robustness and information levels in case based multiple criteria sorting abstract case based preference elicitation methods for multiple criteria sorting problems have the advantage of posing rather small cognitive demands on a decision maker but they may lead to ambiguous results when preference parameters are not uniquely determined we use a simulation approach to determine the extent of this problem and to study the impact of additional case information on the quality of results our experiments compare two decision analysis tools case based distance sorting and the simple additive weighting method in terms of the effects of additional case information on sorting performance depending on problem dimension number of groups number of criteria etc our results confirm the expected benefit of additional case information on the precision of estimates of the decision maker s preferences problem dimension however has some unexpected effects

group decision making model using fuzzy multiple attributes analysis for the evaluation of advanced manufacturing technology selection of advanced manufacturing technology is important for improving manufacturing system competitiveness this study builds a group decision making model using fuzzy multiple attributes analysis to evaluate the suitability of manufacturing technology since numerous attributes have been considered in evaluating the manufacturing technology suitability most information available in this stage is subjective and imprecise and fuzzy sets theory provides a mathematical framework for modeling imprecision and vagueness the proposed approach involved developing a fusion method of fuzzy information which was assessed using both linguistic and numerical scales in addition an interactive decision analysis is developed to make a consistent decision when evaluating the suitability of manufacturing technology it may be necessary to improve upon the technology and naturally advanced manufacturing technology is seen as the best direction for improvement the flexible manufacturing system adopted in the taiwanese bicycle industry is used in this study to illustrate the computational process of the proposed method the results of this study are more objective and unbiased owing to being generated by a group of decision makers

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

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

hessenberg pairs of linear transformations abstract let k denote a field and v denote a nonzero finite dimensional vector space over k we consider an ordered pair of linear transformations a v v and a v v that satisfy i iii below 1 i each of a a is diagonalizable on v 2 ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v 0 v 1 v i 1 0 i d where v 1 0 v d 1 0 3 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v 0 v 1 v i 1 0 i where v 1 0 v 1 0 we call such a pair a hessenberg pair on v in this paper we obtain some characterizations of hessenberg pairs we also explain how hessenberg pairs are related to tridiagonal pairs

the utility efficient set and its interactive reduction we assume a decision situation under risk with incomplete information on preferences modelled as a vector utility function we consider an additive aggregation of its components and partial information on the scaling constants we develop the concept of utility efficiency to identify efficient strategies in discrete problems when the information about the scaling constants of the decision maker is in the form of a polyhedral cone a characterization of the utility efficient set provides a practical way to compute such efficient strategies we then discuss an interactive method based on the assessment of the scaling constants via an interactive paired comparison with its convergence the method is complemented by a procedure to reduce the utility efficient set to aid in the process of reaching a final strategy

the categorical data analysis approach for ratio model of pairwise comparisons abstract in the analytic hierarchy process ahp pairwise comparison values are elicited by a ratio scale representing a set of categories in this article we propose a method of statistical inference in ahp which pertains to cardinal representations of categorical variables relevant estimation and hypothesis testing procedures are addressed in this perspective

compatible weighting method with rank order centroid maximum entropy ordered weighted averaging approach in a situation where imprecise attribute weights such as a rank order are captured various approximate weighting methods have been proposed to aid multiattribute decision analysis among others it is well known that the rank order centroid roc weights result in the highest performance in terms of the identification of the best alternative under the ranked attribute weights in this paper we aim to reinterpret the meaning of the roc weights and to develop a compatible weighting method that is based on other well established academic disciplines the ordered weighted averaging owa method is a nonlinear aggregation method in that the weights are associated with the objects reordered according to their magnitudes in the aggregation process some interesting semantics can be attached to the approximate weights in view of the measure developed in the owa method furthermore the weights generated by the maximum entropy method show equally compatible performance with the roc weights under some condition which is demonstrated by theoretical and simulation analysis

multi attribute decision making a simulation comparison of select methods abstract several methods have been proposed for solving multi attribute decision making problems madm a major criticism of madm is that different techniques may yield different results when applied to the same problem the problem considered in this study consists of a decision matrix input of n criteria weights and ratings of l alternatives on each criterion the comparative performance of some methods has been investigated in a few mostly field studies in this simulation experiment we investigate the performance of eight methods electre topsis multiplicative exponential weighting mew simple additive weighting saw and four versions of ahp original vs geometric scale and right eigenvector vs mean transformation solution simulation parameters are the number of alternatives criteria and their distribution the solutions are analyzed using twelve measures of similarity of performance similarities and differences in the behavior of these methods are investigated dissimilarities in weights produced by these methods become stronger in problems with few alternatives however the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives although less significant the distribution of criterion weights affects the methods differently in general all ahp versions behave similarly and closer to saw than the other methods electre is the least similar to saw except for closer matching the top ranked alternative followed by mew topsis behaves closer to ahp and differently from electre and mew except for problems with few criteria a similar rank reversal experiment produced the following performance order of methods saw and mew best followed by topsis ahps and electre it should be noted that the electre version used was adapted to the common madm problem and therefore it did not take advantage of the method s capabilities in handling problems with ordinal or imprecise information

