A parametric GP model dealing with incomplete information for group decision-making

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

Strong reciprocity and strong consistency in pairwise comparison matrix with fuzzy elements

The decision making problem considered in this paper is to rank n alternatives from the best to the worst, using the information given by the decision maker in the form of an \(n\times n\) pairwise comparison matrix. Here, we deal with pairwise comparison matrices with fuzzy elements. Fuzzy elements of the pairwise comparison matrix are applied whenever the decision maker is not sure about the value of his/her evaluation of the relative importance of elements in question. We investigate pairwise comparison matrices with elements from abelian linearly ordered group (alo-group) over a real interval. The concept of reciprocity and consistency of pairwise comparison matrices with fuzzy elements have been already studied in the literature. Here, we define stronger concepts, namely the strong reciprocity and strong consistency of pairwise comparison matrices with fuzzy intervals as the matrix elements (PCF matrices), derive the necessary and sufficient conditions for strong reciprocity and strong consistency and investigate their properties as well as some consequences to the problem of ranking the alternatives.     

A new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process

The DEAHP method for weight deviation and aggregation in the analytic hierarchy process (AHP) has been found flawed and sometimes produces counterintuitive priority vectors for inconsistent pairwise comparison matrices, which makes its application very restrictive. This paper proposes a new data envelopment analysis (DEA) method for priority determination in the AHP and extends it to the group AHP situation. In this new DEA methodology, two specially constructed DEA models that differ from the DEAHP model are used to derive the best local priorities from a pairwise comparison matrix or a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent. The new DEA method produces true weights for perfectly consistent pairwise comparison matrices and the best local priorities that are logical and consistent with decision makers (DMs)’ subjective judgments for inconsistent pairwise comparison matrices. In hierarchical structures, the new DEA method utilizes the simple additive weighting (SAW) method for aggregation of the best local priorities without the need of normalization. Numerical examples are examined throughout the paper to show the advantages of the new DEA methodology and its potential applications in both the AHP and group decision making.     

Inferring consensus weights from pairwise comparison matrices without suitable properties

Pairwise comparison is a popular method for establishing the relative importance of n objects. Its main purpose is to get a set of weights (priority vector) associated with the objects. When the information gathered from the decision maker does not verify some rational properties, it is not easy to search the priority vector. Goal programming is a flexible tool for addressing this type of problem. In this paper, we focus on a group decision-making scenario. Thus, we analyze different methodologies for getting a collective priority vector. The first method is to aggregate general pairwise comparison matrices (i.e., matrices without suitable properties) and then get the priority vector from the consensus matrix. The second method proposes to get the collective priority vector by formulating an optimization problem without determining the consensus pairwise comparison matrix beforehand.     

Priority estimation in the AHP through maximization of correlation coefficient

This paper proposes a correlation coefficient maximization approach (CCMA) for estimating priorities from a pairwise comparison matrix. The priorities are supposed to be as highly correlated with each column of a pairwise comparison matrix as possible. Such priorities are not unique and can be determined in different ways. Two optimization models are therefore suggested for determining further the priorities, one of which leads to an analytic solution. Theorems about the CCMA are developed and its potential applications are illustrated with two numerical examples.     

A heuristic method to rectify intransitive judgments in pairwise comparison matrices

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.     

Modeling subjective evaluation for fuzzy group multicriteria decision making

This paper presents a new fuzzy multicriteria decision making (MCDM) approach for evaluating decision alternatives involving subjective judgements made by a group of decision makers. A pairwise comparison process is used to help individual decision makers make comparative judgements, and a linguistic rating method is used for making absolute judgements. A hierarchical weighting method is developed to assess the weights of a large number of evaluation criteria by pairwise comparisons. To reflect the inherent imprecision of subjective judgements, individual assessments are aggregated as a group assessment using triangular fuzzy numbers. To obtain a cardinal preference value for each decision alternative, a new fuzzy MCDM algorithm is developed by extending the concept of the degree of optimality to incorporate criteria weights in the distance measurement. An empirical study of aircraft selection is presented to illustrate the effectiveness of the approach.     

A method for approximating pairwise comparison matrices by consistent matrices

In several methods of multiattribute decision making, pairwise comparison matrices are applied to derive implicit weights for a given set of decision alternatives. A class of the approaches is based on the approximation of the pairwise comparison matrix by a consistent matrix. In the paper this approximation problem is considered in the least-squares sense. In general, the problem is nonconvex and difficult to solve, since it may have several local optima. In the paper the classic logarithmic transformation is applied and the problem is transcribed into the form of a separable programming problem based on a univariate function with special properties. We give sufficient conditions of the convexity of the objective function over the feasible set. If such a sufficient condition holds, the global optimum of the original problem can be obtained by local search, as well. For the general case, we propose a branch-and-bound method. Computational experiments are also presented.     

Estimation of missing judgments in AHP pairwise matrices using a neural network-based model

Selecting relevant features to make a decision and expressing the relationships between these features is not a simple task. The decision maker must precisely define the alternatives and criteria which are more important for the decision making process. The Analytic Hierarchy Process (AHP) uses hierarchical structures to facilitate this process. The comparison is realized using pairwise matrices, which are filled in according to the decision maker judgments. Subsequently, matrix consistency is tested and priorities are obtained by calculating the matrix principal eigenvector. Given an incomplete pairwise matrix, two procedures must be performed: first, it must be completed with suitable values for the missing entries and, second, the matrix must be improved until a satisfactory level of consistency is reached. Several methods are used to fill in missing entries for incomplete pairwise matrices with correct comparison values. Additionally, once pairwise matrices are complete and if comparison judgments between pairs are not consistent, some methods must be used to improve the matrix consistency and, therefore, to obtain coherent results. In this paper a model based on the Multi-Layer Perceptron (MLP) neural network is presented. Given an AHP pairwise matrix, this model is capable of completing missing values and improving the matrix consistency at the same time.     

