tradeoff based decomposition and decision making in multiobjective programming to facilitate the evaluation of tradeoffs and the articulation of preferences in multiple criteria decision making a multiobjective decomposition scheme is proposed that restructures the original problem as a collection of smaller sized subproblems with only subsets of the original criteria a priori preferences on objective tradeoffs are integrated into this process by modifying the ordinary pareto order by more general domination cones and decision makers are supported by an interactive decision making procedure to coordinate any remaining tradeoffs using concepts of approximate efficiency a theoretical foundation for this method is provided and an illustrative application to multiobjective portfolio optimization is described in detail

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

using choquet integral as preference model in interactive evolutionary multiobjective optimization we propose an interactive multiobjective evolutionary algorithm that attempts to discover the most preferred part of the pareto optimal set preference information is elicited by asking the user to compare some solutions pairwise this information is then used to curb the set of compatible user s value functions and the multiobjective evolutionary algorithm is run to simultaneously search for all solutions that could potentially be the most preferred compared to previous similar approaches we implement a much more efficient way of determining potentially preferred solutions that is solutions that are best for at least one value function compatible with the preference information provided by the decision maker for the first time in the context of evolutionary computation we apply the choquet integral as a user s preference model allowing us to capture interactions between objectives as there is a trade off between the flexibility of the value function model and the complexity of learning a faithful model of user s preferences we propose to start the interactive process with a simple linear model but then to switch to the choquet integral as soon as the preference information can no longer be represented using the linear model an experimental analysis demonstrates the effectiveness of the approach

preference structure representation using convex cones in multicriteria integer programming a new efficient system of representing the decision maker s preference structure in solving multicriteria integer programming problems is developed the problem is solved by an interactive branch and bound method that employs the procedure of zionts and wallenius zionts s j wallenius 1983 an interactive multiple objective linear programming method for a class of underlying nonlinear utility functions management sci 295 for multicriteria linear programming the decision maker s underlying utility function is assumed to be pseudoconcave and his pairwise comparisons of decision alternatives are used to determine his preference structure in terms of certain convex cones in the objective function space and constraints on the weights on the objectives in the weight space the two forms of preference structure representation are interrelated and their underlying theory is developed the primary objective of a representation scheme is exactness and in this respect it is shown that the constraints on the weights are not adequate for representing nonlinear utility functions on the other hand the convex cones exactly represent any quasiconcave utility function and clearly avoid the approximations and inaccuracies in other utility assessment systems accordingly an efficient ordered representation using convex cones is developed an algorithmic framework for multicriteria integer programming that integrates the representation using convex cones with the branch and bound solution procedure is developed computational experience with bicriteria problems having up to 80 variables and 40 constraints is presented

inertial hegselmann krause systems we derive an energy bound for inertial hegselmann krause hk systems which are a variant of the classic hk model that allows agents to change weights as they please at each step we use the bound to prove the convergence of hk systems with static agents which settles a widely believed conjecture this paper also introduces anchored hk systems and show their equivalence to the symmetric heterogeneous model

solving the discrete multiple criteria problem using linear prospect theory abstract prospect theory developed by kahneman and tversky is a popular model of choice in decision problems under uncertainty prospect theory has recently been extended to multiple criteria choice problems in this paper an interactive method for solving discrete multiple criteria decision problems based on prospect theory type value functions has been developed piecewise linear marginal value functions are assumed to approximate the s shaped value functions of prospect theory therefore the proposed procedure is valid only for convex preferences

eigenfunction approach to the persistent random walk in two dimensions abstract the fourier bessel expansion of a function on a circular disc yields a simple series representation for the end to end probability distribution function w r encountered in a planar persistent random walk where the direction taken in a step depends on the relative orientation towards the preceding step for all but the shortest walks the proposed method provides a rapidly converging numerically stable algorithm that is particularly useful for the precise study of intermediate size chains that have not yet approached the diffusion limit a

a two parameter random walk with approximate exponential probability distribution we study a non markovian random walk in dimension 1 it depends on two parameters r and l the probabilities to go straight on when walking to the right and respectively to the left the position x of the walk after n steps and the number of reversals of direction k are used to estimate r and l we calculate the joint probability distribution pn x k in a closed form and show that approximately it belongs to the exponential family

