UTA-poly and UTA-splines: Additive value functions with polynomial marginals

Additive utility function models are widely used in multiple criteria decision analysis. In such models, a numerical value is associated to each alternative involved in the decision problem. It is computed by aggregating the scores of the alternative on the different criteria of the decision problem. The score of an alternative is determined by a marginal value function that evolves monotonically as a function of the performance of the alternative on this criterion. Determining the shape of the marginals is not easy for a decision maker. It is easier for him/her to make statements such as “alternative a is preferred to b”. In order to help the decision maker, UTA disaggregation procedures use linear programming to approximate the marginals by piecewise linear functions based only on such statements. In this paper, we propose to infer polynomials and splines instead of piecewise linear functions for the marginals. In this aim, we use semidefinite programming instead of linear programming. We illustrate this new elicitation method and present some experimental results.     

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.     

Preference disaggregation and statistical learning for multicriteria decision support: A review

Disaggregation methods have become popular in multicriteria decision aiding (MCDA) for eliciting preferential information and constructing decision models from decision examples. From a statistical point of view, data mining and machine learning are also involved with similar problems, mainly with regard to identifying patterns and extracting knowledge from data. Recent research has also focused on the introduction of specific domain knowledge in machine learning algorithms. Thus, the connections between disaggregation methods in MCDA and traditional machine learning tools are becoming stronger. In this paper the relationships between the two fields are explored. The differences and similarities between the two approaches are identified, and a review is given regarding the integration of the two fields.     

ACUTA: A novel method for eliciting additive value functions on the basis of holistic preference statements

In multiple criteria decision aiding, it is common to use methods that are capable of automatically extracting a decision or evaluation model from partial information provided by the decision maker about a preference structure. In general, there is more than one possible model, leading to an indetermination which is dealt with sometimes arbitrarily in existing methods. This paper aims at filling this theoretical gap: we present a novel method, based on the computation of the analytic center of a polyhedron, for the selection of additive value functions that are compatible with holistic assessments of preferences. We demonstrate the most important characteristics of this technique with an experimental and comparative study of several existing methods belonging to the UTA family.     

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.     

Optimality conditions in preference-based spanning tree problems

Spanning tree problems defined in a preference-based environment are addressed. In this approach, optimality conditions for the minimum-weight spanning tree problem (MST) are generalized for use with other, more general preference orders. The main goal of this paper is to determine which properties of the preference relations are sufficient to assure that the set of ‘most-preferred’ trees is the set of spanning trees verifying the optimality conditions. Finally, algorithms for the construction of the set of spanning trees fulfilling the optimality conditions are designed, improving the methods in previous papers.     

Ranking alternatives using fuzzy numbers

We investigate the problem of employing expert opinion to rank alternatives across a set of criteria. The experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue's fuzzy ranking. This leads to a partition of the alternatives into sets H₁, H₂,… where H₁ contains the highest ranked issues, H₂ has all the second highest ranked alternatives, etc. The total ranking process is shown to possess a number of important properties. An example is presented to illustrate the method.     

Eliciting Electre Tri category limits for a group of decision makers

Multiple criteria sorting aims at assigning alternatives evaluated on several criteria to predefined ordered categories. In this paper, we consider a well known multiple criteria sorting method, Electre Tri, which involves three types of preference parameters: (1) category limits defining the frontiers between consecutive categories, (2) weights and majority level specifying which coalitions form a majority, and (3) veto thresholds characterizing discordance effects. We propose an elicitation procedure to infer category limits from assignment examples provided by multiple decision makers. The procedure computes a set of category limits and vetoes common to all decision makers, with variable weights for each decision maker. Hence, the method helps reaching a consensus among decision makers on the category limits and veto thresholds, whereas finding a consensus on weights is left aside. The inference procedure is based on mixed integer linear programming and performs well even for datasets corresponding to real-world decision problems. We provide an illustrative example of the use of the method and analyze the performance of the proposed algorithms.     

A simulation-based test of stochastic multicriteria acceptability analysis using achievement functions

Stochastic multicriteria acceptability analysis using achievement functions (SMAA-A) is a preference model for discrete-choice decision making that inverts the traditional goal programming process by asking what combinations of aspirations are necessary to make each alternative the preferred one, rather than what alternative is preferred given a set of aspirations. In this paper, we test the ability of the model to discern good-performing alternatives from poorly-performing ones using a simulation study. Simulation results show that a suitably detailed construction of the acceptability index is particularly important, and that the resulting model can be fruitfully applied in the selection of a shortlist of alternatives from a larger set with only very limited decision maker involvement.     

Modeling a multi-attribute utility newsvendor with partial backlogging

Traditional newsvendor models usually focus on single profit maximization or cost minimization approaches. However, making a monetary estimate of the consequences of lost sales as a result of shortages is often a difficult task for many practitioners. Besides, there is still a lack of an explicit account of decision-making judgments on the multiple consequences of making decisions with regard to order quantities. In order to deal with this problem, this paper presents a multi-attribute utility model for the newsvendor problem with regard to profit, the impacts of service level on corporate image and on customers’ goodwill, and the impact on the environment arising from the disposal of unsold products. Demand is partially backlogged according to a decreasing exponential function of the waiting time. The fundamental principles and limitations related to the application of the model built are also discussed.     

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.     

