goal programming for decision making an overview of the current state of the art there have been significant advances in the theory of goal programming gp in recent years particularly in the area of intelligent modelling and solution analysis the intention of this paper is to provide an overview of these developments to detail and assess the current state of the art in the subject and to highlight areas which seem promising for future research modelling techniques such as detection and restoration of pareto efficiency normalisation redundancy checking and non standard utility function modelling are overviewed the connection between gp and other multi objective programming techniques as well as a utility interpretation of gp are examined the rationality of ranking multi criteria decision making techniques and of placing gp in such a ranking is discussed

a parametric model for determining consensus priority vectors from fuzzy comparison matrices abstract we consider a group decision making problem where a set of alternatives have to be ranked according to fuzzy preference judgments given by multiple experts we assume that expert assessments are expressed in the form of fuzzy multiplicative preference relations or fuzzy comparison matrices in this paper we propose a general model to generate crisp priority weights of the alternatives from possibly inconsistent and conflicting fuzzy preference relations we express our model in terms of matrix approximations to address the consistency problem and we use weighted metrics to simulate the group dynamics matrix approximation techniques for deriving a common crisp consistent matrix are extended to work with fuzzy matrices by using the concept of cuts in the aggregation process the importance of each expert is taken into consideration according to the agreement of the group with the expert this results in a parametric optimization problem for which a computational formulation is given

deriving weights from pairwise comparison ratio matrices an axiomatic approach abstract this paper examines the problem of extracting object or attribute weights from a pairwise comparison ratio matrix this problem is approached from the point of view of a distance measure on the space of all such matrices a set of axioms is presented which such a distance measure should satisfy and the uniqueness of the measure is proven the problem of weight derivation is then shown to be equivalent to that of finding a totally transitive matrix which is a minimum distance from the given matrix this problem reduces to a goal programming model finally it is shown that the problem of weight derivation is related to that of ranking players in a round robin tournament the space of all binary tournament matrices is proven to be isometric to a subset of the space of ratio matrices

moments of utility functions and their applications this paper presents moments and cross moments of utility functions and measures of utility dependence we start with an interpretation of the nth moment of a utility function and describe methods for its assessment in practice and consistency checks that need to be satisfied for any assessed moments we then show how moments of a utility function i provide a new method to determine the parameters of a given functional form of a utility function and ii to derive the functional form of a utility function that satisfies some given moment assessments next we derive a fundamental formula that relates the expected utility of a joint distribution to the expected utility of the marginal distributions for multiattribute utility functions we use this formulation to provide an intuitive interpretation for cross moments of utility functions and illustrate their use in i constructing multiattribute utility functions that incorporate utility dependence and ii in providing necessary conditions for utility independence in decisions with multiple attributes we end with a new measure of utility dependence for multiattribute utility functions and work through several examples to illustrate the approach

eigenvector ranking method as a measuring tool formulas for errors abstract the eigenvector method em is well known to derive information from pairwise comparison matrices in decision making processes however this method is logically incomplete since its actual numerical error is unknown and its reliability is doubted by such phenomena as right left asymmetry rank reversal and reversal of order of intensity of preference in this paper we associate em with some standard measuring procedure analyze this procedure from the viewpoint of measurement theory and find the actual em error we show that the above phenomena have the same cause and are eliminated when the em errors are taken into account the full decision support tool which has all components of a standard measuring tool is composed of pairwise comparisons as an initial measuring procedure em as a data processor and the obtained formulas for em errors as an error indicator we consider two versions of this tool based on the right and the left principal eigenvectors of a pairwise comparison matrix both versions are equally suitable to measure and rank any comparable elements with positive numerical values and have the same mean relative errors equal to the square root of the double saaty s consistency index using the mean relative error we find the simple upper estimate for maximum permissible errors not impeding the reliable ranking this estimate imposes tight restrictions on inconsistency of expert judgements in decision making processes with a large number of alternatives these restrictions are much stronger than previously thought