interquad an interactive quad tree based procedure for solving the discrete alternative multiple criteria problem abstract in this paper an interactive procedure based upon a data structure called a quad tree is developed for solving the discrete alternative multiple criteria problem called interquad the procedure has been designed with large discrete alternative problems in mind interquad takes advantage of the ability of a quad tree to identify store and retrieve nondominated criterion vectors then the user interacts with the nondominated criterion vectors stored in the quad tree in a fashion similar to that of the combined tchebycheff aspiration criterion vector procedure of steuer silverman and whisman

probabilistic methods in crystal structure analysis abstract one of the main goals of modern crystallography is the determination of the detailed internal structure of crystalline matter at the atomic level statistical analyses and in particular random walk models play a central role in inferring structural information from crystallographic data such methods are routinely employed by crystallographers in the determination of crystal symmetry from the experimental data and in the solution of the outstandingly important problem for this discipline the phase problem three classes of approaches are discussed a methods based entirely on the central limit theorem b approximate expansions in terms of orthogonal polynomials that have the central limit theorem pdf as their weight function that is gram charlier and edgeworth expansions and c pdf s that are exactly formulated and reduced to computable forms represented as fourier and fourier bessel series both univariate and multivariate pdf s of crystallographic interest are derived and discussed

evolutionary algorithms with preference polyhedron for interval multi objective optimization problems multi objective optimization problems mops with interval parameters are ubiquitous in real world applications existing evolutionary optimization methods however aim to obtain a set of well converged and evenly distributed pareto optimal solutions in this paper we presented a novel evolutionary algorithm ea that interacts with a decision maker dm during the optimization process to obtain the most preferred solution the theory of preference polyhedron for the optimization problem with interval parameters was developed first an interactive evolutionary algorithm iea for mops with interval parameters was then proposed based on the theory of preference polyhedron the algorithm periodically provides a set of non dominated solutions to the dm a preference polyhedron is constructed in the objective space by taking the worst solution chosen by the dm as the vertex the solutions with the same rank are sorted based on the polyhedron constructed finally our method was empirically evaluated on several mops with interval parameters where the value functions were used to simulate the dm s responses the numerical results indicated that our method is simpler and more efficient than the a posteriori method

a multiple criteria method for choosing among discrete alternatives abstract an interactive method is presented for solving a discrete alternative multiple criteria problem under certainty the method is based on the approach of zionts and wallenius a computer program embodying the method has been written and the results to date have been favorable extensions to the case in which some of the criteria are not readily quantifiable are also considered

cone contraction and reference point methods for multi criteria mixed integer optimization abstract interactive approaches employing cone contraction for multi criteria mixed integer optimization are introduced in each iteration the decision maker dm is asked to give a reference point new aspiration levels the subsequent pareto optimal point is the reference point projected on the set of admissible objective vectors using a suitable scalarizing function thereby the procedures solve a sequence of optimization problems with integer variables in such a process the dm provides additional preference information via pair wise comparisons of pareto optimal points identified using such preference information and assuming a quasiconcave and non decreasing value function of the dm we restrict the set of admissible objective vectors by excluding subsets which cannot improve over the solutions already found the procedures terminate if all pareto optimal solutions have been either generated or excluded in this case the best pareto point found is an optimal solution such convergence is expected in the special case of pure integer optimization indeed numerical simulation tests with multi criteria facility location models and knapsack problems indicate reasonably fast convergence in particular under a linear value function we also propose a procedure to test whether or not a solution is a supported pareto point optimal under some linear value function

a sampling based method for generating nondominated solutions in stochastic momp problems chapters 4 and 6 presented methods for incorporating risk in the project selection optimization problem this chapter presents a method for generating nondominated solutions for stochastic multiobjective mathematical programming problems that is applicable to both continuous and zero one variables the method is based on the assumption that the objective function coefficients are random variables with probability distributions that are known or can be approximated the method results in solutions that are nondominated in terms of the expected value of each objective and the probability that each objective meets or exceeds a specified target value a method for generating a set of such solutions is presented and illustrated with examples the method presented here is more general in that it allows for any probability distribution it makes no behavioral assumptions about the decision maker it can accommodate integer zero one or real valued variables and it can accommodate linear or nonlinear constraints and objective functions

