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

preference inference with general additive value models and holistic pair wise statements additive multi attribute value models and additive utility models with discrete outcome sets are widely applied in both descriptive and normative decision analysis their non parametric application allows preference inference by analyzing sets of general additive value functions compatible with the observed or elicited holistic pair wise preference statements in this paper we provide necessary and sufficient conditions for the preference inference based on a single preference statement and sufficient conditions for the inference based on multiple preference statements in our computational experiments all inferences could be made with these conditions moreover our analysis suggests that the non parametric analyses of general additive value models are unlikely to be useful by themselves for decision support in contexts where the decision maker preferences are elicited in the form of holistic pair wise statements

a bounded approximation of weakly compact operators abstract we study bounded approximation properties of operators it is shown that for every weakly compact operator t on a banach space x if t is uniformly approximated on each compact subset of x by finite rank operators then t 2 can be approximated in the same topology by finite rank operators with the operator norm less than or equal to t 2

electre like clustering from a pairwise fuzzy proximity index abstract in this paper we propose an electre like approach for clustering judges from their l valued pairwise proximities in preference judgements the approach is based on the extraction of l valued null kernels from a pairwise l valued proximity index a practical application will concern the clustering of movie critics

an approach for finding the most preferred alternative in the presence of multiple criteria abstract an interactive approach for discrete multiple criteria decision making problems is developed the approach requires the decision maker to compare pairs of presented alternatives these alternatives may be existing alternatives or dummy alternatives created from existing alternatives inferior alternatives are sequentially eliminated by either using the responses of the decision maker directly or by constructing cones of inferior solutions based on these responses the approach converges to the most preferred solution so long as the decision maker has a nondecreasing quasiconcave value function the aim is to keep the number of pairwise comparisons required of the decision maker as small as possible computational experience on randomly generated problems is reported and the results are compared with the results of similar approaches

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

how well does the owa operator represent real preferences we present the results of an empirical study in which we analyze the ability of the ordered weighted average owa operator to model actual preferences in a multi attribute ranking task we compare the owa to a competing model simple additive weighting saw and we also study whether its characteristic feature the ability to represent different attitudes toward compensation among attributes is reflected in the preferences provided by subjects we find that in general the owa model fits slightly less well to empirical data subjects whose preferences are better explained by the saw model are also more consistent in their choice behavior our results furthermore indicate that preferences of most subjects are not fully compensatory thus the ability of the owa operator to represent different attitudes toward compensation can be a useful feature in modeling actual preferences the structure of the weights estimated for the owa operator also suggests that it might provide a good approximation to certain types of decision heuristics

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

rank inclusion in criteria hierarchies this paper presents a method called rank inclusion in criteria hierarchies rich for the analysis of incomplete preference information in hierarchical weighting models in rich the decision maker is allowed to specify subsets of attributes which contain the most important attribute or more generally to associate a set of rankings with a given set of attributes such preference statements lead to possibly non convex sets of feasible attribute weights allowing decision recommendations to be obtained through the computation of dominance relations and decision rules an illustrative example on the selection of a subcontractor is presented and the computational properties of rich are considered

formalizing and solving the problem of clustering in mcda the topic of clustering has been widely studied in the field of data analysis where it is defined as an unsupervised process of grouping objects together based on notions of similarity clustering in the field of multi criteria decision aid mcda has seen a few adaptations of methods from data analysis most of them however using concepts native to that field such as the notions of similarity and distance measures as in mcda we model the preferences of a decision maker over a set of decision alternatives we can find more diverse ways of comparing them than in data analysis as a result these alternatives may also be arranged into different potential structures in this paper we wish to formally define the problem of clustering in mcda using notions that are native to this field alone and highlight the different structures which we may try to uncover through this process following this we propose a method for finding these structures as in any clustering problem finding the optimal result in an exact manner is impractical and so we propose a stochastic heuristic approach which we validate through tests on a large set of artificially generated benchmarks

