Building a set of additive value functions representing a reference preorder and intensities of preference: GRIP method

We present a method called Generalized Regression with Intensities of Preference (GRIP) for ranking a finite set of actions evaluated on multiple criteria. GRIP builds a set of additive value functions compatible with preference information composed of a partial preorder and required intensities of preference on a subset of actions, called reference actions. It constructs not only the preference relation in the considered set of actions, but it also gives information about intensities of preference for pairs of actions from this set for a given decision maker (DM). Distinguishing necessary and possible consequences of preference information on the considered set of actions, GRIP answers questions of robustness analysis. The proposed methodology can be seen as an extension of the UTA method based on ordinal regression. GRIP can also be compared to the AHP method, which requires pairwise comparison of all actions and criteria, and yields a priority ranking of actions. As for the preference information being used, GRIP can be compared, moreover, to the MACBETH method which also takes into account a preference order of actions and intensity of preference for pairs of actions. The preference information used in GRIP does not need, however, to be complete: the DM is asked to provide comparisons of only those pairs of reference actions on particular criteria for which his/her judgment is sufficiently certain. This is an important advantage comparing to methods which, instead, require comparison of all possible pairs of actions on all the considered criteria. Moreover, GRIP works with a set of general additive value functions compatible with the preference information, while other methods use a single and less general value function, such as the weighted-sum.     

Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral

Within the multicriteria aggregation–disaggregation framework, ordinal regression aims at inducing the parameters of a decision model, for example those of a utility function, which have to represent some holistic preference comparisons of a Decision Maker (DM). Usually, among the many utility functions representing the DM’s preference information, only one utility function is selected. Since such a choice is arbitrary to some extent, recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the DM’s preference information. Until now, robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence. In this paper we propose a non-additive robust ordinal regression on a set of alternatives A, whose utility is evaluated in terms of the Choquet integral which permits to represent the interaction among criteria, modelled by the fuzzy measures, parameterizing our approach.     

ELECTRE: Robust ordinal regression for outranking methods

We present a new method, called ELECTRE^GKMS, which employs robust ordinal regression to construct a set of outranking models compatible with preference information. The preference information supplied by the decision maker (DM) is composed of pairwise comparisons stating the truth or falsity of the outranking relation for some real or fictitious reference alternatives. Moreover, the DM specifies some ranges of variation of comparison thresholds on considered pseudo-criteria. Using robust ordinal regression, the method builds a set of values of concordance indices, concordance thresholds, indifference, preference, and veto thresholds, for which all specified pairwise comparisons can be restored. Such sets are called compatible outranking models. Using these models, two outranking relations are defined, necessary and possible. Whether for an ordered pair of alternatives there is necessary or possible outranking depends on the truth of outranking relation for all or at least one compatible model, respectively. Distinguishing the most certain recommendation worked out by the necessary outranking, and a possible recommendation worked out by the possible outranking, ELECTRE^GKMS answers questions of robustness concern. The method is intended to be used interactively with incremental specification of pairwise comparisons, possibly with decreasing confidence levels. In this way, the necessary and possible outranking relations can be, respectively, enriched or impoverished with the growth of the number of pairwise comparisons. Furthermore, the method is able to identify troublesome pieces of preference information which are responsible for incompatibility. The necessary and possible outranking relations are to be exploited as usual outranking relations to work out recommendation in choice or ranking problems. The introduced approach is illustrated by a didactic example showing how ELECTRE^GKMS can support real-world decision problems.     

Robust ordinal regression for value functions handling interacting criteria

We present a new method called UTA^GMS–INT for ranking a finite set of alternatives evaluated on multiple criteria. It belongs to the family of Robust Ordinal Regression (ROR) methods which build a set of preference models compatible with preference information elicited by the Decision Maker (DM). The preference model used by UTA^GMS–INT is a general additive value function augmented by two types of components corresponding to “bonus” or “penalty” values for positively or negatively interacting pairs of criteria, respectively. When calculating value of a particular alternative, a bonus is added to the additive component of the value function if a given pair of criteria is in a positive synergy for performances of this alternative on the two criteria. Similarly, a penalty is subtracted from the additive component of the value function if a given pair of criteria is in a negative synergy for performances of the considered alternative on the two criteria. The preference information elicited by the DM is composed of pairwise comparisons of some reference alternatives, as well as of comparisons of some pairs of reference alternatives with respect to intensity of preference, either comprehensively or on a particular criterion. In UTA^GMS–INT, ROR starts with identification of pairs of interacting criteria for given preference information by solving a mixed-integer linear program. Once the interacting pairs are validated by the DM, ROR continues calculations with the whole set of compatible value functions handling the interacting criteria, to get necessary and possible preference relations in the considered set of alternatives. A single representative value function can be calculated to attribute specific scores to alternatives. It also gives values to bonuses and penalties. UTA^GMS–INT handles quite general interactions among criteria and provides an interesting alternative to the Choquet integral.     

Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions

We present a new method, called UTA^GMS, for multiple criteria ranking of alternatives from set A using a set of additive value functions which result from an ordinal regression. The preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives A^R ⊆ A, called reference alternatives. The preference model built via ordinal regression is the set of all additive value functions compatible with the preference information. Using this model, one can define two relations in the set A: the necessary weak preference relation which holds for any two alternatives a, b from set A if and only if for all compatible value functions a is preferred to b, and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b. These relations establish a necessary and a possible ranking of alternatives from A, being, respectively, a partial preorder and a strongly complete relation. The UTA^GMS method is intended to be used interactively, with an increasing subset A^R and a progressive statement of pairwise comparisons. When no preference information is provided, the necessary weak preference relation is a weak dominance relation, and the possible weak preference relation is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary relation and it is impoverishing the possible relation, so that they converge with the growth of the preference information. Distinguishing necessary and possible consequences of preference information on the complete set of actions, UTA^GMS answers questions of robustness analysis. Moreover, the method can support the decision maker when his/her preference statements cannot be represented in terms of an additive value function. The method is illustrated by an example solved using the UTA^GMS software. Some extensions of the method are also presented.     

DIS-CARD: a new method of multiple criteria sorting to classes with desired cardinality

In this paper, we present a new preference disaggregation method for multiple criteria sorting problems, called DIS-CARD. Real-life experience indicates the need of considering decision making situations in which a decision maker (DM) specifies a desired number of alternatives to be assigned to single classes or to unions of some classes. These situations require special methods for multiple criteria sorting subject to desired cardinalities of classes. DIS-CARD deals with such a problem, using the ordinal regression approach to construct a model of DM’s preferences from preference information provided in terms of exemplary assignments of some reference alternatives, together with the above desired cardinalities. We develop a mathematical model for incorporating such preference information via mixed integer linear programming (MILP). Then, we adapt the MILP model to two types of preference models: an additive value function and an outranking relation. Illustrative example is solved to illustrate the methodology.     

Robust multi-criteria ranking with additive value models and holistic pair-wise preference statements

We consider a problem of ranking alternatives based on their deterministic performance evaluations on multiple criteria. We apply additive value theory and assume the Decision Maker’s (DM) preferences to be representable with general additive monotone value functions. The DM provides indirect preference information in form of pair-wise comparisons of reference alternatives, and we use this to derive the set of compatible value functions. Then, this set is analyzed to describe (1) the possible and necessary preference relations, (2) probabilities of the possible relations, (3) ranges of ranks the alternatives may obtain, and (4) the distributions of these ranks. Our work combines previous results from Robust Ordinal Regression, Extreme Ranking Analysis and Stochastic Multicriteria Acceptability Analysis under a unified decision support framework. We show how the four different results complement each other, discuss extensions of the main proposal, and demonstrate practical use of the approach by considering a problem of ranking 20 European countries in terms of 4 criteria reflecting the quality of their universities.     

Multiple criteria sorting with a set of additive value functions

We present a new multiple criteria sorting method that aims at assigning actions evaluated on multiple criteria to p pre-defined and ordered classes. The preference information supplied by the decision maker (DM) is a set of assignment examples on a subset of actions relatively well known to the DM. These actions are called reference actions. Each assignment example specifies a desired assignment of a corresponding reference action to one or several contiguous classes. The set of assignment examples is used to build a preference model of the DM represented by a set of general additive value functions compatible with the assignment examples. For each action a, the method computes two kinds of assignments to classes, concordant with the DM’s preference model: the necessary assignment and the possible assignment. The necessary assignment specifies the range of classes to which the action can be assigned considering all compatible value functions simultaneously. The possible assignment specifies, in turn, the range of classes to which the action can be assigned considering any compatible value function individually. The compatible value functions and the necessary and possible assignments are computed through the resolution of linear programs.     