modelling pairwise comparisons on ratio scales abstract many real world decision problems have multiple criteria on qualitative domains given the expectation that more and more of such problems will be modelled as mcdm problems involving scoring on subjective qualitative domains then there is a class of significant problems for which the evaluator will require an evaluation framework which handles occurrences of seeming intransitivity and inconsistency these phenomena are likely to appear in qualitative subjective domains where the decision making environment is characterized by ambiguity and vagueness in this paper we present a model for exploring the implications of pairwise comparisons on ratio scales this model provides a mathematical explanation for the phenomena of intransitivity and inconsistency that sometimes appear in situations involving scoring on a subjective qualitative domain using this framework we develop methods for generating consistent intervals on ratio scales we also present a two stage action learning process for generating consistent points on ratio scales that provides both behavioral and mathematical convergence

convergence properties and practical estimation of the probability of rank reversal in pairwise comparisons for multi criteria decision making problems abstract in this paper we address the impact of uncertainty introduced when the experts complete pairwise comparison matrices in the context of multi criteria decision making we first discuss how uncertainty can be quantified and modeled and then show how the probability of rank reversal scales with the number of experts we consider the impact of various aspects which may affect the estimation of probability of rank reversal in the context of pairwise comparisons such as the uncertainty level alternative preference scales and different weight estimation methods we also consider the case where the comparisons are carried out in a fuzzy manner it is shown that in most circumstances augmenting the size of the expert group beyond 15 produces a small change in the probability of rank reversal we next address the issue of how this probability can be estimated in practice from information gathered simply from the comparison matrices of a single expert group we propose and validate a scheme which yields an estimate for the probability of rank reversal and test the applicability of this scheme under various conditions the framework discussed in the paper can allow decision makers to correctly choose the number of experts participating in a pairwise comparison and obtain an estimate of the credibility of the outcome

distance based and ad hoc consensus models in ordinal preference ranking abstract this paper examines the problem of aggregating ordinal preferences on a set of alternatives into a consensus this problem has been the subject of study for more than two centuries and many procedures have been developed to create a compromise or consensus we examine a variety of structures for preference specification and in each case review the related models for deriving a consensus two classes of consensus models are discussed namely ad hoc methods evolving primarily from parliamentary settings over the past 200 years and distance or axiomatic based methods we demonstrate the levels of complexity of the various distance based models by presenting the related mathematical programming formulations for them we also present conditions for equivalence that is for yielding the same consensus ranking for some of the methods finally we discuss various extensions of the basic ordinal ranking structures paying specific attention to partial ranking voting member weighted consensus ranking with intensity of preference and rank correlation methods as alternative approaches to deriving a consensus suggestions for future research directions are given

equality of decomposable symmetrized tensors and matrix groups let g be a subgroup of the full symmetric group sn and a character of g a matrix can be defined as an n n matrix b which satisfies dgx bx dgx x for every n n matrix x they form a multiplicative group denoted s g x which plays a fundamental role in the study of equality of two decomposable symmetrized tensors the main result of this paper theorems 2 2 2 3 and 2 4 is a complete description of the matrices in s g x this description has many consequences that we present there are also results on related questions

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 process for generating quantitative belief functions abstract we present a structured methodology for transforming qualitative preference relationships among propositions into appropriate numeric representations this approach will be useful in the difficult process of knowledge acquisition from experts on the degree of belief in various propositions or the probability of the truthfulness of those propositions the approach implicitly through the qualitative assignments and explicitly through the vague interval pairwise comparisons provides for different levels of preference relationships among its advantages it permits the expert to explore the given problem situation using linguistic quantifiers avoid the premature use of numeric measures and identify input data that are inconsistent with the theory of belief functions

tridiagonal pairs of q racah type abstract let f denote an algebraically closed field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that a w w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d for 0 i d let i resp i denote the eigenvalue of a resp a associated with v i resp v i the pair a a is said to have q racah type whenever i a b q 2 i d c q d 2 i and i a b q 2 i d c q d 2 i for 0 i d where q a b c a b c are scalars in f with q b c b c nonzero and q 2 1 1 this type is the most general one we classify up to isomorphism the tridiagonal pairs over f that have q racah type our proof involves the representation theory of the quantum affine algebra u q sl 2