integrated framework for preference modeling and robustness analysis for outranking based multiple criteria sorting with electre and promethee a novel framework for preference modeling and robustness analysis is proposed we apply a sorting rule deriving from the semantic meaning of an outranking relation we ensure input output correspondence with respect to three sorting perspectives the generality of our framework is illustrated with its two specific implementations preference modeling can be simplified when using model instances providing precise assignments we introduce an integrated framework for preference modeling and robustness analysis in outranking based multiple criteria sorting the preference information supplied by the decision maker dm is composed of parts of desired recommendation and imprecise requirements that the delivered outcomes should satisfy its allowed forms are imprecise assignment examples desired class cardinalities and assignment based pairwise comparisons the exploitation of all instances of the outranking model compatible with these preferences results in three types of outcomes these raise robustness concerns in terms of the stability of a suggested assignment for each alternative an observed size of each class and a comparison between recommendation delivered for pairs of alternatives the correspondence between different types of inputs and outputs facilitates the dialogue with the dm and enhances her his confidence in the suggested recommendation while referring only to a semantic meaning of an outranking relation we present the procedures for translating the preference information into parameters of compatible outranking model instances and for deriving various kinds of sorting results then we implement this general framework in the context of outranking models specific for electre and promethee finally we show how preference modeling and robustness analysis can be performed and greatly simplified with a set of preference model instances providing precise assignments for the alternatives the framework proposed for this case is based on mixed integer linear programming milp being independent of the underlying model and method application of the approaches is demonstrated on the case of classifying polish research units into four quality classes

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

incremental rough set approach for hierarchical multicriteria classification abstract multicriteria classification refers to classify objects evaluated by a set of criteria to preference ordered decision classes dominance based rough set approach has been successfully introduced to express and reason inconsistencies with a dominance principle in multicriteria classification problems hierarchical attribute values exist extensively within many real world applications which provide a hierarchical form to organize view and analyze data from different perspectives for accommodating the preference variability in this study we consider an extension of dominance based rough set approach by applying an incremental learning technique for hierarchical multicriteria classification while attribute values dynamically vary across different levels of granulations we formalize the dynamic characteristics of knowledge granules with the cut refinement and coarsening through attribute value taxonomies in the hierarchical multicriteria decision systems in consequence incremental algorithms for computing dominance based rough approximations of preference ordered decision classes are developed by applying the resulted prior knowledge as the input and only recomputing those outputs which depend on the changed attribute values this paper presents the theoretical foundation of the proposed approach example analysis and experimental evaluation are also provided for illustration of the feasibility and efficiency

a fast heuristic attribute reduction approach to ordered decision systems abstract rough set theory has shown success in being a filter based feature selection approach for analyzing information systems one of its main aims is to search for a feature subset called a reduct which preserves the classification ability of the original system in this paper we consider ordered decision systems where the preference order a fundamental concept in dominance based rough set approach plays a critical role in recent literature based on the greedy hill climbing method many heuristic attribute reduction algorithms are proposed by utilizing significance measures of attributes and they are extended to deal with ordered decision systems unfortunately they are often time consuming especially when applied to deal with large scale data sets with high dimensions to reduce the complexity a novel accelerator is introduced in heuristic algorithms from the perspectives of objects and criteria based on the new accelerator the number of objects and the dimension of criteria are lessened thus making the accelerated algorithms faster than their original counterparts while maintaining the same reducts experimental analysis shows the validity and efficiency of the proposed methods

jie ke versus alphago a ranking approach using decision making method for large scale data with incomplete information abstract ranking historical players in sports is challenging since some players have never played against each other it is even more complex in go because of alphago a project based on artificial intelligence who became the world s number 1 after it defeated the 528th and the 4th human go players alphago is ranked high in the current go ranking system because it is undefeated the objective of this paper is to propose a new ranking method for large scale go players by means of incomplete fuzzy pair wise comparison matrix whose priority vector is derived using a cosine similarity measure using match results provided by go4go net experiments are designed to rank top go players in the past 45 years and examine the change in ranking after alphago faced off against jie ke furthermore the proposed method was applied to rank all 1544 go players available at go4go net to illustrate its efficiency in handling large scale data

