by
Philippe
Vincke
SMG,
Université Libre de Bruxelles
CP 210/01,
1050 Bruxelles, Belgium,
E-mail:
pvincke@ulb.ac.be
Introduction
It
is probably not useful any more to justify the introduction of the concept of
robustness and to emphasise the interest of this concept in decision aiding.
Confronted to the necessity (or, simply, the wish) to help a decision-maker, the
analyst cannot avoid the presence of a lot of uncertainties, at least at three
levels, as illustrated in figure 1. Traditional tools (like probability theory)
or more recent ones (possibility theory, fuzzy sets, rough sets, ...) are useful
but not sufficient to cope with all these uncertainties. Moreover they introduce
themselves new uncertainties at the three levels of figure 1. So, we need a
theoretical framework and methodologies to take into account the irreducible
part of ignorance contained in any decision aiding process.
No
unique definition of robustness has been accepted by the scientific community
until now and this is rather natural: the diversity of situations is so large
that it will probably be necessary to classify the types of decision problems
and the types of uncertainties before proposing different kinds of robustness
which could be operational. In the literature, we can essentially distinguish 4
concepts, which could be starting points for future developments:
1. the concept of robust decision in a dynamic context (Gupta and Rosenhead, 1972; Rosenhead et al., 1972; Rosenhead, 1989) which could also be called flexibility in the sense that a decision at a given time is robust if it keeps open as many ``good'' plans as possible for the future;
2. the concept of robust solution in optimisation problems (Rosenblatt and Lee, 1987, in facility design problem; Mulvey et al., 1994, in mathematical programming; Kouvelis and Yu, 1997, in combinatorial optimisation) where robust means ``good in all or most versions'', a version being a plausible set of values for the data in the model;
3. the concept of robust conclusion (Roy, 1998) where robust means ``valid in all or most versions'', a version being an acceptable set of values for the parameters of the model;
4. the concept of robust method (Vincke, 1999 a, b, Sorensen, 2001) where robust means ``which gives results valid in all or most versions'', a version being a possible set of values for the data of the problem and for the parameters of the method.
Remark
that we have adopted here the term “version”, recently proposed by B. Roy,
instead of “scenario”, in order to avoid any reference to an unknown future
and to the traditional probabilistic approaches.
There
is no contradiction between these definitions: they only illustrate the fact
that different kinds of robustness should be introduced in decision aiding. It
is also important to avoid any confusion between robustness and the traditional
stability property associated to sensitivity analysis. In this last context, a
solution (decision) is determined in a particular version and an a posteriori
study is made of the neighbourhood of that solution. The idea of robustness
leads to consider, a priori, several versions (eventually rather different from
each other) and to look for solutions (decisions, conclusions) which are good
(valid) in all or most versions. In this perspective, the expression
``robustness analysis'' should be avoided because robustness considerations must
be integrated during the decision aiding process and are not the result of an a
posteriori analysis.
In
the case where the decision problem is modelized as an optimisation problem and
where a finite number of versions (sets of values for the data and the
parameters of the model) has to be taken into account, one could argue that
there are some similarities between searching for a good robust solution of the
optimisation problem (that is a solution which is good in most versions and not
too bad in the other) and searching for a good compromise solution of a
multicriteria problem where the versions play the role of criteria. A concept
like efficiency (non-dominance) could be used to select the candidates to the
qualification of robust solutions and multicriteria methodologies could be
applied to determine good robust solutions. The interested reader will find an
illustration of that approach in Hites (2000), where the robustness of a
solution does not only depend on its worst performance (as in Kouvelis and Yu)
but simultaneously on its good an bad performances (without trivially applying
an arithmetic or a weighted mean whose inconvenients were abundantly illustrated
in Bouyssou et al., 2000). See also the concept of generalised Lorenz dominance
used by Perny and Spanjaard (2002) in the same kind of problem.
Despite
the similarities between searching for a good compromise solution of a
multicriteria problem and searching for a good robust solution of a
multiversions optimisation problem, one should avoid to consider that the only
difference is the vocabulary (on this subject, see Hites et al., 2003). In the
formulation of the problem, the family of criteria is built in such a way that
the opinion of the decision-maker is as well represented as possible (cf. the
concept of coherent family of criteria in Roy and Bouyssou, 1993), while the set
of versions is often at least partially imposed by external conditions. Moreover
the number of versions can be infinite (if the values of the parameters are
defined through intervals) and the concepts of relative importance or
preferential independence are not easy to transpose. Finally, most decision
problems are simultaneously multicriteria and multiversions. In conclusion, we
are convinced that the concept of robustness justifies the development of a
specific theoretical framework and of new methodologies.