on the minimum violations ranking of a tournament this paper examines the problem of rank ordering a set of players or objects on the basis of a set of pairwise comparisons arising from a tournament the criterion for deriving this ranking is to have as few cases as possible where player i is ranked above j while i was actually defeated by j in the tournament such a situation is referred to as a violation the objective therefore is to determine the minimum violations ranking mvr while there are situations where this ranking would be allowed to contain ties among subsets of objects we will concern ourselves herein with linear ordering no ties a series of examples are given where this requirement would seem to be appropriate in order to put the mvr problem into proper perspective we introduce the concept of a distance on the set of tournaments a set of natural axioms is presented which any such distance measure should obey and it is proven that in the presence of these axioms a unique such measure exists it is then shown that the mvr problem is equivalent to the minimum distance problem which can be represented in several forms in particular as a problem of determining the minimum feedback edge set in a graph and as a mixed integer generalized network problem this opens up a wide scope of possible solution procedures for the mvr problem an optimal algorithm is presented along with computational results in addition various heuristics are discussed including an improved heuristic referred to as the iterated kendall method

bivariate distributions with exponential conditionals abstract it is frequently easier to visualize conditional distributions of experimental variables rather than joint distributions in this article we consider the most general class of bivariate distributions such that both sets of conditional densities are exponential the class proves to be remarkably simple to describe the joint density must be proportional to exp x y xy where the constant of proportionality depends on the classical exponential integral the joint distribution has marginals that are not exponential and a negative correlation coefficient except in the special case of independence after deriving some distributional results we develop methods for parameter estimation and simulation a simple method of moments estimator appears to give reasonable results we also briefly discuss generalizations to higher dimensions and to distributions with conditionals in a general exponential family

a parametric gp model dealing with incomplete information for group decision making abstract we consider a group decision making problem where preferences given by the experts are articulated into the form of pairwise comparison matrices in many cases experts are not able to efficiently provide their preferences on some aspects of the problem because of a large number of alternatives limited expertise related to some problem domain unavailable data etc resulting in incomplete pairwise comparison matrices our goal is to develop a computational method to retrieve a group priority vector of the considered alternatives dealing with incomplete information for that purpose we have established an optimization problem in which a similarity function and a parametric compromise function are defined associated to this problem a logarithmic goal programming formulation is considered to provide an effective procedure to compute the solution moreover the parameters involved in the method have a clear meaning in the context of group problems

the pearson system of utility functions this paper describes a parametric family of utility functions for decision analysis the parameterization embeds the hara class in a four parameter representation for the risk aversion function the resulting utility functions can have only four shapes concave convex s shaped and reverse s shaped this makes the family suited for both expected utility and prospect theory the paper also describes an alternative technique to estimate the four parameters from elicited utilities which is simpler than standard fitting by minimization of the mean quadratic error

on the efetov wegner terms by diagonalizing a hermitian supermatrix the diagonalization of hermitian supermatrices is studied such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory however it still poses serious problems since until now the calculation of all rothstein contributions known as efetov wegner terms in physics was quite cumbersome we derive the supermatrix bessel function with all efetov wegner terms for an arbitrary rotation invariant probability density function as applications we consider representations of generating functions for hermitian random matrices with and without an external field as integrals over eigenvalues of hermitian supermatrices all results are obtained with all efetov wegner terms which were previously unknown in such an explicit and compact representation

fourier analysis on a hyperbolic supermanifold with constant curvature the fourier inversion theorem is proved for a rank one noncompact homogeneous space the hyperbolic superplane the proof makes use of some novel features of perfectly graded superspaces which are not encountered in classical geometric analysis an application to quasi one dimensional disordered one electron systems is given

a comparison of the superbosonization formula and the generalized hubbard stratonovich transformation recently two different approaches were put forward to extend the supersymmetry method in random matrix theory from gaussian ensembles to general rotation invariant ensembles these approaches are the generalized hubbard stratonovich transformation and the superbosonization formula here we prove the equivalence of both approaches to this end we reduce integrals over functions of supersymmetric wishart matrices to integrals over quadratic supermatrices of certain symmetries

arbitrary rotation invariant random matrix ensembles and supersymmetry orthogonal and unitary symplectic case recently the supersymmetry method was extended from gaussian ensembles to arbitrary unitarily invariant matrix ensembles by generalizing the hubbard stratonovich transformation here we complete this extension by including arbitrary orthogonally and unitary symplectically invariant matrix ensembles the results are equivalent to but the approach is different from the superbosonization formula we express our results in a unifying way we also give explicit expressions for all one point functions and discuss features of the higher order correlations