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

using box indices in supporting comparison in multiobjective optimization because of the conflicting nature of criteria or objectives solving a multiobjective optimization problem typically requires interaction with a decision maker who can specify preference information related to the objectives in the problem in question due to the difficulties of dealing with multiple objectives the way information is presented plays a very important role questions posed to the decision maker must be simple enough and information shown must be easy to understand for this purpose visualization and graphical representations can be useful and constitute one of the main tools used in the literature in this paper we propose to use box indices to represent information related to different solution alternatives of multiobjective optimization problems involving at least three objectives box indices are an intelligible and easy to handle way to represent data they are based on evaluating the solutions in a natural and rough enough scale in order to let the decision maker easily recognize the main characteristics of a solution at a glance and to facilitate comparison of two or more solutions in an easily understandable way

interactive group decision making procedure under incomplete information this paper deals with multiple criteria decision making problem with incomplete information when multiple decision makers multiple criteria group decision making mcgdm are involved usually decision makers dms are willing or able to provide only incomplete information because of time pressure lack of knowledge or data and their limited expertise related with problem domain there have been just a few studies considering incomplete information in group settings this incompletely specified information constructs region of linear constraints and therefore pairwise dominance relationship between alternatives reduces to intractable nonlinear programmings hence to handle this difficulty we suggest a method utilizing individual decision results to form group consensus final group consensus ranking toward more agreement of participants can be built through solving a series of linear programmings using individual decision results under group members possibly different weight constraints

an action learning evaluation procedure for multiple criteria decision making problems abstract in this paper we focus on an extension of the analytic hierarchy process ahp that accommodates ambiguity on the part of the decision maker dm and facilitates the exploration of the decision domain we propose a systematic action learning process that builds confidence as it converges from numeric interval estimates to numeric point estimates our multiple criteria decision making mcdm problem procedure structures the problem as a hierarchy evaluates all objects using pairwise comparisons that accommodate vagueness and ambiguity uses interval prioritization techniques and does synthesis using the linear additive value function this action learning process facilitates the understanding of key stakeholders which is imperative for the successful implementation of the subsequent decision

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

probabilistic judgments specified partially in the analytic hierarchy process the analytic hierarchy process ahp is a decision making tool which yields priorities for decision alternatives this paper proposes a new approach to elicit and synthesize expert assessments for the group decision process in the ahp these new elicitations are given as partial probabilistic specifications of the entries of pairwise comparisons matrices for a particular entry of the matrix the partial probabilistic elicitations could arise in the form of either probability assignments regarding the chance of that entry falling in specified intervals or selected quantiles for that entry a new class of models is introduced to provide methods for processing this partial probabilistic information one advantage of this approach is that it allows to generate as many pairwise comparison matrices of the decision alternatives as one desires this in turn allows us to determine the statistical significance of the priorities of decision alternatives

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

establishing dominance between alternatives with incomplete information in a hierarchically structured attribute tree abstract a model for establishing dominance with incomplete information is proposed in a hierarchically structured attribute tree the linear additive value function under certainty is used in the model the model allows the decision maker to provide any type of linear partial information on the attribute weights and the alternative values the incompletely specified information constructs a feasible region of linear constraints and therefore pairwise dominance relationship between alternatives reduces to intractable non linear programming hence to handle this difficulty we propose a solution technique also to handle the tree structure we break down the attribute tree into sub trees due to the recursive structure of the solution technique the optimization results from sub trees can be utilized in computing the value interval on the topmost attribute based on the solution technique an algorithm for obtaining the value interval is proposed and dominance relation is established

pairwise comparison methods in multiple objective programming with applications in a long term energy planning model abstract we propose to use pairwise comparisons within the framework of ideal point or reference point methods for multi objective programming the decision makers are requested to estimate the ratios which are acceptable for deviations from the ideal vector thereafter we seek the nearest feasible solution using the tchebycheff norm in this paper we sketch the pairwise comparison methods some aspects of magnitude scaling and the ideal point methods we show the results of our numerical experiments in long term energy planning with nine objective functions finally we present a preliminary evaluation of the combined method on the basis of its possible contribution to interactive decision analysis

ranking in tournaments and group decisionmaking this paper fully discusses methods for ranking a set of alternatives in the fairest possible way according to a minimum violations criterion new methods based on finding paths and circuits in graphs are presented for ranking participants in round robin and generalized tournaments and for consensus and group decisionmaking problems the objective of the paper is to review existing methods for tackling these problems and compare them with the new methods according to a fairness criterion and the amount of computing required to reach a solution it is shown that the new methods often exceed the existing methods in both fairness and reduced computing requirements in particular the new methods are generally more versatile than existing methods this allows organizations to obtain the fairest ranking of a number of alternatives according to their managers or employees wishes particular attention is given to incomplete rankings where insufficient exposure of an individual to some alternatives restricts that individual to ranking only the remaining alternatives

