Decision under risk as a multicriteria problem

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Abstract

Most of the approaches to decision problems under uncertainty are based on decision paradigms, generally associated to an optimization process that leads to a final solution. For the Decision Maker, the basic decision is thus what paradigm to choose, the rest of the procedure being mainly technical.

In this paper, a different approach is advocated for this kind of problems. The main idea is to leave prescriptive models in favor of a more flexible approach, where risk related criteria are explicitly considered, conducting to an “equivalent” multicriteria (deterministic) model where decision-aid procedures can be used, with a greater involvement of the Decision Maker.

The paper discusses first the uncertainty model and then reviews existing paradigms for the single criterion problem under uncertainty. Proposed risk and opportunity attributes come mainly from the analysis of those methodologies and from risk perception studies reports.

Some hints about multicriteria aid methods and an illustrative example complete the paper.

Introduction

Decision under uncertainty is certainly one of the most frequent situations in practical decision-making, namely in planning activities in many fields. Modeling has been done mostly with scenarios, probabilistic or stochastic approaches and, in the last years, also with fuzzy set theory. Then, the usual approach for single criterion problems under uncertainty consists of selecting a decision paradigm, or rule, that finds the “optimal” solution of the problem. The expected value paradigm is by far the most frequent rule, but E-V analysis, utility theory, minimax approaches, regret minimization and fuzzy satisfaction have also been used as decision paradigms. Less frequently, minimizing risk is explicitly identified as a criterion, sometimes in a vague way, leading to different attributes and methodologies.

It is interesting to note that, when comparing the results produced by different paradigms, contradictory conclusions appear (this is particularly true between the expected value optimization and minimax, or regret minimization), which could lead to the meta-problem of choosing the “best” paradigm. However, our perspective is different – different paradigms may be considered as different points of view, equally interesting to characterize the uncertain outcome of an alternative, leading to a deterministic multicriteria problem. This is not completely new, so the aim of the paper is mainly to build a unified view over the issue and propose some procedures that may help selecting the indices that can be used as decision criteria.

The main advantage of this approach is to maintain the Decision Maker close to the process, which increases the robustness of the final solution regarding his preferences, even if a decision-aid phase is necessary to reach the preferred solution. We strongly believe that, in many circumstances, the insight gained by the Decision Maker more than compensates the potential disadvantages of increased complexity of the process. The systematic implementation of this concept leads to the identification of risk indices that can be used as attributes, but also of less frequently seen opportunity indices that may be useful to complete the picture.

The structure of this paper is the following. Probabilistic, fuzzy and scenario based models of uncertainty are addressed in Section 2, and decision rules are described in Section 3. Definitions of risk are invoked and compared (Section 4), in order to establish meaningful attributes, suitable for multicriteria analysis and decision-aid (Section 5). An illustrative example (Section 6), the conclusions and references complete the paper.

Section snippets

Uncertainty models

Dealing with uncertainty, or risk, includes building models for uncertainty (what may happen) and decision (what to do). Of course, in a less prescriptive way, we could substitute the latter by a decision-aid model (how to help the decision maker).

So, before going into decision-aid or decision-making models, it seems interesting to review the main approaches to model uncertainty itself. It is not important for the discussion if the problem is discrete, countable or continuous, so we will talk

Decision paradigms

We will begin by revising the main approaches, focusing in the concepts rather than in the mathematical details or algorithms, that can be seen in the references (besides, we are speaking about well-known methodologies). A common characteristic of the methodologies available to a decision maker for this situation is that they are generally prescriptive (with little exceptions), in the sense that, after being chosen, the paradigm goes to a “correct” or “optimal” decision with not too much

Risk indices

From the previous discussion about methodologies, the concept of risk emerges as an important way of qualifying possible decisions. In fact, it is probably the most frequent word when people express their concerns about deciding under uncertainty.

According to Anders et al. (1999), “risk is associated with the lack of certainty of an outcome and how sensitive one is to that outcome and thus to the uncertainty”. Of course, other definitions could be given, but the main point is that risk is a

A multicriteria approach

As stressed before, the approaches to optimization under uncertainty are typically prescriptive. After a “meta-decision” (which paradigm to use?) a final solution is, in most of the cases, immediately obtained, without any further intervention of the Decision Maker.

However, most of the times different aggregated attributes to evaluate the alternatives are identified, some related with central tendencies, others with risk.

We advocate the substitution of the original description of the

An illustrative example

We will give now a sketch of the ideas presented in the paper, by means of the example presented in Table 1, where the cost of 10 alternatives in three scenarios C1, C2 and C3 is presented. The table includes already a central value measure (expected cost) and five possible indices that may be used as attributes in the decision process: the worst cost of each alternative, the standard deviation of the cost, the probability of getting a cost over 60 (an important reference value for the Decision

Conclusions

When dealing with uncertainty, the use of predefined decision paradigms is reassuring for the Decision Maker, but may conduct to undesirable solutions, due to its excessive prescriptive nature, that leaves not much room for the Decision Maker to express his preferences. On the other hand, since paradigms are generally presented as the paradigm, we are not always sure that the Decision Maker is aware of the possibility of considering different points of view.

However, analysis of existing

Acknowledgements

The author is grateful to FCT (Portugal) for its support to the project POSI/SRI/41845/2001.

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