an interactive multiple objective linear programming method for a class of underlying nonlinear utility functions this paper develops a method for interactive multiple objective linear programming assuming an unknown pseudo concave utility function satisfying certain general properties the method is an extension of our earlier method published in this journal zionts s wallenius j 1976 an interactive programming method for solving the multiple criteria problem management sci 22 6 652 663 various technical problems present in predecessor versions have been resolved in addition to presenting the supporting theory and algorithm we discuss certain options in implementation and summarize our practical experience with several versions of the method

cut a multicriteria approach for concavifiable preferences we consider the problem of helping a decision maker dm choose from a set of multiattributed objects when her preferences are concavifiable i e representable by a concave value function we establish conditions under which preferences or preference intensities are concavifiable we also derive a characterization for the family of concave value functions compatible with a set of such preference statements expressed by the dm this can be used to validate dominance relations over discrete sets of alternatives and forms the basis of an interactive procedure we report on the practical use of this procedure with several dms for a flat choice problem and its computational performance on a set of project portfolio selection problem instances the use of preference intensities is found to provide significant improvements to the performance of the procedure

zero one programming with multiple criteria abstract the study of multiple criteria problems is not new but during the last seventeen years more and more study has been done in this area the study of 0 1 multiple criteria problems started much later than the study of continuous multiple criteria problems some of the earliest works on 0 1 problems with multiple criteria was first presented in 1973 quite a few solution methods have been developed since then especially during the last seven years in this paper most of these methods and some applications will be treated

the ipicea g a new hybrid evolutionary multi criteria decision making approach using the brushing technique various preference based multi objective evolutionary algorithms have been developed to help a decision maker search for his her preferred solutions to multi objective problems in most of the existing approaches the decision maker preferences are formulated either by mathematical expressions such as the utility function or simply by numerical values such as aspiration levels and weights however in some sense a decision maker may find it easier to specify preferences visually by drawing rather than using numbers this paper introduces such a method namely the brushing technique using this technique the decision maker can specify his her preferences easily by drawing in the objective space combining the brushing technique with one existing algorithm picea g we present a novel approach named ipicea g for an interactive decision making the performance of ipicea g is tested on a set of benchmark problems and is shown to be good

solving the discrete multiple criteria problem using convex cones an interactive method employing pairwise comparisons of attainable solutions is developed for solving the discrete deterministic multiple criteria problem assuming a single decision maker who has an implicit quasi concave increasing utility or value function the method chooses an arbitrary set of positive multipliers to generate a proxy composite linear objective function which is then maximized over the set of solutions the maximizing solution is compared with several solutions using pairwise judgments asked of the decision maker responses are used to eliminate alternatives using convex cones based on expressed preferences and then a new set of weights is found that satisfies the indicated preferences the requisite theory and proofs as well as a detailed numerical example are included in addition the results of some computational experiments to test the effectiveness of the method are described

a multi criteria agency model with incomplete preference information abstract we analyze an agency model in which the principal has only partial information on the agent s preferences consequently the principal can predict the agent s reaction to an incentive system only stochastically we first present the analytical solution for a simple decision problem in which the agent can choose only between two alternatives from this simple model we derive several hypotheses on how the preference information available to the principal influences her net results these hypotheses are then tested via a simulation model for more complex decision problems and we show that not all results of the simple model hold for the more general case most notably it is not generally the case that more precise information on the agent s preferences improves the principal s situation

an interactive branch and bound algorithm for multiple criteria optimization we present a new interactive algorithm for multiple criteria optimization the algorithm is of the branch and bound type and differs from previous interactive algorithms in several ways first the field of application is wider because it applies to two important classes of multiple criteria decision problems those for which the feasible set is convex and those for which the feasible set is discrete secondly the algorithm does not require a great deal from the decision maker he is merely required to indicate his preference between two vectors whenever the algorithm so demands

a fuzzy programming approach to multiobjective multidimensional 0 1 knapsack problems abstract the objective of this paper is to present an interactive procedure for the multiobjective multidimensional 0 1 knapsack problem that takes into consideration the incorporation of fuzzy goals of the decision maker that is easy to use since it requires from the decision maker to handle only one parameter namely the aspiration level of each objective and that is fast and can treat our problem as a usual 0 1 knapsack problem using already available software namely the primal effective gradient method used primarily to solve the large scale cases to get some statistics on the behavior of the algorithm a number of randomly generated simulation of problems is solved from our numerical experience it is possible to conclude that our proposed method is a worthwhile alternative to existing methods from a practical point of view