Handling imprecise evaluations in multiple criteria decision aiding and robust ordinal regression by <Emphasis Type="Italic">n</Emphasis>-point intervals

We consider imprecise evaluation of alternatives in multiple criteria ranking problems. The imprecise evaluations are represented by n-point intervals which are defined by the largest interval of possible evaluations and by its subintervals sequentially nested one in another. This sequence of subintervals is associated with an increasing sequence of plausibility, such that the plausibility of a subinterval is greater than the plausibility of the subinterval containing it. We explain the intuition that stands behind this proposal, and we show the advantage of n-point intervals compared to other methods dealing with imprecise evaluations. Although n-point intervals can be applied in any multiple criteria decision aiding (MCDA) method, in this paper, we focus on their application in robust ordinal regression which, unlike other MCDA methods, takes into account all compatible instances of an adopted preference model, which reproduce an indirect preference information provided by the decision maker. An illustrative example shows how the method can be applied in practice.     

Inferring robust decision models in multicriteria classification problems: An experimental analysis

Recent research on robust decision aiding has focused on identifying a range of recommendations from preferential information and the selection of representative models compatible with preferential constraints. This study presents an experimental analysis on the relationship between the results of a single decision model (additive value function) and the ones from the full set of compatible models in classification problems. Different optimization formulations for selecting a representative model are tested on artificially generated data sets with varying characteristics.     

Preference Programming with incomplete ordinal information

This paper extends possibilities for analyzing incomplete ordinal information about the parameters of an additive value function. Such information is modeled through preference statements which associate sets of alternatives or attributes with corresponding sets of rankings. These preference statements can be particularly helpful in developing a joint preference representation for a group of decision-makers who may find difficulties in agreeing on numerical parameter values. Because these statements can lead to a non-convex set of feasible parameters, a mixed integer linear formulation is developed to establish a linear model for the computation of decision recommendations. This makes it possible to complete incomplete ordinal information with other forms of incomplete information.     

Constructing a monotonic quadratic objective function in n variables from a few two-dimensional indifferences

We develop a model for constructing quadratic objective functions in n target variables. At the input, a decision maker is asked a few simple questions about his ordinal preferences (comparing two-dimensional alternatives in terms `better', `worse', `indifferent'). At the output, the model mathematically derives a quadratic objective function used to evaluate n-dimensional alternatives.Thus the model deals with some imaginary decisions (criteria aggregates) at the input, and disaggregates the decision maker's preference into partial criteria and their cross-correlations (=a quadratic objective function). Therefore, the model provides an approximation step which is next to the disaggregation of a preference into additively separable linear criteria with weight coefficients.The model is based on least squares fitting a quadratic indifference hypersurface (if n=2, indifference curve) to several alternatives which are supposed to be equivalent in preference. The resulting ordinal preference is independent of the cardinal utility scale used in intermediate computations which implies that the model is ordinal. The monotonicity of the quadratic objective function is implemented by means of a finite number of linear constraints, so that the computational model is reduced to restricted least squares.In illustration, we construct a quadratic objective function of German economic policy in four target variables: inflation, unemployment, GNP growth, and increase in public debt. This objective function is used to evaluate the German economic development in 1980–1994.In another application, we construct a quadratic objective function of ski station customers. Then it is used to adjust prices of 10 ski stations to the South of Stuttgart.In Appendix A we provide an original fast algorithm for restricted least squares and quadratic programming used in the main model.     

Multi-criteria semantic dominance: A linguistic decision aiding technique based on incomplete preference information

A linguistic decision aiding technique for multi-criteria decision is presented. We define a relation between alternatives as multi-criteria semantic dominance (MCSD). It adopts the similar ideal of the stochastic dominance by utilizing the partial information of the decision maker’s preference, which is only ordinal or partially cardinal. The MCSD rules based on three typical types of semanteme functions are introduced and proven. By using these rules, all the alternatives under consideration are divided into two mutually exclusive sets called efficient set and inefficient set. The decision maker who has such a semanteme function will never choose the alternative from the corresponding inefficient set as the optimal one. In such a way, when we analyze the linguistic decision information, the inherent fuzziness of preference can be handled and several controversial operations of the linguistic terms can be avoided. An example is also provided to illustrate the procedure of the proposed method.     

