Showing papers on "Pairwise comparison published in 1986"
TL;DR: In this paper, the authors show that the uncovered set is a subset of the covered set, and that it is a better choice correspondence than the Copeland set and the sophisticated agenda algorithm.
Abstract: A tournament is any complete asymmetric relation over a finite set A of outcomes describing pairwise comparisons. A choice correspondence assigns to every tournament on A a subset of winners. Miller's uncovered set is an example for which we propose an axiomatic characterization. The set of Copeland winners (outcomes with maximal scores) is another example; it is a subset of the uncovered set: we note that it can be a dominated subset. A third example is derived from the sophisticated agenda algorithm; we argue that it is a better choice correspondence than the Copeland set.
247 citations
TL;DR: In this article, an axiomatic approach is shown for deciding which method is the best one for priority vectors from matrices of pairwise comparisons, based on the geometric row means.
Abstract: Amongst the mathematical questions related to the AHP's theory the one of how to derive priority vectors from matrices of pairwise comparisons has found an especially large interest. Following a general discussion of scales for priorities and their evaluation by pairwise comparisons several concrete proposals are reviewed. An axiomatic approach is shown for deciding which method is the “best” one. Invariance principles are motivated and formulated as axioms. The only method which fulfills all these axioms uses the geometric row means. It is often called Logarithmic Least Squares Method (LLSM). However, only one axiom would have to be replaced in order to get the widely used Right Eigenvector Method.
150 citations
TL;DR: It is 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.
Abstract: 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.
109 citations
TL;DR: In this article, a method for solving discrete choice problems characterized by multiple evaluative criteria is presented, known as MAPPAC, which is based on a pairwise comparison of alternatives relative to each pair of criteria, defining the two relations P (preference) and I (indifference), which constitute a complete preorder.
62 citations
TL;DR: In this paper, probabilistic ideal point and vector approaches are used to handle inter-and intra-individual irregularities in paired comparisons preference data and maximum likelihood estimates of the model parameters are computed.
Abstract: The aim of this paper is to add results to the everlasting attempt to find appropriate models for the description and analysis of choice behaviour. As stochastic generalizations within choice models which allow estimation and testing in a straightforward way should be of great interest, probabilistic ideal point and vector approaches are used to handle inter- and intra-individual irregularities in paired comparisons preference data. For choice behaviour analysis maximum likelihood estimates of the model parameters are computed and possibilities of testing various model variants by means of log-likelihood ratio tests are discussed. The proposed models offer information to decide whether an ideal point or a vector approach is more appropriate for the description of the pairwise choice data. Examples of data sets known from previous research in choice behaviour are used for the comparison and for the demonstration of the merits of the proposed procedures.
37 citations
TL;DR: In this paper, it is shown that one cannot simply take columns of numbers, normalize them and add to obtain results corresponding to operations in the AHP, and care needs to be exercised.
Abstract: It is shown here that one cannot simply take columns of numbers, normalize them and add to obtain results corresponding to operations in the AHP. This is what traditional expected value theory using a single scale would lead one to do. Care needs to be exercised. What one must do is to interpret the data represented by each column according to relative importance to a decision maker so that the alternatives under each criterion are pairwise compared according to the fundamental scale used to represent judgments. This procedure then leads to a set of vectors which belong to the same ratio scale and they can now be combined by using the weights of the criteria.
32 citations
TL;DR: In this paper, the optimal location of public facilities under the influence of the land market is examined, which is a spatial analogy to the concept of stochastic-dominance.
32 citations
TL;DR: In this paper, the authors analyzed group decisions in terms of current models of social decision scheme (SDS) research and two new models which simulate pairwise majority voting over all alternatives, the full paired comparison and the reduced paired comparison (RPC) models.
32 citations
TL;DR: This paper shows how the consensus formation problem for strict linear orderings can be modelled as a generalized network and will permit the solution of much larger problems than has been the case previously.
Abstract: One of the best known and most widely referenced models for representing ordinal preferences is that due to Kemeny and Snell Kemeny, J. G., L. J. Snell. 1962. Preference ranking: an axiomatic approach. Mathematical Models in the Social Sciences. Glnn, New York, 9-23.. This model is designed to accommodate pairwise comparison data with an l1 norm used to measure voter disagreement. While this model possesses many of the necessary properties for a social choice function, solution procedures developed to date have been capable of handling only small problems due to the difficulty of modelling the transitivity requirements of an optimal consensus ranking. This paper shows how the consensus formation problem for strict linear orderings can be modelled as a generalized network. Since efficient computer codes already exist for handling this special structure, this approach will permit the solution of much larger problems than has been the case previously.
