ordinal regression revisited multiple criteria ranking using a set of additive value functions abstract we present a new method called utagms 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 ar 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 utagms method is intended to be used interactively with an increasing subset ar 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 utagms 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 utagms software some extensions of the method are also presented

tools for interactive multiattribute decisionmaking with incompletely identified information abstract the aim of this paper is to present tools or techniques for an interactive multiattribute decisionmaking with the decision maker s incomplete information that is 1 we formally describe mathematical models under certainty and uncertainty for establishing pairwise dominance where the incomplete information becomes the set of constraints in the model 2 we present a description on styles of incomplete parameter value information 3 the characteristic of a weak dominance is described when the decision maker is not willing or able to provide more information on parameters the weak dominance technique is useful for a final choicemaking 4 we propose a dominance graph and also present an algorithm of generating the dominance graph based on the information of pairwise dominance the dominance graph compactly displays the dominance structure of all alternatives thus it can be used for aiding the selection of preferable alternatives

deriving rankings from incomplete preference information a comparison of different approaches abstract volume based methods for decision making under incomplete information like the smaa family of methods provide rich probabilistic information to support decision making however they usually do not directly generate a unique ranking of alternatives methods to create such a unique ranking from incomplete preference information typically select one parameter vector either by mathematical programming approaches or by averaging and then apply a preference model using this parameter vector in the present paper we develop several models to infer a complete ranking or a complete preorder of alternatives directly from the probabilistic information provided by volume based methods without singling out a specific parameter vector we compare the results obtained by these models to those obtained with a single parameter approach in a computational study results indicate small but significant differences in the performance of methods as well as in the probability that additional preference information might worsen rather than improve the results

ranking with partial information a method and an application a method is presented for ranking multiattributed alternatives using a weighted additive evaluation function with partial information about the weighting scaling constants the method is applied to evaluate materials for use in nuclear waste containment the paper derives conditions to determine whether a pair of alternatives can be ranked given the partial information about weighting constants and presents an algorithm that partially rank orders the complete set of alternatives based on the pairwise ranking information

selection of a representative value function in robust multiple criteria ranking and choice we introduce the concept of a representative value function in robust ordinal regression applied to multiple criteria ranking and choice problems the proposed method can be seen as a new interactive uta like procedure which extends the utagms and grip methods the preference information supplied by the decision maker dm is composed of a partial preorder and intensities of preference on a subset of reference alternatives robust ordinal regression builds a set of general additive value functions which are compatible with the preference information and returns two binary preference relations necessary and possible they identify recommendations which are compatible with all or at least one compatible value function respectively in this paper we propose a general framework for selection of a representative value function from among the set of compatibles ones there are a few targets which build on results of robust ordinal regression and could be attained by a representative value function in general according to the interactively elicited preferences of the dm the representative value function may emphasize the advantage of some alternatives over the others when all compatible value functions acknowledge this advantage or reduce the ambiguity in the advantage of some alternatives over the others when some compatible value functions acknowledge an advantage and other ones acknowledge a disadvantage the basic procedure is refined by few extensions they enable emphasizing the advantage of alternatives that could be considered as potential best options accounting for intensities of preference or obtaining a desired type of the marginal value functions

robustness analysis for decision under uncertainty with rule based preference model we introduce robust ordinal regression to decision under uncertainty we propose an integrated framework for robustness analysis with the rule based preference model we formulate the procedures for deriving a univocal classification of acts we account for different types of indirect preference information we consider group decision under uncertainty with dominance based rough set approach we consider decision under uncertainty as a multi attribute classification problem where a set of acts is described by outcomes gained with given probabilities the decision maker dm provides desired classification for a small subset of reference acts such preference information is structured using dominance based rough set approach drsa and the resulting lower approximations of the quality class unions are used as an input for construction of an aggregate preference model we induce all minimal cover sets of rules being compatible with the non ambiguous assignment examples and satisfying some additional requirements that may be imposed by the dm applying such compatible instances of the preference model on a set of all acts we draw conclusions about the certainty of recommendation assured by different minimal cover sets of rules in particular we analyze the diversity of class assignments assignment based preference relations and class cardinalities then we solve an optimization problem to get a univocal precise classification for all acts taking into account the robustness concern this optimization problem admits incorporation of additional indirect and imprecise preferences in form of desired class cardinalities and assignment based pairwise comparisons finally we extend the proposed approach to group decision under uncertainty we present a set of indicators and outcomes giving an insight into the spaces of consensus and disagreement between the dms