attribute reduction in ordered decision tables via evidence theory rough set theory and dempster shafer theory of evidence are two distinct but closely related approaches to modeling and manipulating uncertain information it is quite natural to set up a hybrid model based on these two theories in this paper we investigate the problem of attribute reduction for ordered decision tables based on evidence theory belief and plausibility functions which are strongly connected with lower and upper approximation operators in dominance based rough set approach are proposed to define relative belief and plausibility reducts of ordered decision tables relationships among various types of relative reducts are thoroughly studied in consistent and inconsistent ordered decision tables a pair of numeric measures the inner and outer significance measures of a criterion is presented to search for a relative belief plausibility reduct which is meaningful for practical problems some real world tasks taken from the uci repository are employed to verify the feasibility and effectiveness of the proposed technique

preference assessment by imprecise ratio statements the pairs method developed in this paper introduces imprecise preference statements into value trees the assessment of attribute weights in pairs extends the well known smart technique so that in addition to exact statements the decision maker can enter interval judgments which indicate ranges for the relative importance of the attributes the interval judgments and the possibly range valued information about the outcomes of the alternatives are processed with linear programming into value intervals and dominance relations as the decision maker refines the description of his preferences either by entering new statements or by tightening his earlier judgments these results become more detailed and convey more information about which alternatives are preferred throughout the interactive refinement process pairs supports the decision maker by deriving and displaying the consequences of his earlier judgments

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

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

global maximization of uta functions in multi objective optimization abstract the utas utilite additives type methods for constructing nondecreasing additive utility functions were first proposed by jacquet lagreze and siskos in 1982 for handling decision problems of multicriteria ranking in this article by uta functions we mean functions which are constructed by the uta type methods our purpose is to propose an algorithm for globally maximizing uta functions of a class of linear convex multiple objective programming problems the algorithm is established based on a branch and bound scheme in which the branching procedure is performed by a so called i rectangular bisection in the objective outcome space and the bounding procedure by some convex or linear programs preliminary computational experiments show that this algorithm can work well for the case where the number of objective functions in the multiple objective optimization problem under consideration is much smaller than the number of variables

robust ordinal regression for dominance based rough set approach to multiple criteria sorting we present a new multiple criteria sorting method deriving from dominance based rough set approach drsa the preference information supplied by the decision maker dm is a set of possibly imprecise and inconsistent assignment examples on a subset of reference alternatives relatively well known to the dm to structure the data we use drsa and subsequently represent the assignment examples by all minimal sets of rules covering all alternatives from the lower approximations of class unions such a set of rules is called minimal cover set it is one of the instances of the preference model compatible with dm s preference information in this way we implement the principle of robust ordinal regression ror to decision rule preference model for each alternative we derive the necessary and possible assignments specifying the range of classes to which the alternative is assigned by all or at least one compatible set of rules respectively as well as class acceptability indices we also introduce the notion of a representative compatible minimal cover set of rules whose selection builds on the results of ror addressing the robustness concern application of the approach is demonstrated by classifying 69 land zones in 4 classes representing different risk levels

electregkms robust ordinal regression for outranking methods we present a new method called electregkms 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 electregkms 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 electregkms can support real world decision problems

variable consistency dominance based rough set approach to preference learning in multicriteria ranking abstract we present a methodology for non statistical preference learning in multicriteria ranking based on variable consistency dominance based rough set approach vc drsa a finite set of objects to be ranked is evaluated by a set of criteria which are real valued functions with ordinal or cardinal scales given the statement of a multicriteria ranking problem the only objective information one can get is the dominance relation over the set of objects the dominance relation is however too poor because it leaves many objects incomparable to enrich this relation and make the objects more comparable a decision maker dm must supply some preference information revealing her his value system with respect to multicriteria evaluations we are considering a frequent case when the preference information has the form of pairwise comparisons of some objects relatively well known to the dm called reference objects this information is thus composed of some decision examples on the reference objects it is the input data for a method that learns the preferences of the dm since this information is prone to inconsistencies we propose to structure it using vc drsa then the pairs of objects that are sufficiently consistent serve as a basis for induction of a preference model this model has the form of a set of if then decision rules application of these rules on the whole set of objects to be ranked yields a fuzzy preference structure directed weighted graph this preference structure is then exploited using a ranking method so as to work out a final recommendation we propose a list of properties that helps to choose a proper ranking method the methodology is illustrated by an example

