(1) INESC Coimbra, Rua Antero de Quental, 199, 3000-033 Coimbra,
Portugal

(2) Faculdade de Economia, Universidade de Coimbra, Av. Dias da Silva
165, 3004-512 Coimbra, Portugal

(3) LAMSADE, Université Paris-Dauphine, Place du Maréchal
De Lattre de Tassigny, 75775 Paris Cedex 16, France

(4) Escola Superior de Tecnologia e Gestão, Instituto Politécnico
de Leiria, 2401-951 Leiria, Portugal

IRIS (Interactive Robustness analysis and parameters' Inference
for multicriteria Sorting) is a Decision Support Software designed to address
the problem of sorting a set of actions (alternatives, projects, candidates,
etc.) into predefined ordered categories, according to their evaluations
(performances) on multiple criteria. For instance, it may be used to sort
funding requests according to merit categories (e.g. “Very good”, “Good”,
“Fair”, “Not eligible”), or to sort loan applicants into categories (e.g.
“Accept”, “Require more collateral”, “Reject”), or to sort employees in
a company into categories that define incentive packages, etc.

IRIS implements the methodology presented in Dias et al. (2002), using
a pessimistic concordance-only variant of the ELECTRE TRI method. Rather
than demanding precise values for the ELECTRE TRI parameters, IRIS allows
to enter constraints on these values, namely assignment examples that it
tries to restore. It adds a module to identify the source of inconsistency
among the constraints when it is not possible to respect all of them at
the same time, according to a method described in Mousseau et al. (2002).
On the other hand, if the constraints are compatible with multiple assignments
for the actions, IRIS allows drawing robust conclusions by indicating the
range of assignments (for each action) that do not contradict any constraint.

The main characteristics of IRIS are the following:

- IRIS implements a concordance-only variant of the pessimistic ELECTRE TRI.
- IRIS accepts imprecision concerning the criteria weights and the cutting level. The users may indicate intervals for each of these parameters, as well as linear constraints on the weights, rather than being forced to indicate precise values for all these parameters. Furthermore, the constraints may be defined indirectly, as indicated in the next item.
- IRIS accepts assignment examples, where the users indicate minimum and maximum categories for some of the actions, according to their holistic judgement (e.g. “action a-1 is a typical element of C3”, or “action a-2 should be placed in category C3 or higher”, or “I hesitate: action a-2 should be placed in category C3 or C4”). These assignment examples are translated into constraints on the parameter values, meaning that the assignments of ELECTRE TRI should restore these examples.
- When the constraints are inconsistent, IRIS infers a combination of parameter values that least violates the constraints, by minimizing the maximum deviation. Then, it shows the sorting that corresponds to these parameter values (see example in Fig. 1). Furthermore, a module becomes available to determine the alternative subsets of constraints that must be removed to restore the consistency (see example in Fig. 2).
- When the constraints are consistent, IRIS infers a "central" combination of parameter values by minimizing the maximum slack. For each action, it depicts the category corresponding to that combination, as well as the range of categories where the action might be assigned without violating any constraint (robustness analysis). For each category in the range IRIS may also determine a combination of parameter values that assigns the action to that category (see example in Fig. 3).
- Moreover, when the constraints are consistent, IRIS may compute some indicators concerning the precision of the inputs (by estimating the volume of the polyhedron of all feasible combinations of parameter values) and the precision of the outputs (by indicating the geometric mean of the number of possible assignments per action). See example in Fig. 4.
- These features allow decision makers to build sorting models in a progressive and interactive manner, where the output at a given iteration is used to guide the revision of the input for the following iteration. The general idea is to start with few constraints of the parameter values, adding more inequalities as a product of an interactive learning process about the problem and the method. This process should aim at progressively reducing the set of accepted combinations of parameter values, until the end users (decision makers, problem owners) are satisfied with the results’ precision, and yet comfortable with and confident about the constraints introduced.

- a set of constraints and assignment examples defining a set of acceptable combinations of parameter values;
- an inferred combination of parameter values defining a model in a precise manner;
- a precise assignment or range of assignments for each action in A that is robust with respect to the constraints inserted.

Figure 1. The proposed sorting does not restore the assignment example
that a1 belongs to C3 due to inconsistent constraints. It corresponds to
the parameter values indicated on the right bottom of the screen.

Figure 2. Given an inconsistent system of constraints (on the left),
IRIS suggests five alternative ways to restore the consistency by removing
constraints. The first suggestion is to remove constraint no. 2; the fifth
suggestion is to remove constraints no. 7, 8, and 12.

Figure 3. There is a range of categories where each action may be assigned
to without violating any constraint (e.g. a robust conclusion is that a2
is not worse than C3). The proposed assignment (darker cell) corresponds
to the inferred parameter values shown in the last row of the grid on the
right. The parameter values shown in the penultimate line of that grid
lead to the assignment of a28 to C5, corresponding to the selected cell.
If the user chooses another cell these values will change. IRIS also shows
that a28 cannot be assigned to C2, regardless of the parameter values that
are chosen.

Figure 4. (Left:) the constraints define a 7-dimension polytope of
very small volume; from the combinations of parameter values that satisfy
the bounds, about 14.3% also respect the remaining constraints. (Right:)
the geometric mean of the number of categories where each action may be
assigned (respecting all the constraints) is now 1.357, which is less 47.7%
relatively to the previous iteration.

MORE INFORMATION

INESC Coimbra

C/O Luís Dias

Rua Antero de Quental, 199, 3000-033 Coimbra, PORTUGAL

Fax: +351 239 824692, e-mail: LDias@inescc.pt

http://www4.fe.uc.pt/lmcdias/iris.htm

REFERENCES

Mousseau, V. J. Figueira, L. Dias, C. Gomes da Silva, J. Clímaco
(2002), "Resolving inconsistencies among constraints on the parameters
of an MCDA model", to appear in the European Journal of Operational Research.

Dias, L., V. Mousseau, J. Figueira, J. Clímaco (2002), "An Aggregation/Disaggregation
Approach to Obtain Robust Conclusions with ELECTRE TRI", European Journal
of Operational Research, vol 138, 332-348.