Checking the stability property with missing evaluations

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The main formal multicriteria decision philosophies

The purpose of this section is to present the two main philosophies, or schools, for solving multicriteria decision problems, namely the American school and the Eu-ropean school. We present the main outlines of the two approaches, which are mostly composed of two steps: the construction of the evaluation model, which mainly re-sults in defining the parameters of the considered method, and its exploitation with the aim of providing a final recommendation to the decision-maker. These two philosophies may appear as quite similar, especially by the fact that they deal with information of the same kind, but they clearly diff er in the way they envision the construction of the decision-maker’s preference. We will not detail here the consideration of one particular philosophy, done by an analyst with the decision-maker’s agreement: For further details, an interested reader may refer to Simpson [Sim96]. Simply notice that the analyst must ensure the complete match between the considered philosophy and the decision-maker’s thought patterns.

Designing and exploiting an overall value function

In a very intuitive approach, the Multiattribute Value Theory (mavt) [KR76] consists in avoiding the difficulty of the multidimensional evaluation by creating a unique criterion that aggregates every decision criteria in order to construct a numerical representation of the global value of each alternative, often called score or value, based on the decision-maker’s preferences. Such an approach, which assumes that the decision-maker’s preferences can be specified as a weak order over the set of alternatives A (see e.g. [BP05, KLST71]), attempts to model the complete and transitive binary relation ⌫ on A via an overall value function U [Roy71, KR76] such that, for every (x, y) 2 A2: x ⌫ y () U(x) > U(y), 8x, y 2 A.
This overall value function may be of any kind, but the most studied one is the additive form [Fis64, Fis65, Fis70], the overall evaluation being equal to the sum of the whole marginal value functions ui: X U(x) = ui(xi), 8x 2 A, i2F where ui is a function entirely determined by the criterion i . More particularly, its linear form of a weighted sum is fairly studied, where we associate with every criteria i a weight wi: X U(x) = wi.xi, 8x 2 A. i2F.
One classical example is the evaluation of students in a class: Each course may represent one criterion for the evaluation and the applied coefficients define the linear additive value functions allowing the computation of a global score for each student, namely his average grade. In that case, the weights parameters w i are acting like trade-offs among the criteria, allowing to balance a locally weak evaluation on one criterion by a good performance on one or some others.
Two different approaches exist to specify the parameters of a given method: either via direct preference information, where the parameters are first assessed and then the aggregated relation is computed, or via indirect preference information, where some a priori partial knowledge of the resulting aggregated relation is used in order to infer plausible estimators of the parameters. We shall remark that the additive form of U makes an important hypothesis on the independence of the criteria. Indeed, it is possible to have some interactions between some subsets of criteria. For instance, we may consider the use of a Choquet integral [Cho53] or a Sugeno integral [ Sug74], that considers positive interactions on some criteria coalitions, when there is a reinforcement of the impact of one criterion with another one, or some negative ones when there is some redundancy in the different criteria. But, as the number of parameters increases, the complexity of the model, especially the tuning of its parameters, becomes a harder task, such that it does not seem realistic to ask the decision-maker to provide such parameters. Notice that in these cases, the weights are called capacities for the Choquet integral or fuzzy measures for the Sugeno integral.