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

distributional efficiency in multiobjective stochastic linear programming several concepts of distributional efficiency are proposed for the multiobjective stochastic linear programming mslp problem in contexts where the probability distribution of random parameters is known and the decision maker dm has an unknown multi attribute utility function belonging to a given glass u we present a general efficient set the u admissible solutions and two subsets the u unanimous and u advocated solutions the latter being particularly relevant to the case of a single dm we show how advocated solutions can be generated and or tested when u is the class of non decreasing additive concave functions

statistical robustness in utility preference robust optimization models utility preference robust optimization pro concerns decision making problems where information on decision maker s utility preference is incomplete and has to be elicited through partial information and the optimal decision is based on the worst case utility function elicited a key assumption in the pro models is that the true probability distribution is either known or can be recovered by real data generated by the true distribution in data driven optimization this assumption may not be satisfied when perceived data differ from real data and consequently it raises a question as to whether statistical estimators of the pro models based on perceived data are reliable in this paper we investigate the issue which is also known as qualitative robustness in the literature of statistics huber in robust statistics 3rd edn wiley new york 1981 and risk management kratschmer et al in finance stoch 18 271 295 2014 by utilizing the framework proposed by kratschmer et al 2014 we derive moderate sufficient conditions under which the optimal value and optimal solution of the pro models are robust against perturbation of the exogenous uncertainty data and examine how the tail behaviour of utility functions affects the robustness moreover under some additional conditions on the lipschitz continuity of the underlying functions with respect to random data we establish quantitative robustness of the statistical estimators under the kantorovich metric finally we investigate uniform consistency of the optimal value and optimal solution of the pro models the results cover utility selection problems and stochastic optimization problems as special cases

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

stochastic dominance tests for ranking alternatives under ambiguity abstract ambiguity in decision making is a condition associated with the existence of one to many or more generally many to many relations between outcomes and probabilities of outcomes the current use of stochastic dominance tests to rank decision alternatives assumes that outcome probability distributions are unambiguous this paper first establishes procedures to identify an envelope of the ambiguously described probability outcome mapping then extends stochastic dominance theorems to provide tests that partition alternatives into dominated and non dominated sets

criteria importance theory for multicriterial decision making problems with a hierarchical structure abstract we develop a new model of multicriterial decision analysis with a hierarchical criteria structure the suggested methodology is based on a rigorously defined specification of qualitative and quantitative importance of criteria of all levels of the hierarchy and enables the comparison of decision alternatives for different types of criteria scale

ranking and selection for multiple performance measures using incomplete preference information abstract this paper presents two new procedures for ranking and selection r s problems where the best system designs are selected from a set of competing ones based on multiple performance measures evaluated through stochastic simulation in the procedures the performance measures are aggregated with a multi attribute utility function and incomplete preference information regarding the weights that reflect the relative importance of the measures is taken into account a set of feasible weights is determined according to preference statements that are linear constraints on the weights given by a decision maker non dominated designs are selected using two dominance relations referred to as pairwise and absolute dominance based on estimates for the expected utilities of the designs over the feasible weights the procedures allocate a limited number of simulation replications among the designs such that the probabilities of correctly selecting the pairwise and absolutely non dominated designs are maximized the new procedures offer ease of eliciting the weights compared with existing r s procedures that aggregate the performance measures using unique weights moreover computational advantages are provided over existing procedures that identify non dominated designs based on the expected values of the performance measures the new procedures allow to obtain a smaller number of non dominated designs they also identify these designs correctly with a higher probability or require a smaller number of replications for correct selection finally the new procedures allocate a larger number of replications to the non dominated designs that are therefore evaluated with greater accuracy these computational advantages are illustrated through several numerical experiments

establishing strict dominance between alternatives with special type of incomplete information abstract a special model for establishing dominance with decision maker s incomplete information is proposed in multi attribute decision problem under certainty incomplete information is a set of linear inequalities about utilities as well as attribute weights since a general model presented in a prior work uses a linear programming technique a number of linear programs should be solved for checking dominance with the proposed method we can check strict dominance between alternatives by simple matrix operation without solving linear programs thus reducing computational time

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

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

capturing group preferences in a multicriteria decision abstract this paper examines the extent to which methods used for analyzing judgments from a group of decision makers result in preference rankings which are consistent with the group member s subjective beliefs about their preferences in a laboratory group decision setting the analytic hierarchy process a newly developed probabilistic extension to the ahp and two pairwise voting methods are used to develop separate preference rankings based on input from the group members the probabilistic version of the ahp in its first independent test is found to provide more insight into the group decision while requiring fewer a priori assumptions a commonly used and easily implemented pairwise voting procedure is found to be significantly inferior to the other methods in its ability to capture the group s preference rankings this underscores the importance of using appropriate decision models for developing a full understanding of group preferences

computational experience concerning payoff tables and minimum criterion values over the efficient set abstract minimum criterion values from payoff tables have often been used in multiple objective linear programming molp the assumption has often been that the minimum criterion values from payoff tables provide reasonably accurate estimates of the minimum criterion values over the efficient set in this paper however we report computational experience that demonstrates that the discrepancies between the payoff table minimums and the minimums over the efficient set can often be large this tends to imply that the field of multiple objective programming needs a better method than payoff tables for estimating the minimum criterion values over the efficient set the paper concludes with a discussion of a simplex based procedure for deterministically computing the minimum criterion values over the efficient set that has potential in large molp applications