Effects of imprecise weighting in hierarchical preference programming

Preference programming is a general term for multi-criteria decision analytical approaches allowing incomplete preference information. In the PAIRS method, interval judgments are assigned to weight ratios between attributes to model imprecision in multi-attribute value trees. This paper studies the effects of a hierarchical model structure on the overall imprecision, as the form of the hierarchy also affects the form of imprecision that can be assigned to the model. The aim is to find out good procedural practices for reducing overall imprecision descending inherently from the model structure. The study provides simulation results about the ability of various weighting schemes to identify dominated alternatives, which are discussed with respect to other issues related to the weighting process. According to the results, a hierarchical model is structurally somewhat more unable to identify dominances than a corresponding nonhierarchical model, but its cognitive advantages often cancel out this. The results also suggest paying reasonable attention to the precision of the lower level judgments and to identifying possible correlations between the criteria.     

Weight determination for consistently ranking alternatives in multiple criteria decision analysis

One of the most difficult tasks in multiple criteria decision analysis (MCDA) is determining the weights of individual criteria so that all alternatives can be compared based on the aggregate performance of all criteria. This problem can be transformed into the compromise programming of seeking alternatives with a shorter distance to the ideal or a longer distance to the anti-ideal despite the rankings based on the two distance measures possibly not being the same. In order to obtain consistent rankings, this paper proposes a measure of relative distance, which involves the calculation of the relative position of an alternative between the anti-ideal and the ideal for ranking. In this case, minimizing the distance to the ideal is equivalent to maximizing the distance to the anti-ideal, so the rankings obtained from the two criteria are the same. An example is used to discuss the advantages and disadvantages of the proposed method, and the results are compared with those obtained from the TOPSIS method.     

Further results on concordance relations

The purpose of this note is to sharpen the results in an earlier paper [Bouyssou, D., Pirlot, M., 2005. A characterization of concordance relations. European Journal of Operational Research 167 (2), 427–443] giving an axiomatic characterization of concordance relations. We show how the conditions used in this earlier paper can be weakened so as to become independent from the conditions needed to characterize a general conjoint measurement model tolerating intransitive and/or incomplete relations. This leads to a clearer characterization of concordance relations within this general model.     

Approximating Promethee II’s net flow scores by piecewise linear value functions

Promethee II is a prominent method for multi-criteria decision aid (MCDA) that builds a complete ranking on a set of potential actions by assigning each of them a so-called net flow score. However, to calculate these scores, each pair of actions has to be compared, causing the computational load to increase quadratically with the number of actions, eventually leading to prohibitive execution times for large decision problems. For some problems, however, a trade-off between the ranking’s accuracy and the required evaluation time may be acceptable. Therefore, we propose a piecewise linear model that approximates Promethee II’s net flow scores and reduces the computational complexity (with respect to the number of actions) from quadratic to linear at the cost of some wrongly ranked actions. Simulations on artificial problem instances allow us to quantify this time/quality trade-off and to provide probabilistic bounds on the problem size above which our model satisfyingly approximates Promethee II’s rankings. They show, for instance, that for decision problems of 10,000 actions evaluated on 7 criteria, the Pearson correlation coefficient between the original scores and our approximation is of at least 0.97. When put in balance with computation times that are more than 7000 times faster than for the Promethee II model, the proposed approximation model represents an interesting alternative for large problem instances.     

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.     

On the multi-attribute stopping problem with general value functions

We examine a sequential selection problem in which a single option must be selected. Each option's value is a function of its attributes, whose precise values can be ascertained at a given cost. We prove the optimality of a threshold stopping rule for a general class of objective functions.     

An axiomatic analysis of concordance–discordance relations

Outranking methods propose an original way to build a preference relation between alternatives evaluated on several attributes that has a definite ordinal flavor. Indeed, most of them appeal the concordance/non-discordance principle that leads to declaring that an alternative is “superior” to another, if the coalition of attributes supporting this proposition is “sufficiently important” (concordance condition) and if there is no attribute that “strongly rejects” it (non-discordance condition). Such a way of comparing alternatives is rather natural. However, it is well known that it may produce binary relations that do not possess any remarkable property of transitivity or completeness. This explains why the axiomatic foundations of outranking methods have not been much investigated, which is often seen as one of their important weaknesses. This paper uses conjoint measurement techniques to obtain an axiomatic characterization of preference relations that can be obtained on the basis of the concordance/non-discordance principle. It emphasizes their main distinctive feature, i.e. their very crude way to distinguish various levels of preference differences on each attribute. We focus on outranking methods, such as ELECTRE I, that produce a reflexive relation, interpreted as an “at least as good as” preference relation. The results in this paper may be seen as an attempt to give such outranking methods a sound axiomatic foundation based on conjoint measurement.     

A grey relational projection method for multi-attribute decision making based on intuitionistic trapezoidal fuzzy number

With respect to multiple attribute decision making (MADM) problems in which the attribute value takes the form of intuitionistic trapezoidal fuzzy number, and the attribute weight is unknown, a new decision making analysis methods are developed. Firstly, some operational laws and expected values of intuitionistic trapezoidal fuzzy numbers, and distance between two intuitionistic trapezoidal fuzzy numbers, are introduced. Then information entropy method is used to determine the attribute weight, and the grey relational projection method combined grey relational analysis method and projection method is proposed, and to rank the alternatives are done by the relative closeness to PIS which combines grey relational projection values from the positive ideal solution and negative ideal solution to each alternative. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.