21 citations
TL;DR: In this paper, the authors present a method which avoids the uniqueness problem encountered in the construction of utility functions when using either the certainty equivalence method or the probability equivalence methods, and they show that under the assumption of cardinal consistency utility functions are a particular case of these ratio scales.
Abstract: The criticisms of Utility Theory focus on either its axioms or the construction of utility functions. Here we present a method which avoids the problems of uniqueness encountered in the construction of utility functions when using either the certainty equivalence method or the probability equivalence method. The method is based on the construction of ratio scale value functions from reciprocal pairwise comparisons and Saaty's Eigenvector Method. We show that under the assumption of cardinal consistency utility functions are a particular case of these ratio scales. Reciprocal pairwise comparisons allow decision makers to relax the transitivity assumption and help to derive a unique scaling of preferences.
20 citations
01 Dec 1986
TL;DR: The results of an experiment exploring the use of linguistic scales in soliciting responses for Saaty's pairwise comparison method illustrate the usefulness of a subjective evaluation technique in structural engineering.
Abstract: This technical note reviews the application of a simple pairwise comparison method to multi-objective subjective evaluation problems and reports the results of an experiment exploring the use of linguistic scales in soliciting responses for Saaty's pairwise comparison method. The experimental results also illustrate the usefulness of a subjective evaluation technique in structural engineering.
TL;DR: The selection of pairwise MCPs for one-factor ANOVA models is considered, following a discussion of Type I error and power issues as they apply to the testing of multiple hypotheses.
01 Jan 1986
TL;DR: In this paper, the basic types of 3-way data matrices are described and several exploratory techniques of analysis are suggested, based on appropriately defined measures of relationship between individuals or between categories of the investigated characters.
Abstract: The basic types of 3-way data matrices are described. The special case of one or more sets of qualitative characters observed on one or more groups of individuals is then examined. Several exploratory techniques of analysis are suggested, based on appropriately defined measures of relationship between individuals or between categories of the investigated characters.
TL;DR: A paired comparison experiment is an extension of a binomial experiment; the treatments are compared pairwise and the response is binary, and the minimization of the variance of the maximum likelihood estimate of a single parameter and D-optimality is considered.
TL;DR: Recently, it has been proved in a number of studies as discussed by the authors that, under a proper set of assumptions, the optimal group decision rule in pairwise choice situations is a weighted majority rule, with weights that are proportional to the logarithms of the decision makers' odds of choosing the correct alternative.
Abstract: Recently it has been proved in a number of studies, that, under a proper set of assumptions, the optimal group decision rule in pairwise choice situations is a weighted majority rule, with weights that are proportional to the logarithms of the decision makers' odds of choosing the correct alternative.
01 Jan 1986
TL;DR: In this article, the authors extend previous work to define importance measures for a pair of components of the system (accident sequence, core damage frequency, or health risks as appropriate), and illustrate the usefulness of these pairwise importance measures in nuclear power plants.
Abstract: Importance measures in systems unreliability (or unavailability) analysis provide useful information in identifying components which are critical with regard to the availability or reliability of a system. Various importance measures known in the reliability literature are defined for a single component. This paper extends previous work to define importance measures for a pair of components of the system (accident sequence, core damage frequency, or health risks as appropriate), and illustrates the usefulness of these pairwise importance measures in nuclear power plants. The pairwise importance measures are immediately applicable to risk-based evaluation of the technical specifications; in addition, pairwise importances could play an important role in systems interaction studies by highlighting pairs of events between which a coupling would be significant if it existed.
TL;DR: In this paper, a three-step procedure for analyzing multi-dimensional contingency tables is introduced, where the first step is to catch the relationship structure between the variables connected by the contingency table, and the second step analyses the significant conditional pairs in regard to the question for what attribute level combinations of the condition complexes the relations remain significant.
Abstract: By the aid of analysing a medical example a three—step procedure for analysing multi—dimensional contingency tables is introduced. This procedure has some good properties.
