Showing papers on "Pairwise comparison published in 2000"

A fuzzy programming method for deriving priorities in the analytic hierarchy process

Abstract: The estimation of the priorities from pairwise comparison matrices is the major constituent of the Analytic Hierarchy Process (AHP). The priority vector can be derived from these matrices using different techniques, as the most commonly used are the Eigenvector Method (EVM) and the Logarithmic Least Squares Method (LLSM). In this paper a new Fuzzy Programming Method (FPM) is proposed, based on geometrical representation of the prioritisation process. This method transforms the prioritisation problem into a fuzzy programming problem that can easily be solved as a standard linear programme. The FPM is compared with the main existing prioritisation methods in order to evaluate its performance. It is shown that it possesses some attractive properties and could be used as an alternative to the known prioritisation methods, especially when the preferences of the decision-maker are strongly inconsistent.

Structure Comparison and Structure Patterns

Abstract: This article investigates aspects of pairwise and multiple structure comparison, and the problem of automatically discover common patterns in a set of structures. Descriptions and representation of structures and patterns are described, as well as scoring and algorithms for comparison and discovery. A framework and nomenclature is developed for classifying different methods, and many of these are reviewed and placed into this framework.

Spatial Social Networks

Abstract: We introduce a spatial cost topology in the network formation model analyzed by Jackson and Wolinsky, Journal of Economic Theory (1996), 71: 44–74. This cost topology might represent geographical, social, or individual differences. It describes variable costs of establishing social network connections. Participants form links based on a cost-benefit analysis. We examine the pairwise stable networks within this spatial environment. Incentives vary enough to show a rich pattern of emerging behavior. We also investigate the subgame perfect implementation of pairwise stable and efficient networks. We construct a multistage extensive form game that describes the formation of links in our spatial environment. Finally, we identify the conditions under which the subgame perfect Nash equilibria of these network formation games are stable.

Mathematical structure of voting paradoxes

Abstract: A theory is developed to identify, characterize, and explain all possible positional and pairwise voting outcomes that can occur for any number of alternatives and any profile. This paper describes pairwise voting where new results include explanations for all paradoxes, cycles, conflict between Borda and Condorcet rankings, differences among procedures using pairwise votes (such as the Borda Count, Kemeny's method, and the Arrow-Raynaud rule), and discrepancies among the societal rankings as candidates are dropped or added. Other new results include new relationships among the Borda and Condorcet “winners” and “losers.” The theory also shows how to construct all supporting profiles. The following companion paper does the same for positional methods.

Development of a methodology for selecting criteria and indicators of sustainable forest management: a case study on participatory assessment.

Abstract: This paper describes an application of multiple criteria analysis (MCA) in assessing criteria and indicators adapted for a particular forest management unit. The methods include: ranking, rating, and pairwise comparisons. These methods were used in a participatory decision-making environment where a team representing various stakeholders and professionals used their expert opinions and judgements in assessing different criteria and indicators (C&I) on the one hand, and how suitable and applicable they are to a forest management unit on the other. A forest concession located in Kalimantan, Indonesia, was used as the site for the case study. Results from the study show that the multicriteria methods are effective tools that can be used as structured decision aids to evaluate, prioritize, and select sets of C&I for a particular forest management unit. Ranking and rating approaches can be used as a screening tool to develop an initial list of C&I. Pairwise comparison, on the other hand, can be used as a finer filter to further reduce the list. In addition to using these three MCA methods, the study also examines two commonly used group decision-making techniques, the Delphi method and the nominal group technique. Feedback received from the participants indicates that the methods are transparent, easy to implement, and provide a convenient environment for participatory decision-making.

Extension Of The Rough Set Approach To Multicriteria Decision Support

Abstract: The original version of the rough sets theory has proved to be particularly useful in the analysis of multiattribute classification problems under inconsistency following from information granulation, i.e. objects having the same description but belonging to different classes. It fails, however, when attributes with preference-ordered domains (criteria) have to be taken into account. In order to deal with problems of multicriteria decision analysis (MCDA), such as sorting, choice or ranking, the authors have extended the original rough sets theory in a number of directions. The main extension is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered decision classes in multicriteria sorting. Second extension was necessary to approximate preference relations in multicriteria choice and ranking problems; it requires substitution of the data table by a pairwise comparison table, where each row corresponds to a pair of actions described by bina...