Designing and exploiting an outranking relation

In response to the assumptions of the valued methods, often considered as too difficult to ensure, the European School, lead by Roy’s work [Roy68], suggests a different approach based on the use of less precise information, but with a stronger support. Indeed, instead of building complex value functions that rank every alter-natives on a common scale, such an approach constructs a binary relation, called the outranking relation, by comparing the alternatives systematically by pairs. The main purpose of these methods is not to provide a complete preorder on the alternatives, but to support the decision-maker on his preferences and his choices, in order to explicit them. Literature on mcda methods suggests different ways of constructing the outrank-ing relations. Among the most famous ones, you can find the electre-like methods (see for example [KR76, RB93] with their detailed description) or the promethee-like methods (an extensive presentation can be found in [BM02]), and also the rubis method [BMR08]. Again, as it is not in the scope of this thesis, we will not detail them.
The outranking paradigm is the following: we consider that an alternative x outranks an alternative y when there is sufficient support amongst the criteria to validate the fact that x is at least as good as y. In a formal manner, it translates the fact that there is a qualified majority of weighted criteria on which x is performing at least as good as y and there is no criterion where y seriously outperforms x [RB93]. Notice that the outranking relation is neither transitive nor reflexive.
Unlike in mavt, the outranking methods permit three types of alternatives com-parisons, wich are: preference , indifference and also incomparability. According to Roy [Roy90], incomparability allows to represent decision-maker’s hesitations which may result from phenomena like uncertainty, conflicts and/or contradictions. Most of the time, it results from the comparison of two alternatives stating some very contrasted advantages, describing two opportunities completely opposed. An alternative x outranks another alternative y when x is at least as good as y. Logically, x and y are indifferent when both alternatives outranks the other one (namely x is at least as good as y and y is at least as good as x). In a similar manner, x is said to be preferred to y when x outranks y and y does not outrank x.

Quality of the expressed preferential information

To better describe the “quality” of the expressed information, we should make a clear distinction between a precise information and an accurate information:
Definition 2.1 (Preciseness) An information, given by the decision-maker, is said to be precise when it constraints the value of one parameter, or the ratio be-tween some parameters into reduced intervals (the intervals may be reduced to a unique value). An imprecise information is then a less restrictive constraint.
Definition 2.2 (Accuracy) An information may be also viewed as accurate when it can be considered in total accordance with the decision-maker’s mind. On the contrary, an inaccurate information is going against the decision-maker’s thoughts.
An example of precise information can be the association of a criterion weight with a unique value (e.g. “the weight associated with criterion i is 0.2”), or the fact that two criteria must be associated with the same weight. An accurate information may be the clear consideration by the decision-maker of a criterion more important
than another one, without any precision on their relative importance degree. Notice that this last information is accurate but also imprecise. Also notice that we can have precise information that are inaccurate, when for instance a decision-maker is asked to give a precise value for a criterion weight, but he may not be totally confident about the expressed value.
In a quite intuitive manner, one can conceive that the more precise the infor-mation are, the more questionable their accuracy is. In that case, assuming the modeling of a decision problem, it is appropriate to consider that only an expert decision-maker is comfortable in the expression of precise and accurate preferential information, due to his experience in the domain on which he uses to take this particular decision.
For a novice decision-maker, the expression of precise preferential information on the parameters may appear quite arbitrary. Indeed, he may be able to provide an accurate partial preorder between some criteria (for instance, when comparing some cars, the fact that the color is less important than the security), but he probably cannot express the exact relative importance between two criteria (for instance, the security is three times more important than the color in the decision). Asking for such precise, but inaccurate, input-oriented information may result in the setting of a method that will not reflect the decision-maker’s expectations. In consequence, it seems more advisable to focus on less precise information, but with an incontestable accuracy (i.e. a stronger support from the decision-maker).

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Table of contents :