an approach to using the analytic hierarchy process for solving multiple criteria decision making problems abstract in this paper we present a procedure for solving multiple criteria decision making problems our procedure which is an extension of the analytic hierarchy process addresses the issue of fuzziness in the weighing information provided by the decision maker the consequent stability of the ahp ranking of the alternatives and the sensitivity of such rankings to small perturbations these issues are especially important when the concept of optimality incorporates parameters or weights that reflect subjective preferences the procedure provides a consistent and systematic method for carrying out goal seeking sensitivity analysis that captures the decision maker s preference structure using his her indifference region

thin hessenberg pairs and double vandermonde matrices abstract a square matrix is called hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero let v denote a nonzero finite dimensional vector space over a field k we consider an ordered pair of linear transformations a v v and a v v which satisfy both i ii below i there exists a basis for v with respect to which the matrix representing a is hessenberg and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is hessenberg we call such a pair a thin hessenberg pair or th pair by the diameter of the pair we mean the dimension of v minus one there is an oriented version of a th pair called a th system in this paper we investigate a connection between th systems and double vandermonde matrices we have two main results for the first result we give a bijection between any two of the following three sets the set of isomorphism classes of th systems over k of diameter d the set of normalized west south vandermonde systems in mat d 1 k the set of parameter arrays over k of diameter d for the second result we give a bijection between any two of the following five sets the set of affine isomorphism classes of th systems over k of diameter d the set of isomorphism classes of rth systems over k of diameter d the set of affine classes of normalized west south vandermonde systems in mat d 1 k the set of normalized west south vandermonde matrices in mat d 1 k the set of reduced parameter arrays over k of diameter d

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

an aggregation disaggregation approach to obtain robust conclusions with electre tri electre tri is a well known method to assign a set of alternatives to a set of predefined categories considering multiple criteria using this method requires setting many parameters which is often a difficult task we consider the case where the decision makers dms in the decision process are unsure of which values should each parameter take which may result from uncertain imprecise or inaccurately determined information as well as from lack of consensus among them this paper discusses the synergy between two approaches developed independently to deal with this difficulty the first approach infers the value of parameters from assignment examples provided by the dms as an elicitation aid each assignment example originates mathematical constraints that the parameter values should satisfy the second approach considers a set of constraints on the parameter values reflecting the imprecise information that the dms are able to provide then it computes the best and worst categories for each alternative compatible with constraints in order to present robust conclusions both approaches avoid asking for precise values for the parameters rather they proceed to solve the problem in a way that requires from the dms much less effort by integrating these two approaches this paper proposes a new interactive approach where the insight obtained during robustness analyses guides the dms during the elicitation phase

defining managerial problems a framework for prescriptive theorizing reflective problem solving activity presumes the existence of a problem definition or representation often the precursor of a formal model although problem definition has been addressed by theorists from several disciplines it has not been adequately conceptualized limiting the potential value of descriptive and prescriptive research this article places problem definition in the overall problem solving process and delineates the various ways in which problems are commonly defined it identifies the central dilemma of problem definition and proposes a prescriptive framework responding to that dilemma

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

multi attribute decision aid under incomplete information and hierarchical structure abstract this paper presents methods for dealing with incomplete information about both attribute weights and values under a hierarchically structured value tree incomplete information in this paper covers arbitrary linear inequalities and is hence to treat a more general situation than a previous restrictive definition of incomplete information that typically includes interval judgment this may give decision makers chances that is enhanced freedom of choice and comforts of specification we propose two techniques for prioritizing alternatives by strict dominance relationship one is the extension of a previous method for operating flat structured value trees to hierarchical ones the other approach propagates pairwise dominance values from leaf nodes to topmost node consecutively which is also an extension of a previous method because the strict dominance rule fails to fully prioritize alternatives as is usual the case under incomplete information we suggest a new method a measure of preference strength which can provide decision makers with a single optimal alternative or full rank of alternatives without any further interaction with decision makers

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

sources of bias in assessment procedures for utility functions utility functions are an important component of normative decision analysis in that they characterize the nature of people s risk taking attitudes in this paper we examine various factors that make it difficult to speak of the utility function for a given person similarly we show that it is questionable to pool risk propensity data across studies for descriptive purposes that differ in the elicitation methods employed the following five sources of bias or indeterminacy are hypothesized and demonstrated first certainty equivalence methods generally yield greater risk seeking than probability equivalence methods second the probability and outcome levels used in reference lotteries induce systematic bias third combining gain and loss domains yields different utility measures than separate examinations of the two domains fourth whether a risk is assumed or transferred away exerts a significant influence on people s preferences in ways counter to expected utility theory finally context or frami

computation of ideal and nadir values and implications for their use in mcdm methods abstract in this paper we investigate the problem of finding the nadir point for multicriteria optimization problems mop the nadir point is characterized by the componentwise maximal values of efficient points for mop it can be easily computed in the bicriteria case however in general this problem is very difficult we review some existing methods and heuristics and also discuss some new ones we propose a general method to compute nadir values based on theoretical results on pareto optimal solutions of subproblems with fewer criteria the general scheme is valid for any number of criteria and practical for the case of three objectives we also investigate the use of the nadir point for compromise programming when the goal is to be as far away as possible from the worst outcomes we prove some results about weak pareto optimality of the resulting solutions and present examples which show the limitations of this idea and therefore limitations of multicriteria methods using this type of problems