Measurement scales and scale independence in the Analytic Hierarchy Process

Abstract: One approach to evaluate the relative performance of decision alternatives with respect to multiple criteria is provided by the analytic hierarchy process. The method is based on pairwise comparisons between attributes, and several numerical measurement scales for the ratio statements have been proposed. The choice of measurement scale is re-examined, and new arguments supporting the measurement scale of geometric progression are derived. Separately from the measurement scale considerations, the effects of the scale parameter in geometric measurement scale are also studied. By using a regression model for pairwise comparisons data, it is shown that the statistical inference does not depend on the value of the scale parameter in the case of a single pairwise comparison matrix. It is also shown when the scale independence of statistical inference can be achieved in a decision hierarchy. This requires the use of the geometric-mean aggregation rule instead of the traditional arithmetic-mean aggregation. The results of the case study demonstrate that the measurement scale and the aggregation rule have potentially large impacts on decision support. Copyright © 2000 John Wiley & Sons, Ltd.

A Survey of Sample Size Formulas for Pairwise and Many-one Multiple Comparisons in the Parametric, Nonparametric and Binomial Case

Abstract: We present a survey of sample size formulas derived in other papers for pairwise comparisons of k treatments and for comparisons of k treatments with a control. We consider the calculation of sample sizes with preassigned per-pair, any-pair and all-pairs power for tests that control either the comparison-wise or the experimentwise type I error rate. A comparison exhibits interesting similarities between the parametric, nonparametric and binomial case.

Method and system for network-based decision processing and for matching requests for proposals to responses

Abstract: A method of assisting a user over a network (120) that enables users to evaluate various products and services (135) (collectively 'products') is provided. The products are described in one or more dynamically generated data table accessible through the network. In a preferred embodiment, a user provides over the network filtering decisions that enable the system to filter product records to identify a subset of relevant products. In addition, the user preferably performs graphical pairwise comparisons of the characteristics of the desired product to indicate the relative importance of such characteristics creating a unique user profile or a user may choose a predefined profile which corresponds to their user group. The data generated as a result of the pairwise comparisons is converted into weights by applying an analytic hierarchy process to create an accurate user profile which is used to rank a set of alternatives. The system applies the profile to perform synthesis to rank the products with respect to the user's wants and needs.

Some comparisons for Gaussian processes

Abstract: Extensions and variants are given for the well-known comparison principle for Gaussian processes based on ordering by pairwise distance.

An extension of Zermelo's model for ranking by paired comparisons

Abstract: In 1929, Zermelo proposed a probabilistic model for ranking by paired comparisons and showed that this model produces a unique ranking of the objects under consideration when the outcome matrix is irreducible. When the matrix is reducible, the model may yield only a partial ordering of the objects. In this paper, we analyse a natural extension of Zermelo’s model resulting from a singular perturbation. We show that this extension produces a ranking for arbitrary (nonnegative) outcome matrices and retains several of the desirable properties of the original model. In addition, we discuss computational techniques and provide examples of their use. Suppose that n objects are compared a pair at a time, and that for each comparison one of the two objects in the pair is judged superior to the other. (A common example would be athletic teams engaged in pairwise competitions.) The results of the comparisons can be summarized in the outcome matrix A =( aij), where aij is the number of comparisons in which object i is judged to be superior to object j. If all of the o-diagonal elements in A +A T are the same (i.e. there has been roundrobin competition) then the natural way to rank objects would be according to their scores si = P n=1aij. If, on the other hand, the outcome matrix lacks this symmetry, it is reasonable to suspect that ranking by score is not necessarily the best possible choice. (Contrary to common mathematical usage, we will often use the word tournament when referring to this asymmetric case; when we wish to emphasize the possible lack of symmetry, we may use the phrase generalized tournament.) A wide variety of methods have been proposed for ranking generalized tournaments. (See, for example, [9, 10].) In 1929, Zermelo [33] derived the functional

An Iterative Procedure for Evaluating Digraph Competitions

Abstract: A competition which is based on the results of (partial) pairwise comparisons can be modelled by means of a directed graph.Given initial weights on the nodes in such digraph competitions, we view the measurement of the importance (i.e., the cardinal ranking) of the nodes as an allocation problem where we redistribute the initial weights on the basis of insights from cooperative game theory.After describing the resulting procedure of redistributing the initial weights, we describe an iterative process is described which repeats this procedure: at each step the allocation obtained in the previous step determines the new input weights.Existence and uniqueness of the limit is established for arbitrary digraphs.Applications to the evaluation of e.g. sport competitions and paired comparison experiments are discussed.