assessing a set of additive utility functions for multicriteria decision making the uta method abstract the purpose of the method presented in this paper is to assess additive utility functions which aggregate multiple criteria in a composite criterion using the information given by a subjective ranking on a set of stimuli or actions weak order comparison judgments and the multicriteria evaluations of these actions it is an ordinal regression method using linear programming to estimate the parameters of the utility function stability and sensitivity analysis leads to the assessment of a set of utility functions by means of post optimality analysis techniques in linear programming finally a simple illustrative example is presented and some extensions of the method are proposed

estimating linear models with ordinal qualitative regressors abstract the theory and application of ordinal qualitative dependent variable models have been given considerable attention in the social science and statistical literatures linear models with ordinal qualitative regressors have however been neglected in this paper a simple specification for such models is developed and a consistent asymptotically normal estimator is offered the estimator is compared to the conventional dummy variable approach using simulated data the simulation results indicate that substantial gains with regard to bias and efficiency can be achieved relative to estimation using a conventional dummy variable scheme said gains appear to increase as the number of ordinal categories increases the asymptotic properties of the estimator are detailed in the appendices

hit and run enables efficient weight generation for simulation based multiple criteria decision analysis in our previous work published in this journal we showed how the hit and run har procedure enables efficient sampling of criteria weights from a space formed by restricting a simplex with arbitrary linear inequality constraints in this short communication we note that the method for generating a basis of the sampling space can be generalized to also handle arbitrary linear equality constraints this enables the application of har to sampling spaces that do not coincide with the simplex thereby allowing the combined use of imprecise and precise preference statements in addition it has come to our attention that one of the methods we proposed for generating a starting point for the markov chain was flawed to correct this we provide an alternative method that is guaranteed to produce a starting point that lies within the interior of the sampling space

implementing stochastic multicriteria acceptability analysis abstract stochastic multicriteria acceptability analysis smaa is a family of methods for aiding multicriteria group decision making in problems with inaccurate uncertain or missing information these methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one or that would give a certain rank for a specific alternative the main results of the analysis are rank acceptability indices central weight vectors and confidence factors for different alternatives the rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative the central weight vectors represent the typical preferences favouring each alternative and the confidence factors measure whether the criteria measurements are sufficiently accurate for making an informed decision the computations in smaa require the evaluation of multidimensional integrals that must in practice be computed numerically in this paper we present efficient methods for performing the computations through monte carlo simulation analyze the complexity and assess the accuracy of the presented algorithms we also test the efficiency of these methods empirically based on the tests the implementation is fast enough to analyze typical sized discrete problems interactively within seconds due to almost linear time complexity the method is also suitable for analysing very large decision problems for example discrete approximations of continuous decision problems

extended methods for identifying dominance and potential optimality in multi criteria analysis with imprecise information abstract many of the recent prescriptive approaches to multi attribute decision analysis have dealt with situations in which trade off weights are known imprecisely while marginal values are known exactly a way to handle such problems is to utilize a linear programming technique from which non dominated and or potentially optimal score vectors can be obtained in some attributes however it is not easy for decision makers to provide the marginal value functions that are explicit and exact the purpose of this paper is to address problems such that the both attribute weights and marginal values are known imprecisely we then assume without loss of generality that these imprecise information on both weights and values are in the form of inequalities and or equalities such as rankings and bounds the first formulations for checking dominance and potential optimality become non linear programming problems hard to be solved generally we thus present how these non linear problems are reduced to linear programming equivalents