a high order fast direct solver for singular poisson equation we present a fourth order numerical solution method for the singular neumann boundary problem of poisson equations such problems arise in the solution process of incompressible navier stokes equations and in the time harmonic wave propagation in the frequence space with the zero wavenumber the equation is first discretized with a fourth order modified collatz difference scheme producing a singular discrete equation then an efficient singular value decomposition svd method modified from a fast poisson solver is employed to project the discrete singular equation into the orthogonal complement of the null space of the singular matrix in the complement of the null space the projected equation is uniquely solvable and its solution is proven to be a solution of the original singular discrete equation when the original equation has a solution analytical and experimental results show that this newly proposed singular equation solver is efficient while retaining the accuracy of the high order discretization c 2001 academic press

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

dominance based rough set approach for group decisions the objective of this paper is to propose an approach to support group multicriteria classification the approach is composed of three phases the first phase exploits the knowledge provided by each decision maker to individually approximate the decision classes using rough approximation the second phase seeks to combine the outputs of individual approximation phase into a collective decision table by using an appropriate aggregation procedure the third phase uses the collective decision table in order to infer a set of collective decision rules which synthesize the judgements and perspectives of the different decision makers and to permit the classification of all decision objects the proposed approach relies on the dominance based rough set approach drsa which is used at two different levels first the drsa is used during the first phase to approximate the input data relative to each decision maker second the drsa is used during the third phase to approximate the collective decision table and generate the collective decision rules this paper presents the theoretical foundation of the proposed approach three case studies using real world data and a comparative study of recent similar proposals

an interactive approach for placing alternatives in preference classes abstract in this paper we consider the multiple criteria decision making problem of partitioning alternatives into preference classes we develop an interactive procedure based on the assumption that the decision maker dm has a linear utility function the approach requires the dm to place an alternative in a preference class from time to time based on the preference information derived from the dm s placement as well as from dominance and linearity we try to place other alternatives we present results on the performance of the algorithm applied to four different sets of alternatives

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 mono tonically 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

partially monotonic decision trees abstract in multicriteria decision tasks certain features are linearly ordered according to the decision and are called criteria whereas others called regular attributes are not in practice regular attributes and criteria coexist in most classification tasks in this paper we propose a rank inconsistent rate that distinguishes attributes from criteria furthermore it represents the directions of the monotonic relationships between criteria and decisions we design a partially monotonic decision tree algorithm to extract decision rules for partially monotonic classification tasks experimental results show that the proposed algorithm is effective and efficient

a group decision making approach based on evidential reasoning for multiple criteria sorting problem with uncertainty a new group decision making approach is developed to address a multiple criteria sorting problem with uncertainty the uncertainty in this paper refers to imprecise evaluations of alternatives with respect to the considered criteria the belief structure and the evidential reasoning approach are employed to represent and aggregate the uncertain evaluations in our approach the preference information elicited from a group of decision makers is composed of the assignment examples of some reference alternatives the disaggregation aggregation paradigm is utilized to infer compatible preference models from these assignment examples to help the group reach an agreement on the assignment of alternatives we propose a consensus reaching process in this process a consensus degree is defined to measure the agreement among the decision makers opinions when the decision makers are not satisfied with the consensus degree possible solutions are explored to help them adjust assignment examples in order to improve the consensus level if the consensus degree arrives at a satisfactory level a linear program is built to determine the collective assignment of alternatives the application of the proposed approach to a customer satisfaction analysis is presented at the end of the paper

mining diagnostic rules from clinical databases using rough sets and medical diagnostic model since rule induction methods generate rules whose lengths are the shortest for discrimination between given classes they tend to generate rules too short for medical experts thus these rules are difficult for the experts to interpret from the viewpoint of domain knowledge in this paper the characteristics of experts rules are closely examined and a new approach to generate diagnostic rules is introduced the proposed method focuses on the hierarchical structure of differential diagnosis and consists of the following three procedures first the characterization of decision attributes given classes is extracted from databases and the classes are classified into several generalized groups with respect to the characterization then two kinds of sub rules classification rules for each generalized group and rules for each class within each group are induced finally those two parts are integrated into one rule for each decision attribute the proposed method was evaluated on a medical database the experimental results of which show that induced rules correctly represent experts decision processes