A theory of rational choice under ignorance

Abstract: This paper contributes to a theory of rational choice for decision-makers with incomplete preferences due to partial ignorance, whose beliefs are representable as sets of acceptable priors. We focus on the limiting case of `Complete Ignorance' which can be viewed as reduced form of the general case of partial ignorance. Rationality is conceptualized in terms of a `Principle of Preference-Basedness', according to which rational choice should be isomorphic to asserted preference. The main result characterizes axiomatically a new choice-rule called `Simultaneous Expected Utility Maximization'. It can be interpreted as agreement in a bargaining game (Kalai-Smorodinsky solution) whose players correspond to the (extremal) `acceptable priors' among which the decision maker has suspended judgment. An essential but non-standard feature of Simultaneous Expected Utility choices is their dependence on the entire choice set. This is justified by the conception of optimality as compromise rather than as superiority in pairwise comparisons.

A Nonparametric Approach to Identifying Latent Relationships in Hierarchical Models

Abstract: This paper provides a method for nonparametrically modeling the relationship between consumer preference for product features, such as reliability or durability, and covariates that describe consumers and how they use the product. This relationship is of interest to firms designing and delivering products to a market because the extent to which consumers are sensitive to particular features determines the potential profitability of product offerings, and affects decisions relating to appropriate distribution outlets and advertising strategies. The successful identification of these relationships also aids in efficiently targeting marketing activities to specific segments of the consumer population. The relationship between consumer preference for product features and observable covariates is important but is typically unknown. In addition, these relationships are often deeply embedded in a model hierarchy and are not observed directly. For example, in models of household choice, the observed outcomes are multinomial with probabilities driven by latent utilities or values that consumers place on the choice alternatives. These utilities are in turn a function of characteristics, such as price and product features, which are differentially valued. Of primary interest is the relationship between consumer sensitivity to product characteristics and readily observed covariates such as household demographics or aspects of product usage. Because the relationships of interest are not directly observed, it is difficult to draw inferences about them without formal statistical models. This paper presents a three-level hierarchical Bayes model for modeling binary consumer preferences as a function of observable covariates. The hierarchical model nonparametrically estimates the relationships between consumer preferences for product features and the covariates without assuming a specific functional form. A nonparametric model is particularly useful in the exploratory analysis of consumer data in which the primary purpose of the analysis is to generate further questions rather than provide specific answers to well-posed questions. This type of analysis is frequently encountered in marketing where a series of studies are commissioned to better understand the nature of demand. The first level of the hierarchy in the Bayesian model relates the binary consumer choice to the sensitivities of the consumer to product attributes such as brand name, price, reliability, and durability. The second level of the hierarchy models the heterogeneity across consumers using functions that relate attribute sensitivities to observable covariates. This level of the hierarchy also allows each respondent to have unique demand coefficients by introducing random effect components. The third level of the hierarchy specifies a smoothness prior for each of the unknown functions used in the second level. The approach is flexible and works well both when the unknown function can be closely approximated by a linear function and when it cannot be. A Bayesian model selection technique is used to determine which functions can be modeled using a linear function and which ones should be modeled nonparametrically to provide the necessary flexibility to estimate the function accurately. The proposed methodology is illustrated using data from a survey of consumer preferences for features of marine outboard engines that was collected as part of a consulting project. Our analysis focuses on measuring consumer preferences for engine features and their relationships to two variables related to boat length and engine size. Consumer preferences for engine features were obtained through a national survey conducted over the telephone. Preferences were elicited by means of a pairwise evaluation in which respondents chose between two engines that were identical in every respect except for two engine features. The methodology can be modified to allow for more complex comparisons such as conjoint data collected in full profiles. The application of a Bayesian model selection procedure indicates that 4 of the 28 covariate relationships in the model are nonlinear, while the other 24 are linear. The preferences associated with these four functions are involved in 56% of the pairwise comparisons in the study. Therefore, in practice, if the nonlinear functions are not properly estimated there is the potential to draw misleading inferences regarding 56% of the pairwise choices. Firms can use the estimates of the functions relating preferences to covariates in a number of ways. First, they can use the covariates to determine the total number of consumers who have high demand for a particular product feature, and then they can target communication efforts to those individuals. Alternatively, the empirical results can be used as a basis of subsequent analysis to obtain a more complete characterization of a market segment.