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

preference programming through approximate ratio comparisons abstract in the context of hierarchical weighting this paper operationalizes interval judgments which allow the decision maker to enter ambiguous preference statements by indicating the relative importance of factors as intervals of values on a ratio scale through such judgments the decision maker can capture the subjective uncertainty in his preferences and thus avoid the often cumbersome elicitation of exact ratio estimates after each new statement the interval judgments are synthesized into dominance relations on the alternatives by solving a series of linear programming problems this leads to an interactive process of preference programming which provides more detailed results as the decision maker gradually enters a more specific preference description moreover the overall effort of preference elicitation is smaller than in the analytic hierarchy process because the most preferred alternative can usually be identified before all possible comparisons between pairs of factors have been completed

the use of domains in multicriteria decision making abstract this paper describes a domain criterion for a multicriteria problem given decision maker it then outlines possibilities for aggregating individual preference structures in the case of multiple decision makers

median regression for ordered discrete response abstract we introduce a semiparametric median regression approach to ordered discrete responses generalizing the maximum score estimation of manski 1985 for binary response the identification and a s consistency are proved we assume that there exists a latent continuous variable and several variations of the median regression approach depending on the restriction on the thresholds for the latent variable are examined for some cases bounds for the scale are derived we also derive bounds for the finite sample breakdown point of the median regression method to show its robustness

operational characteristics of maximum score estimation abstract this paper reports on the operational characteristics of maximum score estimation of a linear model from binary response data a series of previous articles have shown that in theory the maximum score method makes possible binary response analysis under very weak distributional assumptions here we present evidence on the properties of maximum score estimation in practice after reviewing the known asymptotic theory of maximum score estimation the paper describes an algorithm for maximum score estimation and characterizes its performance then findings from a monte carlo study comparing maximum score and logit maximum likelihood estimation are reported finally the accuracy of bootstrap estimation of maximum score root mean square errors is evaluated

smaa 2 stochastic multicriteria acceptability analysis for group decision making stochastic multicriteria acceptability analysis smaa is a multicriteria decision support method for multiple decision makers in discrete problems in smaa the decision makers need not express their preferences explicitly or implicitly instead the method is based on exploring the weight space in order to describe the valuations that would make each alternative the preferred one inaccurate or uncertain criteria values are represented by probability distributions from which the method computes confidence factors describing the reliability of the analysis in this paper we introduce the smaa 2 method which extends the original smaa by considering all ranks in the analysis in situations where the elitistic smaa may assess large acceptability only for extreme alternatives without sufficient majority support the more holistic smaa 2 analysis can be used to identify good compromise candidates the results are presented graphically we consider also situations where partial preference information is available we demonstrate the new method using a real life decision problem

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

decision making with incomplete information abstract decision situations with incomplete information are characterized by a decision maker without a precisely defined stable preference structure by probability distributions not known completely or by an inexact evaluation of consequences within the paper a general framework for decision making with incomplete information is presented which shows how to solve problems from descriptive as well as prescriptive decision theory based on this framework an overview of existing methods which are particularly suitable for handling incomplete information is given

generating random vectors uniformly distributed inside and on the surface of different regions abstract an acceptance rejection algorithm for generating random vectors uniformly distributed over inside or on the surface of a complex region inserted in a minimal multidimensional rectangle is considered for regions having simple forms simplex hypersphere hyperellipsoid several algorithms are presented as well

smaa stochastic multiobjective acceptability analysis stochastic multiobjective acceptability analysis smaa is a multicriteria decision support technique for multiple decision makers based on exploring the weight space inaccurate or uncertain input data can be represented as probability distributions in smaa the decision makers need not express their preferences explicitly or implicitly instead the technique analyses what kind of valuations would make each alternative the preferred one the method produces for each alternative an acceptability index measuring the variety of different valuations that support that alternative a central weight vector representing the typical valuations resulting in that decision and a confidence factor measuring whether the input data is accurate enough for making an informed decision

partial information dominance and potential optimality in multiattribute utility theory when a multiattribute utility function is only partially specified by prior preference statements what can be said about the relative desirability of actual alternatives this question is addressed for the cases of additively separable cardinal utility with unknown scaling constants multiplicatively separable cardinal utility with unknown scaling constants and additively separable ordinal utility we review previous approaches and treat issues of consistency is the prior information consistent dominance does the prior preference information imply that one outcome is preferred to another and potential optimality are there utility functions of the given form consistent with prior preference information under which a particular outcome is preference optimal in the additive cases a key relationship between dominance and potential optimality may be derived the paper concludes by presenting an example application to a well known nuclear siting study