a note on deriving weights from pairwise comparison ratio matrices abstract this paper addresses the cook and kress method regarding the use of mathematical programming to derive weights from pairwise comparison ratio matrices we note that the cook and kress formulation has the shortcoming that alternative optimal solutions may occur which would lead to an infinite set of possible weights this paper proposes to resolve the problem by formulating a phase ii optimization procedure to be solved using quadratic programming this mathematical formulation is based on the determination of a weight vector that is statistically most likely to cause the associated pairwise comparison matrix it is concluded that the cook and kress method accompanied with the proposed phase ii approach can always derive a unique set of weights from a pairwise comparison matrix

a characterization of the logarithmic least squares method abstract we provide an axiomatic characterization of the logarithmic least squares method sometimes called row geometric mean used for deriving a preference vector from a pairwise comparison matrix this procedure is shown to be the only one satisfying two properties correctness in the consistent case which requires the reproduction of the inducing vector for any consistent matrix and invariance to a specific transformation on a triad that is the weight vector is not influenced by an arbitrary multiplication of matrix elements along a 3 cycle by a positive scalar

computing interval weights for incomplete pairwise comparison matrices of large dimension a weak consistency based approach multiple criteria decision making and evaluation problems dealing with a large number of objects are very demanding particularly when the use of pairwise comparison pc techniques is required a major drawback arises when it is not possible to obtain all the pcs due to time or cost limitations or to split the given problem into smaller subproblems in such cases two tools are needed to find acceptable weights of objects an efficient method for partially filling a pairwise comparison matrix pcm and a suitable method for deriving weights from this incomplete pcm this paper presents a novel interactive algorithm for large dimensional problems guided by two main ideas the sequential optimal choice of the pcs to be performed and the concept of weak consistency the proposed solution significantly reduces the number of needed pcs by adding information implied by the weak consistency after the input of each pc providing sets of feasible values for all missing pcs interval weights of objects are computed from the resulting incomplete weakly consistent pcm adapting the methodology for calculating fuzzy weights from fuzzy pcms the computed weight intervals thus cover all possible weakly consistent completions of the incomplete pcm the algorithm works both with saaty s pcms and fuzzy preference relations the performance of the algorithm is illustrated by a numerical example and a real life case study the performed simulation demonstrates that the proposed algorithm is capable of reducing the number of pcs required in pcms of dimension 15 and greater by more than 60 on average

integration of grassmann variables over invariant functions on flat superspaces we study integration over functions on superspaces these functions are invariant under a transformation which maps the whole superspace onto the part of the superspace which only comprises purely commuting variables we get a compact expression for the differential operator with respect to the commuting variables which results from berezin integration over all grassmann variables also we derive cauchy like integral theorems for invariant functions on supervectors and symmetric supermatrices this extends theorems partly derived by other authors as a physical application we calculate the generating function of the one point correlation function in random matrix theory furthermore we give another derivation of supermatrix bessel functions for u k1 k2

asymptotic coincidence of the statistics for degenerate and non degenerate correlated real wishart ensembles the correlated wishart model provides the standard benchmark when analyzing time series of any kind unfortunately the real case which is the most relevant one in applications poses serious challenges for analytical calculations often these challenges are due to square root singularities which cannot be handled using common random matrix techniques we present a new way to tackle this issue using supersymmetry we carry out an anlaytical study which we support by numerical simulations for large but finite matrix dimensions we show that statistical properties of the fully correlated real wishart model generically approach those of a correlated real wishart model with doubled matrix dimensions and doubly degenerate empirical eigenvalues this holds for the local and global spectral statistics with monte carlo simulations we show that this is even approximately true for small matrix dimensions we explicitly investigate the k point correlation function as well as the distribution of the largest eigenvalue for which we find a surprisingly compact formula in the doubly degenerate case moreover we show that on the local scale the k point correlation function exhibits the sine and the airy kernel in the bulk and at the soft edges respectively we also address the positions and the fluctuations of the possible outliers in the data

characteristic functions and invariants of supermatrices the characteristic functions and the invariants of supermatrices are studied it is shown that the euclidean algorithm is useful in obtaining a system of invariants