An
important feature of robustness, in our mind, is its subjective dimension.
The
fact that a decision (solution, conclusion) can be considered as robust depends
on the more or less great margin the decision-maker is ready to concede in the
information he wants to receive form the analyst. Let us consider an
optimisation problem and suppose that the decision-maker is not affected by a
difference of 5% between the values of different solutions. In this case, a
solution whose value differs by less than 5% from the optimum in each version
could be called robust (in the sense «good in all the versions »).
Replacing 5% by another value will change the set of robust solutions. In
another context, if you aggregate preferences in an outranking relation by using
weights for the criteria, the robustness of the final relation (the versions
being the sets of values for the weights) will depend on which modification of
the relation is considered as negligible by the decision-maker. If he is very
severe and considers that any modification is important, then imposing the
robustness of the result will lead to a very poor relation (as it must be the
same for all the sets of weights). But if he accepts some modifications (for
example the replacement of some strict preferences by indifferences) then other
robust results will be possible. More details on these examples and a
proposition of theoretical framework in this direction were proposed in Vincke,
1999b.
Thirty years ago, the scientific community in decision aiding was confronted to the challenge of solving problems where several criteria were present. This led to the development of MCDA and to a lot of new concepts and tools. We are now facing to the challenge of taking into account the uncertainties, which are irremediably present in any decision aiding process. This probably justifies the development of a specific vocabulary, a specific theoretical framework, a new typology of the decision problems and new methodologies.
It is an open field for the future.
References
N.B.:
A list of references on robustness is maintained by Romina Hites at the
following address http://smg.ulb.ac.be/ then choice Research / Robustness. Every
suggestion of new reference is welcome.
D. Bouyssou, Th. Marchant, M. Pirlot, P. Perny, A. Tsoukias, and Ph. Vincke. Evaluation and decision models: a critical perspective. Kluwer Academic Publishers, 2000.
S.K. Gupta and J. Rosenhead. Robustness in sequential investment decisions, Management Science, 15(2):18-29, 1972.
R. Hites. The Robust Shortest Path Problem. Thesis. Université Libre de Bruxelles, 2000.
R. Hites, Y. De Smet, N. Risse, M. Salazar, and Ph. Vincke. A comparison of multicriteria and robustness frameworks. (To appear), 2003.
P. Kouvelis and G. Yu. Robust Discrete Optimisation and Its Applications. Kluwer Academic Publishers, Netherlands, 1997.
J.M. Mulvey, M.J. Verderbel, and S.A. Zenios. Robust optimisation of large scale systems. Statistics and Operations Research, Princeton University, Princeton, NJ, Report SOR-91-13, 1994.
P. Perny and O. Spanjaard. Preference-based search in state space graphs. AAAI, 02 - Proceedings of the Eighteenth National Conference on Artificial Intelligence: 751‑756, 2002.
M.J. Rosenblatt and H.L. Lee. A robustness approach to facilities design. International Journal of Production Research, 25:479-486, 1987.
M.J. Rosenhead. Rational Analysis for a Problematic World. Wiley, New York, 1989.
M.J. Rosenhead, M. Elton, and S.K. Gupta. Robustness and optimality as criteria for strategic decisions. Operational Research Quarterly, 23(4):413-430, 1972.
B. Roy. A missing link
in operational research decision aiding: robustness analysis. Found. Comput. Decis. Sci.,
23(3):141-160, 1998.
B. Roy and D. Bouyssou. Aide Multicritère à la Décision : Méthodes et Cas. Economica, Paris, 1993.
K. Sorensen. Tabu searching for robust solutions. Porto, July 2001. 4^{th} Metaheuristics International Conference.
Ph. Vincke. Robust and neutral methods for aggregating preferences into an outranking relation. European Journal of Operational Research, 112(2):405-412, 1999a.
Ph. Vincke. Robust solutions and methods in decision aid. Journal of Multicriteria Decisions Analysis, 8:181-187, 1999b.
EWG-MCDA Newsletter, Series 3, No.8, Fall 2003