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

a method of multiattribute decision making with incomplete information a precise determination of a multiattribute utility function of a decision maker or a group of decision makers requires considerable information that may not be available in many decision situations in this paper the method hopie which allows one to determine a set of utility functions consistent with the incomplete information received from the decision maker is proposed the method is based on holistic judgments of hypothetical alternatives defined by a certain factorial design it requires that the evaluation of the alternatives be given on an interval scale the method can also accommodate other types of additional information such as pairwise comparisons

enumerating all connected maximal common subgraphs in two graphs we represent a new method for finding all connected maximal common subgraphs in two graphs which is based on the transformation of the problem into the clique problem we have developed new algorithms for enumerating all cliques that represent connected maximal common subgraphs these algorithms are based on variants of the bron kerbosch algorithm in this paper we explain the transformation of the maximal common subgraph problem into the clique problem we give a short summary of the variants of the bron kerbosch algorithm in order to explain the modification of that algorithm such that the detected cliques represent connected maximal common subgraphs after introducing and proving several variants of the modified algorithm we discuss the runtimes for all variants by means of random graphs the results show the drastical reduction of the runtimes for the new algorithms

elicitation of multiattribute value functions through high dimensional model representations monotonicity and interactions abstract this work addresses the early phases of the elicitation of multiattribute value functions proposing a practical method for assessing interactions and monotonicity we exploit the link between multiattribute value functions and the theory of high dimensional model representations the resulting elicitation method does not state any a priori assumption on an individual s preference structure we test the approach via an experiment in a riskless context in which subjects are asked to evaluate mobile phone packages that differ on three attributes

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

evidential reasoning based preference programming for multiple attribute decision analysis under uncertainty multiple attribute decision analysis mada problems having both quantitative and qualitative attributes under uncertainty can be modelled and analysed using the evidential reasoning er approach several types of uncertainty such as ignorance and fuzziness can be consistently modelled in the er framework in this paper both interval weight assignments and interval belief degrees are considered which could be incurred in many decision situations such as group decision making based on the existing er algorithm several pairs of preference programming models are constructed to support global sensitivity analysis based on the interval values and to generate the upper and lower bounds of the combined belief degrees for distributed assessment and also the expected values for ranking of alternatives a post optimisation procedure is developed to identify non dominated solutions examine the robustness of the partial ranking orders generated and provide guidance for the elicitation of additional information for generating more desirable assessment results a car evaluation problem is examined to show the implementation process of the proposed approach

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

sequential covering rule induction algorithm for variable consistency rough set approaches we present a general rule induction algorithm based on sequential covering suitable for variable consistency rough set approaches this algorithm called vc domlem can be used for both ordered and non ordered data in the case of ordered data the rough set model employs dominance relation and in the case of non ordered data it employs indiscernibility relation vc domlem generates a minimal set of decision rules these rules are characterized by a satisfactory value of the chosen consistency measure we analyze properties of induced decision rules and discuss conditions of correct rule induction moreover we show how to improve rule induction efficiency due to application of consistency measures with desirable monotonicity properties

resolving inconsistencies among constraints on the parameters of an mcda model 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 model s 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