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

the logarithmic least squares and the generalized pseudoinverse in estimating ratios abstract this paper concerns the pairwise comparison method used in analytic hierarchy process ahp the logarithmic least square method is one of the methods used to rank a finite number of stimuli based on their pairwise comparison in the case of one decision maker the problem can be solved using the geometric mean method it is then assumed that the solution is geometrically normalized in the case of multiple decision makers a set of linear equations is obtained and if we have a different number of judgments for each pair of the compared objects the geometric normalization assumption can not be used directly the aim of this paper is to show that applying the generalized pseudoinverse we obtain the solution that is geometrically normalized and consistent with the case of one decision maker to define the pseudoinverse the spectral decomposition is used the structure of the general solution is presented and the existence of the general solution is discussed

backpropagation multi layer perceptron for incomplete pairwise comparison matrices in analytic hierarchy process abstract analytic hierarchy process ahp is a widely used decision making method in many areas such as management sciences the performance ratings of multiple criteria and alternatives can be elicited from pairwise comparisons obtained by expressing the decision maker s perceive however it may be difficult for the decision maker to prudently assign values to comparisons for large number of criteria or alternatives since there exist distinct relationships between any two elements in the real world the relationship between the missing comparison and the assigned comparisons should be taken into account the aim of this paper is to propose a novel method using a well known regression tool namely the backpropagation multi layer perceptron i e mlp to realize the above implicit relationship so as to estimate a missing pairwise judgment from the other assigned entries a computer simulation is employed to demonstrate that the proposed method can effectively find a missing entry of an incomplete pairwise matrix such that its consistency index is minimized

the integral theorem for supersymmetric invariants a supersymmetric integral theorem that extends results of parisi sourlas efetov wegner and others is rigorously proved in particular arbitrary generators are allowed in the integrand instead of canonical ones and the invariance condition is very much relaxed the connection with cauchy s integral formula is made transparent in passing the unitary lie supergroup is studied by using elementary methods applications in the theory of disordered systems are discussed

bbtopsis a bag based technique for order preference by similarity to ideal solution abstract this contribution explores the way of accomplishing a topsis ranking of various submitted variants taking into account a our newly developed notions of variant superiority inferiority and noninferiority multiattribute canonical fuzzy bags defined on attributes and b making use of bag metrics to compute the relative closeness of the different variants to an ideal solution in particular we prove the existence and the uniqueness of a bag metric fulfilling a number of interesting properties

pairwise comparisons simplified this study examines the notion of generators of a pairwise comparisons matrix such approach decreases the number of pairwise comparisons from n n 1 to n 1 an algorithm of reconstructing of the pc matrix from its set of generators is presented

arbitrary unitarily invariant random matrix ensembles and supersymmetry we generalize the supersymmetry method in random matrix theory to ensembles which are unitarily invariant but otherwise arbitrary our exact approach extends a previous contribution in which we constructed a supersymmetric representation for the class of norm dependent random matrix ensembles here we derive a supersymmetric formulation under very general circumstances a reduced probability density and a projector are identified that map the probability density from ordinary to superspace furthermore it is demonstrated that setting up the theory in fourier superspace has considerable advantages general and exact expressions for the correlation functions are given we also show how the use of hyperbolic symmetry can be circumvented in the present context in which the nonlinear model is not used we construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the green s functions

robust decision making using a general utility set abstract we address the problem of ambiguity and inconsistency in a decision maker s dm assessed utility function by using a maxmin framework in this framework dm s utility function belongs to a set of functions the member functions of the set are nondecreasing and satisfy additional boundary and auxiliary conditions the maxmin framework provides robustness in decision making alternatively it allows us to perform sensitivity or parametric analysis for the optimal decision when a reference utility function is known for this purpose we use a cost of ambiguity concept and show that this cost is increasing and concave when extent of ambiguity is parametrically increased next we develop a lagrangian based solution approach for the decision problem we show that under suitable conditions a sample average approximation saa of the lagrangian model can be solved using a mixed integer linear program milp we also show that the set of optimal solutions of the saa converges to that of its true counterpart and the optimum objective value of the saa converges to the true objective value at an exponential rate we use this convergence property to develop a heuristic for identifying a solution of the saa milp with increasing sample size we illustrate the properties of the maxmin model using two examples of portfolio investment and streaming bandwidth we provide a discussion on the performance of a commercial solver used to solve saa milps and the quality of solutions generated by the proposed heuristic