preference convex unanimity in multiple criteria decision making the approach which results in the definition of stochastic dominance orders for risky decision making can also be applied to the nonrisky case that is one may postulate that alternative x dominates alternative y if x is preferred to y by every possible decision maker with specified preference characteristics one such unanimity order is the standard componentwise vector order of multiple objective optimization in this paper a unanimity order is constructed based upon characteristics of preference convexity in conjunction with finitely many elicited preference responses since the new order properly includes the vector order the corresponding efficient frontier may be smaller

extreme point based multi attribute decision analysis with incomplete information abstract in this paper we present a simple method for finding the extreme points of various types of incomplete attribute weights incomplete information about attribute weights is transformed by a sequence of change of variables to a set whose extreme points are readily found this enhanced method fails to derive the extreme points of every type of incomplete attribute weights nevertheless it provides us with a flexible method for finding the extreme points including widely used forms of incomplete attribute weights finally incomplete attribute values expressed in various forms are also analyzed to find their characterizing extreme points by applying similar procedures carried out in the incomplete attribute weights

constraint based optimization and utility elicitation using the minimax decision criterion in many situations a set of hard constraints encodes the feasible configurations of some system or product over which multiple users have distinct preferences however making suitable decisions requires that the preferences of a specific user for different configurations be articulated or elicited something generally acknowledged to be onerous we address two problems associated with preference elicitation computing a best feasible solution when the user s utilities are imprecisely specified and developing useful elicitation procedures that reduce utility uncertainty with minimal user interaction to a point where approximately optimal decisions can be made our main contributions are threefold first we propose the use of minimax regret as a suitable decision criterion for decision making in the presence of such utility function uncertainty second we devise several different procedures all relying on mixed integer linear programs that can be used to compute minimax regret and regret optimizing solutions effectively in particular our methods exploit generalized additive structure in a user s utility function to ensure tractable computation third we propose various elicitation methods that can be used to refine utility uncertainty in such a way as to quickly i e with as few questions as possible reduce minimax regret empirical study suggests that several of these methods are quite successful in minimizing the number of user queries while remaining computationally practical so as to admit real time user interaction

stochastic dominance with incomplete information on probabilities this paper discusses different possibilities to check stochastic dominance if the information on probabilities is incomplete we show that risk profiles can serve as a useful tool to understand stochastic dominance in addition we extend the approach recently proposed by pearman and kmietowicz 1986 to test for stochastic dominance with incomplete information on probabilities

the shape of some linear transformations abstract recently ito and terwilliger proposed a problem about some linear transformations a a a a on a finite dimensional vector space v over a field f which can be seen as a generalization of the notions of tridiagonal pair and q inverting pair see problem 1 2 in this paper we will introduce the notion of the shape of the linear transformations a a a a our main results are the properties of the shape i the shape is symmetric and unimodal ii let denote an anti isomorphism from end v to end v with v a vector space over f with the same dimension as v assume that the linear transformations a a a a on v satisfy all conditions of problem 1 2 then the linear transformations a a a a also satisfy all conditions of problem 1 2 moreover a a a a have the same shape as a a a a

generalized golden sections repeated bisections and aesthetic pleasure abstract we generalize the golden section in the sense that we partition a given line segment into a finite sequence of diminishing subsegments in such a way that the ratio of any pair of adjacent subsegments equals the ratio of the whole and the first subsegment when the number of subsegments increases the partitioning approximates the subdivision of the line segment via repeated bisections this is the basis of the geometric scales in multi criteria decision analysis introduced in order to model the gradations of comparative preferential judgement since the golden section plays an important role in the analysis of beauty and harmony we suppose that it is an aesthetic pleasure to observe the resulting pattern of preference modelling

improving the consistency of conditional probability assessments for forecasting and decision making public agencies are very keen on amassing statistics they collect them add them raise them to the nth power take the cube root and prepare wonderful diagrams but what you must never forget is that every one of those figures comes in the first instance from the village watchman who just puts down what he damn pleases sir josiah stamp the assessment of the conditional probabilities of events is useful and needed for forecasting planning and decision making in this paper the difficulties associated with the assessment of these conditional probabilities are examined the necessary and sufficient conditions that the elicited information on conditional probabilities must satisfy are evaluated against actual assessments in several different controlled settings a high frequency of implicit violations of the probability calculus was observed the consistency of the assessments is affected by the causal diagnostic and positive negative relationships of the events use of a judgmental aid in the form of a joint probability table reduces the number of inconsistent responses significantly using the probability axioms it is also shown that only the first order conditional probabilities need be assessed as higher order probabilities are robust to the unconditional and first order conditional assessments