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

a new decision making approach for multiple criteria sorting with an imbalanced set of assignment examples abstract we propose a novel approach to address a multiple criteria sorting mcs problem with an imbalanced set of assignment examples the approach employs a piecewise linear additive value function as the preference model and adopts the disaggregation aggregation paradigm to infer a sorting model from provided assignment examples on a set of reference alternatives we utilize a hierarchical clustering algorithm and several linear programming models to identify reference alternatives that are active to develop the sorting model so that inactive ones are eliminated from the whole set of reference alternatives then in order to construct a balanced set of assignment examples a balancing algorithm is proposed to balance active reference alternatives across categories finally the sorting model is obtained by minimizing the sum of violations between values of active reference alternatives and corresponding category thresholds furthermore the performance of the proposed approach is investigated on a hypothetical problem and several real data sets the experimental results show that our approach is efficient to address the mcs problem with an imbalanced set of assignment examples

a way to deal with fuzzy preferences in multi criteria decision problems this paper presents a special multiple criteria decision making approach for solving problems in context with fuzzy individual preferences at first we briefly expose the proposed methodology the individual preferences are explicitly given by a complete transitive relation r on a set of reference actions the modelling of the decision maker s preferences is obtained by means of fuzzy outranking relations these fuzzy relations are based on a system of additive utility functions which are estimated by means of ordinal regression methods analysing the preference relation r this is followed by a presentation of two real multicriteria problems which the proposed methodology has been applied to i e a highway plan choice problem and a problem in marketing research dealing with the launching of a new product in each application we tried to specify this method according to the specific structure of the problem considered

modeling assignment based pairwise comparisons within integrated framework for value driven multiple criteria sorting abstract we introduce a new preference disaggregation modeling formulations for multiple criteria sorting with a set of additive value functions the preference information supplied by the decision maker dm is composed of 1 possibly imprecise assignment examples 2 desired class cardinalities and 3 assignment based pairwise comparisons the latter have the form of imprecise statements referring to the desired assignments for pairs of alternatives but without specifying any concrete class additionally we account for preferences concerning the shape of the marginal value functions and desired comprehensive values of alternatives assigned to a given class or class range the exploitation of all value functions compatible with these preferences results in three types of results 1 necessary and possible assignments 2 extreme class cardinalities and 3 necessary and possible assignment based preference relations these outputs correspond to different types of admitted preference information by exhibiting different outcomes we encourage the dm in various ways to enrich her his preference information interactively the applicability of the framework is demonstrated on data involving the classification of cities into liveability classes

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

towards multicriteria clustering an extension of the k means algorithm abstract the research within the multicriteria classification field is mainly focused on the assignment of actions to pre defined classes nevertheless the building of multicriteria categories remains a theoretical question still not studied in detail to tackle this problem we propose an extension of the well known k means algorithm to the multicriteria framework this extension relies on the definition of a multicriteria distance based on the preference structure defined by the decision maker thus two alternatives will be similar if they are preferred indifferent and incomparable to more or less the same actions armed with this multicriteria distance we will be able to partition the set of alternatives into classes that are meaningful from a multicriteria perspective finally the examples of the country risk problem and the diagnosis of firms will be treated to illustrate the applicability of this method

preference programming with incomplete ordinal information abstract 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

the strong approximation property and the weak bounded approximation property abstract we show that the strong approximation property strong ap respectively strong cap and the weak bounded approximation property respectively weak bcap are equivalent for every banach space this gives a negative answer to oja s conjecture as a consequence we show that each of the spaces c 0 and l 1 has a subspace which has the ap but fails to have the strong ap

using classification as an aggregation tool in mcdm abstract in this paper we propose a method to obtain the set of best alternatives when data involved in mcdm are of different types e g numbers linguistic labels or truth values this method consists of a classification of the alternatives according to criteria and then an ordering of the classes obtained in this way we study each alternative in relation with the other ones the groups formed indicate that their members are equally prefered over the others otherwise we describe some specialized methods to be used in case that all criteria have the same type of values