a monte carlo investigation of incomplete pairwise comparison matrices in ahp the analytic hierarchy process ahp is a decision analysis technique used to evaluate complex multiattribute alternatives among one or more decision makers it imposes a hierarchical structure on any complex multicriterion problem however a major drawback of the ahp is that a large number of pairwise comparisons is needed to calibrate the hierarchy when there are a few levels and sublevels the ahp can be applied in a straightforward manner to derive the weights relative preference for each alternative as the size of the hierarchy increases the number of pairwise comparisons increases rapidly it is well established in the marketing and consumer behavior literature that in a very long interview even under the best circumstances the respondent is likely to suffer from information overload recognition of this problem was the motivation which led to the investigation of a modification of ahp which required less data collection i e a reduction in the threat of information overload the first question to be answered is the effect on ahp weights due to different patterns of missing data likely to result from reallife data collection in this study a monte carlo simulation was conducted which uses the incomplete pairwise comparisons ipc algorithm 14 to investigate the effect of reduced sets of pairwise comparisons in the ahp data for the study were generated with known structure and comparisons made between complete and incomplete matrices the results of the simulation suggest that incomplete sets of pairwise comparison matrices can capture the attribute level weights without significant loss of accuracy and independent of decision model form and amount of error considered

the supersymmetry method for chiral random matrix theory with arbitrary rotation invariant weights in the past few years the supersymmetry method has been generalized to real symmetric hermitian and hermitian self dual random matrices drawn from ensembles invariant under the orthogonal unitary and unitary symplectic groups respectively we extend this supersymmetry approach to chiral random matrix theory invariant under the three chiral unitary groups in a unifying way thereby we generalize a projection formula providing a direct link and hence a short cut between the probability density in ordinary space and that in superspace we emphasize that this point was one of the main problems and shortcomings of the supersymmetry method since only implicit dualities between ordinary space and superspace were known before to provide examples we apply this approach to the calculation of the supersymmetric analogue of a lorentzian cauchy ensemble and an ensemble with a quartic potential moreover we consider the partially quenched partition function of the three chiral gaussian ensembles corresponding to four dimensional continuum quantum chromodynamics we identify a natural splitting of the chiral lagrangian in its lowest order into a part for the physical mesons and a part associated with source terms generating the observables e g the level density of the dirac operator

preference elicitation from inconsistent judgments using multi objective optimization abstract several decision making techniques involve pairwise comparisons to elicit the preferences of a decision maker dm this paper proposes a new approach for prioritization from pairwise comparisons using the concept of indirect judgments no method exists that simultaneously minimizes deviations from both direct and indirect judgments in order to estimate preferences it is sensible to consider both the acquired judgments and the other judgments latent in the dm s mind hence a technique is developed here to minimize the deviations from both types of judgments estimated preferences are generally evaluated based on two criteria their deviation from the provided judgments and the number of judgments that have been ordinally violated here it is proposed to optimize three objectives simultaneously the deviations from both direct and indirect judgments and the number of judgments violated a prototype application has been developed to generate all non dominated solutions using a multiple objective evolutionary algorithm the new approach is shown to offer users greater flexibility than all other tested methods

dyson s correlation functions and graded symmetry a new derivation of dyson s k level correlation functions of the gaussian unitary ensemble gue is given the method uses matrices with graded symmetry the number of integrations needed for the ensemble average becomes independent of the level number n for arbitrary level number n the k level correlation function is expressed as an integral involving the eigenvalues of a 2k 2k graded matrix the limit of infinitely many levels n is calculated by a simple saddle point approximation of this integral avoiding the introduction of hermite polynomials and oscillator wave functions