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

thin hessenberg pairs abstract a square matrix is called hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero let v denote a nonzero finite dimensional vector space over a field k we consider an ordered pair of linear transformations a v v and a v v which satisfy both i and ii below i there exists a basis for v with respect to which the matrix representing a is hessenberg and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is hessenberg we call such a pair a thin hessenberg pair or th pair this is a special case of a hessenberg pair which was introduced by the author in an earlier paper we investigate several bases for v with respect to which the matrices representing a and a are attractive we display these matrices along with the transition matrices relating the bases we introduce an oriented version of a a called a th system we classify the th systems up to isomorphism

decision making under uncertainty with unknown utility function and rank ordered probabilities abstract we consider the ranking of decision alternatives in decision analysis problems under uncertainty under very weak assumptions about the type of utility function and information about the probabilities of the states of nature namely the following two assumptions are required for the suggested method the utility function is in the class of increasing continuous functions and the probabilities of the states of nature are rank ordered we develop a simple analytical method for the partial ranking of decision alternatives under the stated assumptions this method does not require solving optimization programs and is free of the rounding errors

the aspiration level interactive method aim reconsidered robustness of solutions abstract in this paper we explore theory and practice for the aspiration level interactive model aim a useful decision tool that takes advantage of the concepts of satisficing as well as other concepts of multiple criteria decision making mcdm we examine the relationship between aspiration levels and their mapped to solutions in the mcdm context using aim we extend the concept of robustness in decision making by defining a solution to be robust if many suitably defined aspiration levels map to it we use simulation to help explore robustness by generating three groups of random test problems each problem has twenty alternatives for each group of problems the top 5 most mapped to alternatives out of 20 are mapped to by at least 50 of the aspiration levels we also relate the concept of robustness in decision making to the ideas of simply ranking alternatives using equal weights there is a strong correlation between the robustness ranking and the equal weights ranking based on our analyses we then randomly generate additional problems to explore certain other factors we also discuss practical aspects of robustness

a heuristic method to rectify intransitive judgments in pairwise comparison matrices abstract this paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a monte carlo simulation which confirms that relatively high percentage of comparison matrices satisfying saaty s cr criterion are ordinally inconsistent it is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process in contrast with cardinal inconsistency a heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices the proposed algorithm generates near optimal solutions and outperforms other tested approaches with respect to computation time

the structure of some linear transformations abstract let f denote an algebraically closed field and let v denote a finite dimensional vector space over f recently ito and terwilliger considered a system of linear transformations a a a a on v which generalizes the notions of a tridiagonal pair and a q inverting pair in their paper they mentioned some open problems about this system in this paper we solve problem 1 2 with the following results let v i i 0 d denote the common eigenspaces of a a and let v i i 0 d denote the common eigenspaces of a a we show that each of a a a a is determined up to affine transformation by the sequences v i i 0 d v i i 0 d we also show that the following are equivalent i there exists a nonzero bilinear form on v such that a u v u a v and a u v u a v for all u v v ii there exist scalars in f with nonzero such that a a i and a a i and iii both a a and a a are tridiagonal pairs

using fuzzy sets in operational research abstract during the last four decades a large number of mainly mathematical tools have been developed in and for operational research they are primarily devices to find optimal solutions after a problem has been modelled formally hardly any new tools have been developed for proper modelling of problems which are not well structured and which can not easily be cast into classical mathematical models fuzzy set theory as proposed by l zadeh in 1965 promises to bridge part of this gap it is much more adaptable to different problem structures and better suited to model human evaluation and decision making processes than classical mathematics this paper will primarily focus attention on four questions why fuzzy sets in or what are fuzzy sets where can they be used and how are they being used

some issues in measuring and reporting solution quality of interactive multiple objective programming procedures issues in measuring and reporting solution quality are examined when value functions are used in computational experiments of interactive multiple objective programming procedures they include value functions used weights assigned to the objective functions in the value functions the size of the efficient set and the number of objective functions the feasibility and existence of the ideal and nadir points are also discussed detailed examples are presented to demonstrate these issues neither the users nor in fact the researchers may discern these issues even though they have strong impacts on the reported solution qualities common practices in the computational experiments of interactive multiple objective programming procedures are reviewed

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

generalization of the rcgm and lslrpairwise comparison methods pairwise comparison methods are convenient procedures for predicting a sound weight vector from a set of relative comparisons between elements to be weighted several pairwise comparison methods exist after a brief presentation of the least squares logarithmic regression lslr method of de graan 1 and lootsma 2 and the recent row and column geometric mean rcgm of koczkodaj and orlowski 3 this paper proposes a common mathematical formulation for these two approaches this common formulation leads to two generalized methods the glslr is now able to process nonreciprocal comparison matrices and the grcgm is extended to several decision makers expressing different opinions per pairwise comparison it also results in an explicit formulation of the weights that generalizes koczkodaj and orlowski s formulation of the closest consistent comparison matrix