pruning of decision alternatives in multiple criteria decision making based on the uta method for estimating utilities abstract a new interactive method for the progressive elimination of elements from a finite set of decision alternatives is proposed a sequence of alternatives is presented to the decision maker who places each new alternative presented in rank order relative to the earlier alternatives evaluated this ranking of elements in a subset of the decision space is used to eliminate other alternatives from further consideration the approach is broadly based on the uta method for utility assessment the emphasis is not however on assessing a utility function as such instead the class of utility functions is used to eliminate elements of the decision space when the optimality of such elements is inconsistent with the presumed properties of the utility function and the rank orderings given in this way the decision maker need only evaluate a relatively small subset of the decision space before all remaining alternatives are eliminated the new procedure is illustrated by two numerical examples

a multiple criteria decision model with ordinal preference data abstract this paper develops a model for aggregating ordinal preference ranking across multiple criteria criteria weights are developed as part of the optimization process the model utilizes three levels of ordinal priorities 1 preferences among alternatives on the basis of each criterion 2 preferences among the criteria to reflect the importance of those criteria and 3 preferences among the criteria which express the degree to which the respondent can clearly distinguish among the alternatives the model is based upon a data envelopment analysis structure which permits a flexible weighting mechanism for each alternative being ranked

aspiration level approach in stochastic mcdm problems the paper considers a discrete stochastic multiple criteria decision making problem this problem is defined by a finite set of actions a a set of attributes x and a set of evaluations of actions with respect to attributes e in stochastic case the evaluation of each action with respect to each attribute takes form of a probability distribution thus the comparison of two actions leads to the comparison of two vectors of probability distributions in the paper a new procedure for solving this problem is proposed it is based on three concepts stochastic dominance interactive approach and preference threshold the idea of the procedure comes from the interactive multiple objective goal programming approach the set of actions is progressively reduced as the decision maker specifies additional requirements at the beginning the decision maker is asked to define preference threshold for each attribute next at each iteration the decision maker is confronted with the set of considered actions if the decision maker is able to make a final choice then the procedure ends otherwise he she is asked to specify aspiration level a didactical example is presented to illustrate the proposed technique

meaning and validity of interactive procedures as tools for decision making this papers examines procedures based on answers to questions an entity q asks an actor d in accordance with a protocol p the aim is either to select an alternative a considered by d to be the optimal choice among a set of alternatives a or to formulate a utility function u considered relevant by d for comparing alternatives in a what gives meaning to the final result of this type of interactive procedure on which criteria can its validity be assessed in analysing three questions of a parlor game we can produce evidence that the answers to such questions do not depend on p alone but that the way in which d produces his answers plays just a central role threee different hypotheses concerning this points led us to a new approach to this type of question an approach which point out the illusive nature of certain types of answers

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

dual cone approach to convex cone dominance in multiple criteria decision making abstract in a paper published in management science in 1984 korhonen wallenius and zionts presented the idea and method based on convex cone dominance in the discrete multiple criteria decision making framework in our current paper we revisit the old idea from a new standpoint and provide the mathematical theory leading to a dual cone based approach to solving such problems our paper makes the old results computationally more tractable the results provided in the present paper also help extend the theory

a data envelopment analysis approach to multicriteria decision problems with incomplete information abstract as a decision aid for discrete multicriteria decision problems this paper proposes a multilevel graph of alternatives to represent the ranking to the extent that this is possible when incomplete information on weights is available under the assumption of the additive value function to construct it the nested decomposition of the set of alternatives is established along the lines of data envelopment analysis dea a numerical example is given to illustrate a multilevel graph based on the nested decomposition and compare it with the hierarchical dominance graph based on dominance relations proposed by park and kim

joint densities for random walks in the plane abstract we point out the existence of computationally convenient techniques for calculating the joint probability density for the position of a pearson random walk after n steps a new fourier bessel function expansion for p n r is developed for this purpose which does not require radial symmetry but does require that p n r 0 when r exceeds some maximum radius r