Convergence properties and practical estimation of the probability of rank reversal in pairwise comparisons for multi-criteria decision making problems

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.     

Incomplete pairwise comparison and consistency optimization

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.     

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

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis of compatibility between interdependent matrices in ANP

Analytic network process (ANP) addresses multi-attribute decision-making where attributes exhibit dependencies. A principal characteristic of such problems is that pairwise comparisons are needed for attributes that have interdependencies. We propose that before such comparison matrices are used—in addition to a test that assesses the consistency of a pairwise comparison matrix—a test must also be conducted to assess ‘consistency’ across interdependent matrices. We call such a cross-matrix consistency test as a compatibility test. In this paper, we design a compatibility test for interdependent matrices between two clusters of attributes. We motivate our exposition by addressing compatibility in Sinarchy, a special form of ANP where interdependency exists between the last and next-to-last level. The developed compatibility test is applicable to any pair of interdependent matrices that are a part of an ANP.     

Fuzzified AHP in the evaluation of scientific monographs

Fuzzification of the analytic hierarchy process (AHP) is of great interest to researchers since it is a frequently used method for coping with complex decision making problems. There have been many attempts to fuzzify the AHP. We focus particularly on the construction of fuzzy pairwise comparison matrices and on obtaining fuzzy weights of objects from them subsequently. We review the fuzzification of the geometric mean method for obtaining fuzzy weights of objects from fuzzy pairwise comparison matrices. We illustrate here the usefulness of the fuzzified AHP on a real-life problem of the evaluation of quality of scientific monographs in university environment. The benefits of the presented evaluation methodology and its suitability for quality assessment of R&D results in general are discussed. When the task of quality assessment in R&D is considered, an important role is played by peer-review evaluation. Evaluations provided by experts in the peer-review process have a high level of subjectivity and can be expected in a linguistic form. New decision-support methods (or adaptations of classic methods) well suited to deal with such inputs, to capture the consistency of experts’ preferences and to restrict the subjectivity to an acceptable level are necessary. A new consistency condition is therefore defined here to be used for expertly defined fuzzy pairwise comparison matrices.     

Representing the strengths and directions of pairwise comparisons

We consider the problem of finding weights that well represent a set of pairwise multiplicative comparisons of a set of objects (as in the AHP and other methods). Our main contribution is a method for deriving such weights that takes into consideration not only the strengths of the pairwise comparisons, but also their directions. For example, if the comparison directions satisfy transitivity, then the weights produced by our method also satisfy transitivity (this is not always true for other methods). We also present a set of reasonable axioms for which our method is the (essentially) unique solution. Our method and axioms are closely related to those of Cook and Kress [Eur. J. Oper. Res. 37 (1988) 335]. Our method, like theirs, reduces to solving a linear program (hence it is different from the approach used in the AHP). For the special case that the comparison directions satisfy transitivity, our method is quite simple and reduces to performing a forward pass as in the critical path method.     

The expected likelihood of transitivity for a probabilistic chooser

May [18] developed an algebraic choice model of pairwise preference comparison in which subjects respond precisely to ordinal information on attributes of comparison. This study considers a probabilistic choice model variation of May's model, in which subjects respond with various degrees of precision in comparison to May's model. This precision can be viewed as an indirect measure of the subject's level of perception of the attributes of comparison. The purpose of the study is to examine the expected likelihood with which subjects will have transitive responses, as the degree of precision is varied. Closed form representations are obtained for the expected likelihood of transitivity for three alternatives for each different level of precision. Results indicate that a relatively small change in this precision can lead to substantial changes in the expected likelihood of transitivity.This research was supported through a fellowship from the Center for Advanced Study of the University of Delaware. Helpful comments from Sven Berg on a draft version of this paper are gratefully acknowledged.     

A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations

Decision makers (DMs)’ preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations. This paper proposes a chi-square method (CSM) for obtaining a priority vector from multiplicative and fuzzy preference relations. The proposed CSM can be used to obtain a priority vector from either a multiplicative preference relation (i.e. a pairwise comparison matrix) or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures. Theorems and algorithm about the CSM are developed. Three numerical examples are examined to illustrate the applications of the CSM and its advantages.     

Preference programming and inconsistent interval judgments

The problem of derivation of the weights of alternatives from pairwise comparison matrices is long standing. In this paper, Lexicographic Goal Programming (LGP) has been used to find out weights from pairwise inconsistent interval judgment matrices. A number of properties and advantages of LGP as a weight determination technique have been explored. An algorithm for identification and modification of inconsistent bounds is also provided. The proposed technique has been illustrated by means of numerical examples.     

成对比较矩阵的一种逼近

§1.问题的陈述 令R~(n×n)表示所有n×n阶实矩阵构成的线性空间,并定义其子集如下: P={p=(p_(ij))∈R~(n×n)|p_(ij)>0,p_(ik)=p_(ki)~(-1)}, Q={q=(qi_(ij))∈R~(n×n)|q_(ij)>0,q_(ik)q_(kj)=q_(ij)}.把P叫做正的互反矩阵(或判断矩阵)的集合,而称Q为相容性矩阵的集合.显然,Q为P的子集,且两者都不是R~(n×n)中的凸集.任取a,b∈R~(n×n),定义内积和范数如下:     

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.