theoretical estimation of the probability of weight rank reversal in pairwise comparisons abstract pairwise comparison is a key component in multi criteria decision making the probability of rank reversal is a useful measure for evaluating the impact of uncertainty on the final outcome in the context of this paper the type of uncertainty considered is related to the fact that experts have different opinions or that they may perform inconsistent pairwise comparisons we provide a theoretical model for estimating the probability of the consequent rank reversal using the multivariate normal cumulative distribution function the model is applied to two alternative weight extraction methods frequently used in the literature the geometric mean and the eigenvalue method we introduce a reasonable framework for incorporating uncertainty in the decision making process and calculate the mean value and cross correlation of the average weights which are required in the application of the model the theoretical results are compared against numerical simulations and a very good agreement is observed we further show how our model can be extended in applications of a full multi criteria decision making analysis such as the analytic hierarchy process we also discuss how the theoretical model can be used in practice where the statistical properties of the uncertainty induced perturbations are unknown and the only information provided by the pairwise comparison matrices of a small group of experts the methodology presented here can be used to extend the pairwise comparison framework in order to provide some information on the credibility of its outcome

two linear transformations each tridiagonal with respect to an eigenbasis of the other comments on the split decomposition let k denote a field and let v denote a vector space over k with finite positive dimension we consider an ordered pair of linear transformations a v v and a v v that satisfy both conditions below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v referring to the above leonard pair it is known there exists a decomposition of v into a direct sum of one dimensional subspaces on which a acts in a lower bidiagonal fashion and a acts in an upper bidiagonal fashion this is called the split decomposition in this paper we give two characterizations of a leonard pair that involve the split decomposition

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

pairs of matrices that preserve the value of a generalized matrix function on the set of the upper triangular matrices let h be a subgroup of the symmetric group of degree m and let be an irreducible character of h in this paper we give conditions that characterize the pairs of matrices that leave invariant the value of a generalized matrix function associated with h and on the set of the upper triangular matrices

rank based statistical procedures in analytic hierarchy process rank based procedures which do not require any assumption on statistical distributions of pairwise comparisons entries are proposed for a few relevant questions arising in the context of analytic hierarchy process one of them relates to the issues of finding out whether various groups of individuals are alike in judgments or not in case not all groups are alike ways are suggested to form cluster of homogeneous groups another concern is that of establishing a particular order of preferences for alternatives in a homogeneous group of individuals

shortest path problems with partial information models and algorithms for detecting dominance abstract in this work we focus on partial information models for the well known shortest path problem where we consider multiple instances of values for the parameters that determine the cost of each arc this allows coping with the uncertainty about the future the imprecision of data the arbitrariness of some options the evolving values of the decision makers dms and or the multiplicity of dms in group decision making this paper proves some results and presents detailed algorithms to identify the set of non dominated paths a concept from decision theory under partial information we first address problems with a finite set of instances then problems with a general eventually not discrete set of instances and finally we study a particular case of the latter which complies with a condition that may hold in some situations to deal with these partial information problems we propose a new use for existing multicriteria algorithms based on the ranking of shortest paths

matrices that preserve the value of the generalized matrix function of the upper triangular matrices abstract let h be a subgroup of the symmetric group s n and an irreducible character of h in this paper we give conditions that characterize matrices that leave invariant the value of a given generalized matrix function on the set of upper triangular matrices in some cases we describe completely these matrices

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

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

decision analysis with incomplete utility and probability information an approach to the solution of decision analysis problems under uncertainty with imprecise and incomplete information is presented the methodology is designed for cases in which payoffs conditional on the state of nature are known precisely but only limited or imprecise probability and utility information is available regarding a decision maker s beliefs and tastes a decision maker provides conditional payoffs optionally bounds on state probabilities bounds on the certainty equivalent for a simple lottery any known relationships between probabilities of states of nature and a series of strict preferences between pairs of vectors of conditional payoffs we assume an exponential utility function with unknown parameter the method proceeds by sequentially eliciting preferences new bounds on probabilities and or the certainty equivalent and new relationships among probabilities until the problem is solved the methodology is demonstrated through a two state and a three state example which illust

an analysis of distributions of priority values from alternative comparison scales within ahp the analytic hierarchy process ahp introduced by t l saaty is a well known and popular method of multi criteria decision making central to this method are the pairwise comparisons between criteria and decision alternatives made using a 9 unit scale the appropriateness of saaty s original one to nine 1 9 scale has been the subject of much debate and cause for concern this paper contrasts the appropriateness of the 1 9 scale with other alternative 9 unit scales also used in ahp by looking at the probability distributions of the associated priority values for large problems estimated probability distributions are found for the priority values through using the method of parzen windows

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

an elementary proof of paelinck s theorem on the convex hull of ranked criterion weights abstract paelinck s theorem provides a simple characterization of the set of possible criterion weights when only the rank order of the criteria is known methods for multi criteria analysis based on this theorem are sometimes able to identify a preferred alternative without numerical estimates of these weights we provide a new elementary proof of the theorem