multicriteria analysis survey and new directions abstract before the advent of multicriteria analysis decision problems often took the form of the optimization of an objective function or a utility function this approach has the advantage of resulting in well defined mathematical problems but is not always representative of reality in fact the comparison of several possible decisions is rarely made according to a single point of view and the preferences on a point of view are in many cases only modelled with difficulty by a function the aim of this paper is to survey the different directions in which multicriteria analysis has developed multiattribute utility theory outranking relations interactive methods the methodology recently proposed by b roy is also presented

estimating aspiration levels from discrete choices computational techniques and experiences abstract this paper discusses the problem of estimating aspiration or reference levels in criteria space from choices between alternatives characterized by multiple criteria several methods for such estimations are developed and compared taking into account both solution quality and computational efficiency methods based on mixed integer linear programming which provide optimal solutions require unacceptable computing times for application in interactive systems an alternative method based on a direct search algorithm is shown to be an effective way of generating high quality estimates with small computational effort

testing the efficiency of pairwise comparison methods in discrete multiple criteria problems we test the efficiency of two pairwise comparison methods the methods studied are salminen s piecewise linear prospect theory plp method and the convex cone method by korhonen wallenius and zionts kwz the plp method is based on a piecewise linear difference function the kwz method assumes a quasi concave utility or value function these methods are tested using randomly generated data sets with 50 100 150 and 200 alternatives and from 2 up to 6 criteria the dm s preference statements are simulated using linear quadratic and chebycheff value functions and piecewise linear difference functions both methods lose their efficiency very rapidly with increasing number of criteria our results indicate that pairwise comparison methods are not as efficient as reported in literature

interactive approaches for discrete alternative multiple criteria decision making with monotone utility functions in this paper we develop interactive approaches for the discrete alternative multiple criteria decision making problem we develop an algorithm that finds the most preferred alternative of a decision maker dm assuming only that the dm has a monotonic utility function the algorithm divides the criteria space into a number of smaller subspaces and then uses the ideal points of these subspaces to eliminate alternatives we also develop a more efficient version of the algorithm for the more restrictive case of a monotonic quasiconcave utility function we present favorable computational results in terms of the required number of pairwise comparisons for both versions of the algorithm we then develop a general algorithm that first identifies the type of the dm s utility function and then employs the approach that is compatible with the identified utility function type we also present computational results for the general algorithm

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

a progressive algorithm for modeling and solving multiple criteria decision problems we consider a decision maker dm who has a set of possible decision alternatives from which one a best is to be chosen however all decision alternatives are not at the dm s disposal initially nor is full knowledge of his her utility value function therefore the dm evaluates only the available subset of all decision alternatives from which he she chooses a most preferred one obviously this decision is not necessarily globally best two natural questions arise how good is the most preferred solution what are the chances of finding better solutions by considering additional alternatives we describe and illustrate a general progressive algorithm and the supporting theory for modeling and solving this problem when alternatives are introduced dynamically

interactive algorithms for a broad underlying family of preference functions abstract in multi criteria decision making approaches it is typical to consider an underlying preference function that is assumed to represent the decision maker s preferences in this paper we introduce a broad family of preference functions that can represent a wide variety of preference structures we develop the necessary theory and interactive algorithms for both the general family of the preference functions and for its special cases the algorithms guarantee to find the most preferred solution point of the decision maker under the assumed conditions the convergence of the algorithms are achieved by progressively reducing the solution space based on the preference information obtained from the decision maker and the properties of the assumed underlying preference functions we first demonstrate the algorithms on a simple bi criteria problem with a given set of available points we also test the performances of the algorithms on three criteria knapsack problems and show that they work well

interactive evolutionary multi objective optimization for quasi concave preference functions we present a new hybrid approach to interactive evolutionary multi objective optimization that uses a partial preference order to act as the fitness function in a customized genetic algorithm we periodically send solutions to the decision maker dm for her evaluation and use the resulting preference information to form preference cones consisting of inferior solutions the cones allow us to implicitly rank solutions that the dm has not considered this technique avoids assuming an exact form for the preference function but does assume that the preference function is quasi concave this paper describes the genetic algorithm and demonstrates its performance on the multi objective knapsack problem