A General Weighted Two-Way Dissimilarity Coefficient

Abstract: Choosing dissimilarity/similarity (proximity) measures for mixed data is not always easy. Gower (1971) has suggested a measure which has become popular in practical applications. This paper suggests an extension to his measure for calculating pairwise dissimilarities for individuals (objects) which also simultaneously produces a set for the variables. The definition of the dissimilarities incorporates the automatic weighting of individuals and variables. The dissimilarities can be used for multidimensional scaling or other statistical techniques that use proximities for both the individuals and variables, with the weights providing information on the relative contribution of each individual and variable in the analysis. The new method is tested by a simulation exercise and then used successfully on three empirical data sets.

A statistical model selection strategy applied to neural networks.

Abstract: In statistical modelling, an investigator must often choose a suitable model among a collection of viable candidates. There is no consensus in the research community on how such a comparative study is performed in a methodologically sound way. The ranking of several methods is usually performed by the use of a selection criterion, which assigns a score to every model based on some underlying statistical principles. The fitted model that is favoured is the one corresponding to the minimum (or the maximum) score. Statistical significance testing can extend this method. However, when enough pairwise tests are performed the multiplicity effect appears which can be taken into account by considering multiple comparison procedures. The existing comparison procedures can roughly be categorized as analytical or resampling based. This paper describes a resampling based multiple comparison technique. This method is illustrated on the estimate of the number of hidden units for feed-forward neural networks.

Pairwise conjoint analysis of activity engagement choice

Abstract: Information overload is a well-known problem of conjoint choice models when respondents have to evaluate a large number of attributes and/or attribute levels. In this paper we develop an alternative conjoint modelling approach, called pairwise conjoint analysis. It differs from conventional conjoint choice and preference models in that the attributes of choice alternatives or choice contexts are not varied simultaneously, but in pairs. Properties of the design strategy are discussed. The new approach is illustrated by using activity engagement choice as an example.

Critical point representations of electron density maps for the comparison of benzodiazepine-type ligands.

Abstract: A procedure for the comparison of three-dimensional electron density distributions is proposed for similarity searches between pharmacological ligands at various levels of crystallographic resolution. First, a graph representation of molecular electron density distributions is generated using a critical point analysis approach. Pairwise as well as multiple comparisons between the obtained graphs of critical points are then carried out using a Monte Carlo/simulated annealing technique, and results are compared with genetic algorithm solutions.

Fast Feature Selection Using Partial Correlation for Multi-vaslued Attributes

Abstract: We propose a fast feature selection method in supervised learning for multi-valued attributes. The main idea is to rewrite the multi-valued problem in the space of examples into a boolean problem in the space of pairwise examples. On basis of this approach, we can use point correlation coefficient which is null in the case of conditional independence, and verifies a formula connecting partial coefficients with marginal coefficients. This property allows to reduce considerably the computing times because a single pass over the database is necessary to compute all coefficients. We test our algorithm on benchmark databases.

The nearest adjoining order method for pairwise comparisons in the form of difference of ranks

Abstract: The problem of ranking of elements from some finite set on the basis of nearest adjoining order method for pairwise comparisons is investigated in this paper. It is assumed that in the set under consideration there exists a weak preference relation, which is to be identified (estimated) on the basis of pairwise comparisons in the form of difference of ranks. Moreover, the results of comparisons may be disturbed with random errors; the assumptions made about error distributions are not restrictive. The paper comprises: the problem formulation (definitions, assumptions, and optimisation problem, which provides the NAO solution) and the theoretical background – the form of distributions of random variables which make it possible to determine the properties of NAO solution, in particular, evaluation of the probability, that the NAO solution is equivalent to the errorless one. The approach presented in the paper can be extended to the case of more than one comparison for each pair of elements, i.e., completely formalised multi-experts ranking procedure.

Feature selection for multi-class classification using pairwise class discriminatory measure and covering concept

Abstract: An algorithm is presented for selecting a suboptimal set of features which classify given data into classes as effectively as the entire set of features. The algorithm is useful for reducing the number of features in a multi-class problem. In this algorithm, for each pair of classes a feature is successively selected which best discriminates the pair. The algorithm stops when all the pairs are covered. Preliminary experimental results are good.