the decision. However, EVPI is in most cases a theoretical construct, as we cannot trade the unpleasing here-and-now decision problem for the…

In Section 12.2.1 we dealt with a production planning problem within an assembly-to-order (ATO) framework. There, we disregarded demand uncertainty and built a deterministic LP model. Now, in order to make the model a bit more realistic, we represent demand uncertainty by a scenario tree and adopt a two-stage stochastic linear programming framework: Of course, we cannot…

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…

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…

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…

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…