Step one is due to catch the relationship structure between the variables connected by the contingency table. Hereby only so—called graphical models, a subclass of hierarchical models in regard of the parameters of the log—linear model, are admitted. The models can be generated by combination of hypotheses of pairwise conditional independence. Hereby a so-called Extended Combination Procedure is proposed using the position of the Chain of (hierarchical) Hypotheses. A useful symbolic notation for ‘Dependence Models’ in addition to that in form of ‘Independence Models’ and ‘Minimal Sets’ is proposed. Step two analyses the significant conditional pairs in regard to the question for what attribute level combinations of the condition complexes the relations remain significant.
Step three investigates those tables recognised as significant in step two more closely to get ideas about the ‘sources’ of dependencies and possibilities of collapsing parts of the table.
The procedure is mostly used in explorative data analysis although the simple steps can be used to test hypotheses, too.
01 Jan 1986
TL;DR: A model for the probabilistic analysis of the eigenvector sealing problem for dominance matrices of unit rank and an analytical procedure for investigating the robustness of the decision making procedure to variations in the pairwise preference distribution are presented.
Abstract: In this paper we present a model for the probabilistic analysis of the eigenvector sealing problem for dominance matrices of unit rank. We also address the problem of subjectively assessing the decision maker's pairwise preference distribution and present an analytical technique for deciding the following types of ultimate decision questions under uncertainty: (1) What is the most (least) preferred course of action? and (2) What is the preferred ordering or ranking of the available courses of action? We also provide an analytical procedure for investigating the robustness of the decision making procedure to variations in the pairwise preference distribution used to model the subjectively assessed distribution.
01 Jan 1986
TL;DR: In the context of social choice, the relaxation of the transitive rationality requirement becomes the central aspect of the social choice literature as mentioned in this paper, and it is now well established that in the absence of the condition of independence of irrelevant alternatives an aggregation procedure is vulnerable to strategic manipulation, that is, an individual by misrevealing his true or sincere preferences can influence the social outcome in his own advantage.
Abstract: Arrow’s celebrated general possibility theorem essentially shows that certain value judgments which we might find fair to incorporate in a social choice mechanism are logically inconsistent. Since the Pareto condition (which simply says that given two alternative social states, a and b, if every individual strictly prefers a to b, then they collectively also strictly prefer a to b), is ethically a compelling normative condition especially in the framework where social choice is based only on the individual’s utility information, one must look at the remaining criteria, i.e., independence of irrelevant alternatives (what Blair and Pollak in the previous chapter called pairwise determination) and transitive rationality. It is now well established that in the absence of the condition of independence of irrelevant alternatives an aggregation procedure is vulnerable to the strategic manipulation, that is, an individual by misrevealing his true or sincere preferences can influence the social outcome in his own advantage. Hence, the relaxation of the transitive rationality requirement becomes the central aspect of the social choice literature.
TL;DR: In this article, a new methodology for estimating the objective function in a multiple objective mathematical programming model is presented, where a decision maker is required to provide pairwise preferences, or rank orders, of a set of solutions to the multiple objective problem.
TL;DR: In this article, the authors describe a microcomputer program that computes all pairwise comparisons of means after analysis of covariance using the Tukey-Kramer test, and the program consists of the means of the covariate, the adjusted criterion means, the sample size, mean square error from a one-way analysis of covariances and the 95th percentile point on the Studentized range distribution.
Abstract: This paper describes a BASIC microcomputer program that computes all pairwise comparisons of means after analysis of covariance using the Tukey-Kramer test. Input to the program consist of the means of the covariate, the adjusted criterion means, the sample size, mean square error from a one-way analysis of covariance, and the 95th percentile point on the Studentized range distribution.
TL;DR: An attempt is made to formulate the optimum aseismic design using the concept of fuzzy mathematical programming, during which several sets of questionnaires were distributed among experienced engineers to find their preference in the design of earthquake-resistant structures.
Abstract: In this paper, an attempt is made to formulate the optimum aseismic design using the concept of fuzzy mathematical programming. While the fuzzy mathematical programming can be applied to optimum aseismic design, it is necessary to consider carefully the aggregation of contradictory components for practical purposes. To find an adequate way of aggregating objective functions, we conducted a survey, during which several sets of questionnaires were distributed among experienced engineers to find their preference in the design of earthquake-resistant structures. Results of these questionnaires are collected and analyzed by using the pairwise comparison method. By using several numerical examples, the present method is illustrated and compared with the previous results using the minimum operator.