Resolving inconsistencies among constraints on the parameters of an MCDA model

Abstract

We consider a framework where decision makers (DMs) interactively define a multicriteria evaluation model by providing imprecise information (i.e., a linear system of constraints to the modelÕs parameters) and by analyzing the consequences of the information provided. DMs may introduce new constraints explicitly or implicitly (results that the model should yield). If a new constraint is incompatible with the previous ones, then the system becomes inconsistent and the DMs must choose between removing the new constraint or removing some of the older ones. We address the problem of identifying subsets of constraints which, when removed, lead to a consistent system. Identifying such subsets would indicate the reason for the inconsistent information given by DMs. There may exist several possibilities for the DMs to resolve the inconsistency. We present two algorithms to identify such possibilities, one using {0,1} mixed integer linear programming and the other one using linear programming. Both approaches are based on the knowledge that the system was consistent prior to introducing the last constraint. The output of these algorithms helps the DM to identify the conflicting pieces of information in a set of statements he/she asserted. The relevance of these algorithms for MCDA is illustrated by an application to an aggregation/disaggregation procedure for the Electre Tri method.

Dealing with inconsistent judgments in multiple criteria sorting models

“…In [10] two algorithms are proposed to compute alternative ways to restore consistency by constraints deletion. We consider here the case where consistency can be solved by relaxing constraints rather than deleting them.…”

“…If we apply the algorithms proposed by [10] (i.e., inconsistency resolution via constraints deletion) to the set S, the solutions correspond to constraints relaxation and/or deletion. It follows from the preceding remark that it is possible to use the algorithm proposed by [10] to solve inconsistencies by relaxation (rather than deletion) of assignment examples.…”

“…It follows from the preceding remark that it is possible to use the algorithm proposed by [10] to solve inconsistencies by relaxation (rather than deletion) of assignment examples. Hence, in the rest of the paper, we will talk in terms of constraints deletion, knowing that it embraces the case of constraints relaxation.…”

“…Note that it is possible to design an algorithm for the same purpose using only linear programming (see the second algorithm presented in [10]). …”

“…Section 1 defines inconsistency relaxation and shows that the algorithms proposed by [10] still apply when considering constraints relaxation rather than constraints deletion. The section 2 considers the case where the DM is able to provide confidence levels associated to the assignment examples.…”