Introduction
I An overview of robust multiple criteria decision aid 
1 Multi criteria decision aid 
1.1 The decision aiding approach
1.1.1 Making a decision
1.1.2 Aiding the decision
1.1.3 Involving the decision-maker
1.2 Modeling the decision aid process
1.2.1 Defining the fundamental decision objects
1.2.2 Formulating the decision aid problem
1.2.3 Modeling and exploiting preferences
1.3 The main formal multicriteria decision philosophies
1.3.1 Designing and exploiting an overall value function
1.3.2 Designing and exploiting an outranking relation
2 Preference elicitationprocesses 
2.1 Decision-makers’ profiles and preferential information
2.1.1 Decision-makers’ profiles
2.1.2 Expressing some preferential information
2.1.3 Quality of the expressed preferential information
2.2 Setting up an iterative preference elicitation process
2.2.1 Principles of an iterative preference elicitation approach
2.2.2 An overview of disaggregation approaches
2.2.3 Knowing the potential pitfalls
2.3 Analysing the robustness of preferential results
2.3.1 Sensitivity analysis within multi attribute valued theory
2.3.2 Credibility level cutting technique
2.3.3 Dealing with imprecise but accurate information
2.3.4 Stability of outranking relations
II On the stability of median-cut outranking relations 
3 Stabilityof themedian-cut outrankingdigraph 
3.1 Preliminary definitions
3.1.1 Construction of a weighted outranking relation
3.1.2 Weights preorder
3.1.3 Defining the preferable relation
3.2 Defining the stability of valued outranking relations
3.2.1 Stability
3.2.2 Extensible stability
3.2.3 !-stability
3.3 Additional properties
3.3.1 Limitation of the stability
3.3.2 Stability of the preferable relation
3.3.3 Stability within the context of the sorting problem
3.3.4 Checking the stability property with missing evaluations
3.3.5 Properties on the discrimination of the preorder
4 Stable elicitationof criteriaweights 
4.1 Stability constraints
4.1.1 Auxiliary variables and constraints
4.1.2 Modeling of the stability constraints
4.1.3 Constraint relaxation using slack variables
4.2 Taking into account decision-maker’s preferences
4.2.1 Types of preferential information
4.2.2 Preferences on alternatives
4.2.3 Preferences on criteria
4.3 Mathematical programs
4.3.1 Control algorithm (acon)
4.3.2 milp with real relaxed stability constraints (stab1)
4.3.3 milp with boolean relaxed stability constraints (stab2) .
5 Elicitationofweights andotherparameters 
5.1 Elicitation of weights and thresholds
5.1.1 Modeling of the constraints on the thresholds
5.1.2 Additional preferential information
5.1.3 The complete models
5.2 Elicitation of weights and categories profiles
5.2.1 Modeling of the constraints on the profiles
5.2.2 Ensuring a stable assignment of an alternative
5.2.3 The complete models
5.2.4 Stable assignment of the other alternatives
III A progressive method for a robust parameters elicitation 
6 Empirical validationof the algorithms 
6.1 Parameters elicitation from a complete set of information
6.1.1 Elicitation of criteria weights
6.1.2 Elicitation of criteria weights and discrimination thresholds .
6.1.3 Elicitation of criteria weights and categories profiles
6.2 Iterative recovering of the median-cut outranking relation
6.2.1 Iterative elicitation of criteria weights
6.2.2 Iterative elicitation of both criteria weights and thresholds .
6.3 Impact of the stability constraints on the preference modeling
6.3.1 Impact on the weights preorder
6.3.2 Impact on the preference discriminating thresholds
7 rewat: Robust elicitation of the weights and thresholds 
7.1 Designing a robust elicitation protocol
7.1.1 Stage i: Initialising the outranking preference model
7.1.2 Stage ii: Validating the criteria weights preorder
7.1.3 Stage iii: Tuning the numerical values of the criteria weights
7.2 Tools for supporting a robust elicitation
7.2.1 Dynamic pairwise performance comparison table
7.2.2 Display of the elicited weights preorder
8 Case study: Applying for a Ph.D. thesis 
8.1 Applying the rewat process
8.1.1 Stage i: Initialising the outranking preference model
8.1.2 Stage ii: Validating the criteria weights preorder
8.1.3 Stage iii: Tuning the numerical values of the criteria weights
8.2 Critical review of the case study
8.2.1 Encountered difficulties
8.2.2 Perspectives for future methodological enhancement
Summary of the main achievements
Perspectives
Concluding remarks
A Annex 
A.1 Mathematical proof of Proposition 3.8
A.2 Complete mathematical models
A.2.1 acon
A.2.2 stab1
A.2.3 stab2
A.2.4 acon’
A.2.5 stab’1
A.2.6 stab’2
A.2.7 acon?
A.2.8 stab?1
A.2.9 stab?2
A.3 Data for the case study
A.4 Local concordance values for the case study
A.4.1 Initial local concordance values
A.4.2 After the comparison of alternatives sw2 and ca2
A.4.3 After the comparison of alternatives nl2 and us2
A.4.4 After the comparison of alternatives nl1 and us1
A.4.5 After the comparison of alternatives ca2 and fr2
A.4.6 After the comparison of alternatives fr2 and en2
A.4.7 Validation of the new preorder >w2
A.4.8 Validation of the preorder and the thresholds
Bibliography 

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