Kernel-based learning methods for preference aggregation

The mathematical representation of human preferences has been a subject of study for researchers in different fields. In multi-criteria decision making (MCDM) and fuzzy modeling, preference models are typically constructed by interacting with the human decision maker (DM). However, it is known that a DM often has difficulties to specify precise values for certain parameters of the model. He/she instead feels more comfortable to give holistic judgements for some of the alternatives. Inference and elicitation procedures then assist the DM to find a satisfactory model and to assess unjudged alternatives. In a related but more statistical way, machine learning algorithms can also infer preference models with similar setups and purposes, but here less interaction with the DM is required/allowed. In this article we discuss the main differences between both types of inference and, in particular, we present a hybrid approach that combines the best of both worlds. This approach consists of a very general kernel-based framework for constructing and inferring preference models. Additive models, for which interpretability is preserved, and utility models can be considered as special cases. Besides generality, important benefits of this approach are its robustness to noise and good scalability. We show in detail how this framework can be utilized to aggregate single-criterion outranking relations, resulting in a flexible class of preference models for which domain knowledge can be specified by a DM.      

Interactive outranking approaches for multicriteria decision-making problems with imprecise information

PROMETHEE is a powerful method, which can solve many multiple criteria decision making (MCDM) problems. It involves sophisticated preference modelling techniques but requires too much a priori precise information about parameter values (such as criterion weights and thresholds). In this paper, we consider a MCDM problem where alternatives are evaluated on several conflicting criteria, and the criterion weights and/or thresholds are imprecise or unknown to the decision maker (DM). We build robust outranking relations among the alternatives in order to help the DM to rank the alternatives and select the best alternative. We propose interactive approaches based on PROMETHEE method. We develop a decision aid tool called INTOUR, which implements the developed approaches.     

Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem

The Choquet integral preference model is adopted in Multiple Criteria Decision Aiding (MCDA) to deal with interactions between criteria, while the Stochastic Multiobjective Acceptability Analysis (SMAA) is an MCDA methodology considered to take into account uncertainty or imprecision on the considered data and preference parameters. In this paper, we propose to combine the Choquet integral preference model with the SMAA methodology in order to get robust recommendations taking into account all parameters compatible with the preference information provided by the Decision Maker (DM). In case the criteria are on a common scale, one has to elicit only a set of non-additive weights, technically a capacity, compatible with the DM’s preference information. Instead, if the criteria are on different scales, besides the capacity, one has to elicit also a common scale compatible with the preferences given by the DM. Our approach permits to explore the whole space of capacities and common scales compatible with the DM’s preference information.     

Reasoning with various kinds of preferences: logic,non-monotonicity,and algorithms

As systems dealing with preferences become more sophisticated, it becomes essential to deal with various kinds of preference statements and their interaction. We introduce a non-monotonic logic distinguishing sixteen kinds of preferences, ranging from strict to loose and from careful to opportunistic, and two kinds of ways to deal with uncertainty, either optimistically or pessimistically. The classification of the various kinds of preferences is inspired by a hypothetical agent comparing the two alternatives of a preference statement. The optimistic and pessimistic way of dealing with uncertainty correspond on the one hand to considering either the best or the worst states in the comparison of the two alternatives of a preference statement, and on the other hand to the calculation of least or most specific “distinguished” preference orders from a set of preference statements. We show that each way to calculate distinguished preference orders is compatible with eight kinds of preferences, in the sense that it calculates a unique distinguished preference order for a set of such preference statements, and we provide efficient algorithms that calculate these unique distinguished preference orders. In general, optimistic kinds of preferences are compatible with optimism in calculating distinguished preference orders, and pessimistic kinds of preferences are compatible with pessimism in calculating distinguished preference orders. However, these two sets of eight kinds of preferences are not exclusive, such that some kinds of preferences can be used in both ways to calculate distinguished preference orders, and other kinds of preferences cannot be used in either of them. We also consider the merging of optimistically and pessimistically constructed distinguished preferences orders.     

Multiphase support vector regression for function approximation with break-points

In this paper, we propose a novel multiphase support vector regression (mp-SVR) technique to approximate a true relationship for the case where the effect of input on output changes abruptly at some break-points. A new formulation for mp-SVR is presented to allow such structural changes in regression function. And then, we present a new hybrid-encoding scheme in genetic algorithms to select the best combination of the kernel functions and to determine both break-points and hyperparameters of mp-SVR. The proposed method has a major advantage over the conventional ones that different kernel functions can be possibly adapted to different regions of the data domain. Computational results in two examples including a real-life data demonstrate its capability in capturing the local characteristics of the data more effectively. Consequently, the mp-SVR has a high potential value in a wide range of applications for function approximations.