multi criteria classification a new scheme for application of dominance based decision rules abstract we are considering the problem of multi criteria classification in this problem a set of if then decision rules is used as a preference model to classify objects evaluated by a set of criteria and regular attributes given a sample of classification examples called learning data set the rules are induced from dominance based rough approximations of preference ordered decision classes according to the variable consistency dominance based rough set approach vc drsa the main question to be answered in this paper is how to classify an object using decision rules in situation where it is covered by i no rule ii exactly one rule iii several rules the proposed classification scheme can be applied to both learning data set to restore the classification known from examples and testing data set to predict classification of new objects a hypothetical example from the area of telecommunications is used for illustration of the proposed classification method and for a comparison with some previous proposals

fourier matrix decomposition methods for the least squares solution of singular neumann and periodic hermite bicubic collocation problems the use of orthogonal spline collocation with piecewise hermite bicubics is examined for the solution of poisson s equation on a rectangle subject to either pure neumann or pure periodic boundary conditions emphasis is placed on finding a least squares solution of these singular collocation problems the technique of matrix decomposition is applied and explicit formulas for the requisite eigensystems corresponding to two point neumann and periodic collocation boundary value problems are presented the resulting algorithms use fast fourier transforms for efficiency and are highly parallel in nature on an n times n partition a fourth order accurate least squares solution is computed at a cost of o n 2 log n operations the results of numerical experiments are provided that demonstrate that the implementations compare very favorably with recent fourth order accurate finite difference and finite element galerkin codes

a linear programming approach for learning non monotonic additive value functions in multiple criteria decision aiding abstract a new framework for preference disaggregation in multiple criteria decision aiding is introduced the proposed approach aims to infer non monotonic additive preference models from a set of indirect pairwise comparisons the preference model is presented as a set of marginal value functions and the discriminatory power of the inferred preference model is maximized against its complexity to infer a value function that is compatible with the supplied preference information the proposed methodology leads to a linear programming optimization problem that is easy to solve the applicability and effectiveness of the new methodology is demonstrated in a thorough experimental analysis covering a broad range of decision problems

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

decision quality using ranked attribute weights three published approximation formulae for selecting the best multiattribute alternative based on rank ordered weights are evaluated all formulae are surprisingly efficacious in determining the best multiattribute alternative rank order centroid roc weights are more accurate than the other rank based formulae furthermore the roc formula generalizes to incorporate both other forms of partial information about attribute weights and partial rank order information as well because a roc based analysis is so straightforward and efficacious it provides an appropriate implementation tool

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

handling multicriteria preferences in cluster analysis in the framework of multicriteria decision aid a lot of interest has been devoted to sorting problems in which the set of categories is pre defined besides preference oriented multicriteria clustering has received little attention usual geometric and related metrics are not well suited for this problem here we propose a clustering method based on a valued indifference relation inspired by outranking methods we suggest a method based on comparing cluster centers and an average net flow score of clusters to build a complete ranking of the set of clusters that is a way of defining a set of ordered categories for sorting purposes the new approach performs very well in some examples

corrigendum and addendum to the strong approximation property and the weak bounded approximation property j funct anal 266 2014 5439 5447 abstract we correct a mistake in the proof of 2 theorem 1 1 and a proof of the main result in 2 is presented without depending on 2 theorem 1 1

ordinal judgments in multiattribute decision analysis abstract the article discusses the contradiction between the ambiguity of human judgment in a multicriterion environment and the exactness of the assessments required in the majority of the decision making methods preferential information from the decision makers in the ordinal form e g more preferable less preferable etc is argued to be more stable and more reliable than cardinal input ways of obtaining and using ordinal judgments for rank ordering of multiattribute alternatives are discussed the effectiveness of the step wise procedure of using ordinal tradeoffs for comparison of alternatives is evaluated we introduce the notion of ordinal tradeoffs presentation of ordinal tradeoffs as a flexible three stage process a paired joint ordinal scale pjos and evaluation of the effectiveness of the three stage process simulation results examine the sensitivity of the number of pairwise comparisons required for given numbers of criteria and categories within criteria as well as the number of alternatives analyzed this simulation shows that ordinal pairwise comparisons provide sufficient power to discriminate between 75 and 80 of the alternatives compared while the proportional number of pairwise comparisons relative to the maximum possible decreases with the number of criteria and categories the method is relatively insensitive to the number of alternatives considered