use of piecewise linear value functions in interactive multicriteria decision support a monte carlo study this paper describes a monte carlo study conducted to evaluate the effects of modelling assumptions and design parameters on the behaviour of interactive methods for the discrete choice mcdm problem based on explicit value function models the purpose of the study is to identify those assumptions and parameters which lead to the most efficient use of preference judgements made by the decision maker and to the greatest robustness to judgmental errors it is concluded that nonlinearities in the value function need to be modelled achieved here by use of a piecewise linear form it was also found that search for indifference points rather than using simple preference judgements alone is of great advantage best realized by expressing judgements in terms of pairwise trade offs methods incorporating these features are highly robust to judgmental errors interactive methods of this class are compared with a priori fitting of similar value functions and found to give a very similar quality of solution

extension of the topsis method for decision making problems with fuzzy data decision making problem is the process of finding the best option from all of the feasible alternatives in this paper from among multicriteria models in making complex decisions and multiple attribute models for the most preferable choice technique for order preference by similarity to ideal solution topsis approach has been dealt with in real word situation because of incomplete or non obtainable information the data attributes are often not so deterministic there for they usually are fuzzy imprecise therefore the aim of this paper is to extend the topsis method to decision making problems with fuzzy data in this paper the rating of each alternative and the weight of each criterion are expressed in triangular fuzzy numbers the normalized fuzzy numbers is calculated by using the concept of a cuts finally a numerical experiment is used to illustrate the procedure of the proposed approach at the end of this paper

joint distribution function for position and rotation angle in plane random walks abstract the angle by which the tangent to the trajectory rotates in the course of a plane random walk is studied the joint probability distribution function for position and rotation angle in a random walk of n steps is obtained in closed form this result is a generalization of kluyver s integral representation for the distribution function of the position alone explicit expressions are presented for n 1 2 and n in the latter case both by application of the central limit theorem and the steepest descent method the distribution of the rotation angle modulo 2 is also discussed and turns out to be particularly simple

an interactive procedure using domination cones for bicriterion shortest path problems abstract in this paper an interactive procedure is developed for the bicriterion shortest path problem it is assumed that the decision maker s inherant utility function is quasi concave and non increasing and that the network consists of non negative integer valued arc lengths the proposed procedure uses the concept of domination cones which it develops from pairwise comparisons of alternatives these domination cones are used to fathom a large number of pareto optimal solutions extensive computational testing was performed on large grid networks simulating the decision maker s response using polynomial utility functions the results indicate that our proposed procedure is able to converge to the optimal solution in a reasonably small number of pairwise comparisons even for those problems with a large number of pareto optimal solutions

multiple criteria decision making multiattribute utility theory the next ten years management science and decision science have grown exponentially since midcentury two closely related fields central to this growth are multiple criteria decision making mcdm and multiattribute utility theory maut this paper comments on the history of mcdm and maut and discusses topics we believe are important in their continued development and usefulness to management science over the next decade our aim is to identify exciting directions and promising areas for future research

graphically structured value function compilation classical work on eliciting and representing preferences over multi attribute alternatives has attempted to recognize conditions under which value functions take on particularly simple and compact form making their elicitation much easier in this paper we consider preferences over discrete domains and show that for a certain class of simple and intuitive qualitative preference statements one can always generate compact value functions consistent with these statements these value functions maintain the independence structure implicit in the original statements for discrete domains these representation theorems are much more general than previous results however we also show that it is not always possible to maintain this compact structure if we add explicit ordering constraints among the available outcomes

preference relations on actions and criteria in multicriteria decision making abstract let a be a finite set of feasible actions which are judged following several criteria an outranking relation is defined on a by considering preference of the decision maker as a weak order on each criterion and the relation among criteria as a semi order on the given set of criteria several ways of constructing outranking relations have been proposed one of the most popular introduced by b roy for instance electre s is based on the use of weights related to criteria in our approach the knowledge of weights is replaced by the existence of a semi order a case study is developed it deals with a computer selection problem

the pearson random walk with unequal step sizes there have been many analyses of pearson random walks with equal step sizes several applications suggest the importance of these walks with unequal step sizes numerical comparisons are made of approximations obtained using the central limit theorem and the steepest descents method these suggest that the latter generally but not always leads to more accurate results the authors also derive an approximation that is extremely accurate at near maximal extensions

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