norm dependent random matrix ensembles in external field and supersymmetry the class of norm dependent random matrix ensembles is studied in the presence of an external field the probability density of the ensemble depends on the trace of the squared random matrices but is otherwise arbitrary an exact mapping to superspace is performed a transformation formula is derived which gives the probability density in superspace as a single integral over the probability density in ordinary space this is done for orthogonal unitary and symplectic symmetries in the case of unitary symmetry some explicit results for the correlation functions are derived

approximation of positive matrices by transitive matrices a newton kantorovich algorithm used in the approximation of symmetrically reciprocalpositive matrices by transitive matrices is found to be more generally useful in the approximation of merely positive matrices the occurrence of multiple stationary values when the approximated matrix has certain symmetries is investigated

certainty equivalents for three point discrete distribution approximations three point discrete distribution approximations are often used in decision and risk analyses to represent probability distributions for continuous random variables e g as probability nodes in decision or probability trees performance evaluations of such approximations have generally been based on their accuracies in estimating moments for the underlying distributions however moment based comparisons for recently proposed approximations have been limited moreover very little research has addressed the accuracies of any such approximations in representing expected utilities or certainty equivalents of the underlying distributions this is of potential concern since recent research shows three point discrete distribution approximations exist that match the first several moments of an underlying distribution exactly while counterintuitively approximating its certainty equivalents poorly this paper compares the best two approximations for estimating means and variances identified in an earlier study with promising approximations proposed more recently specifically it examines how accurately six simple general purpose three point approximations represent certainty equivalents for continuous random variables as the level of risk aversion is varied as well as how accurately they estimate means and variances the results show that several of these are quite accurate over a variety of test distributions when the level of risk aversion and the characteristics of the distributions are within reasonable bounds their robust performance is significant for decision analysis practice

analyticity of density of states in a gauge invariant model for disordered electronic systems then orbital gauge invariant model of disordered electronic systems proposed by wegner is studied in the regime of dominant diagonal disorder analyticity of the density of states is established in two cases a when the number of orbitals is small b when the number of orbitals is large and the energy is in the expected extended states region

combining minsum and minmax a goal programming approach a number of methods for multiple objective optimization problems mop give as solution to mop the set of optimal solutions for some single objective optimization problems associated with it well known examples of these single objective optimization problems are the minsum and the minmax in this note we propose a new parametric single objective optimization problem associated with mop by means of goal programming ideas we show that the minsum and minmax are particular instances so we are somehow combining minsum and minmax by means of a parameter moreover such parameter has a clear meaning in the value space applications of this parametric problem to classical models in locational analysis are discussed

maximum entropy utility this paper presents a method to assign utility values when only partial information is available about the decision makers preferences we introduce the notion of a utility density function and a maximum entropy principle for utility assignment the maximum entropy utility solution embeds a large family of utility functions that includes the most commonly used functional forms we discuss the implications of maximum entropy utility on the preference behavior of the decision maker and present an application to competitive bidding situations where only previous decisions are observed by each party we also present minimum cross entropy utility which incorporates additional knowledge about the shape of the utility function into the maximum entropy formulation and work through several examples to illustrate the approach

efficient weight vectors from pairwise comparison matrices abstract pairwise comparison matrices are frequently applied in multi criteria decision making a weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix and strictly better in at least one position a weight vector is weakly efficient if the pairwise ratios cannot be improved in all non diagonal positions we show that the principal eigenvector is always weakly efficient but numerical examples show that it can be inefficient the linear programs proposed test whether a given weight vector is weakly efficient and in case of strong inefficiency an efficient strongly dominating weight vector is calculated the proposed algorithms are implemented in pairwise comparison matrix calculator available at pcmc online

fourier bessel analysis for ordinary and graded 2 2 hermitian matrices chainlike integrals in matrix spaces play an important part in high energy and solid state physics and in general random matrix theory in the special case of ordinary and graded 2 2 hermitian matrices a method is proposed to integrate out all angular variables the essence of this method is a fourier bessel analysis in these matrix spaces which is formulated in this paper close formal similarities are found between the ordinary and the graded case the main differences arise from the fact that the ordinary case can be reduced to the study of a vector space whereas no analogous feature is present in the graded case