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.     

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.     

Further Discussions on Induced Bias Matrix Model for the Pair-Wise Comparison Matrix

The inconsistency issue of pairwise comparison matrices has been an important subject in the study of the analytical network process. Most inconsistent elements can efficiently be identified by inducing a bias matrix only based on the original matrix. This paper further discusses the induced bias matrix and integrates all related theorems and corollaries into the induced bias matrix model. The theorem of inconsistency identification is proved mathematically using the maximum eigenvalue method and the contradiction method. In addition, a fast inconsistency identification method for one pair of inconsistent elements is proposed and proved mathematically. Two examples are used to illustrate the proposed fast identification method. The results show that the proposed new method is easier and faster than the existing method for the special case with only one pair of inconsistent elements in the original comparison matrix.     

A new approach for weight derivation using data envelopment analysis in the analytic hierarchy process

Recently, some researches have been carried out in the context of using data envelopment analysis (DEA) models to generate local weights of alternatives from pairwise comparison matrices used in the analytic hierarchy process (AHP). One of these models is the DEAHP. The main drawback of the DEAHP is that it generates counter-intuitive priority vectors for inconsistent pairwise comparison matrices. To overcome the drawbacks of the DEAHP, this paper proposes a new procedure entitled Revised DEAHP, and it will be shown that this procedure generates logical weights that are consistent with the decision maker's judgements and is sensitive to changes in data of the pairwise comparison matrices. Through a numerical example, it will be shown that the Revised DEAHP not only produces correct weights for inconsistent matrices but also does not suffer from rank reversal when an irrelevant alternative is added or removed.     

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.     

Efficient weight vectors from pairwise comparison matrices

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

A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP

Tests of consistency for the pair-wise comparison matrices have been studied extensively since AHP was introduced by Saaty in 1970s. However, existing methods are either too complicated to be applied in the revising process of the inconsistent comparison matrix or are difficult to preserve most of the original comparison information due to the use of a new pair-wise comparison matrix. Those methods might work for AHP but not for ANP as the comparison matrix of ANP needs to be strictly consistent. To improve the consistency ratio, this paper proposes a simple method, which combines the theorem of matrix multiplication, vectors dot product, and the definition of consistent pair-wise comparison matrix, to identify the inconsistent elements. The correctness of the proposed method is proved mathematically. The experimental studies have also shown that the proposed method is accurate and efficient in decision maker’s revising process to satisfy the consistency requirements of AHP/ANP.     

A goal programming method for obtaining interval weights from an interval comparison matrix

Crisp comparison matrices lead to crisp weight vectors being generated. Accordingly, an interval comparison matrix should give an interval weight estimate. In this paper, a goal programming (GP) method is proposed to obtain interval weights from an interval comparison matrix, which can be either consistent or inconsistent. The interval weights are assumed to be normalized and can be derived from a GP model at a time. The proposed GP method is also applicable to crisp comparison matrices. Comparisons with an interval regression analysis method are also made. Three numerical examples including a multiple criteria decision-making (MCDM) problem with a hierarchical structure are examined to show the potential applications of the proposed GP method.     

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.     

Data Perturbations of Matrices of Pairwise Comparisons

This paper deals with data perturbations of pairwise comparison matrices (PCM). Transitive and symmetrically reciprocal (SR) matrices are defined. Characteristic polynomials and spectral properties of certain SR perturbations of transitive matrices are presented. The principal eigenvector components of some of these PCMs are given in explicit form. Results are applied to PCMs occurring in various fields of interest, such as in the analytic hierarchy process (AHP) to the paired comparison matrix entries of which are positive numbers, in the dynamic input–output analysis to the matrix of economic growth elements of which might become both positive and negative and in vehicle system dynamics to the input spectral density matrix whose entries are complex numbers.     

Comparison of statistical procedures in analytic hierarchy process using a ranking test

Several statistical procedures for estimation of the priority parameters in the setup of the Analytic Hierarchy Process (AHP) exist in the literature. The purpose of this article is to make appropriate comparisons of such statistical methods. Pairwise comparison matrices are simulated using different statistical distributions of the error part used in the procedures. Priority parameters are estimated for each simulated pairwise comparison matrix using the method suggested. Standard nonparametric statistical procedures are applied to check whether the order of the priority estimates is consistent with that of their parameter values irrespective of the choice of particular statistical procedure. Statistical procedures based on the reciprocal matrices are also compared with the eigenvalue method.     

基于基本回路修正的AHP一致性调整方法研究

为解决AHP一致性问题,提出一种基于基本回路修正的调整方法,能够同时解决数值不一致和逻辑不一致问题,同时保证对原始信息的修改量最小。数值不一致和逻辑不一致均由决策者的不准确判断引起,其中数值不一致可以通过降低一致性比率(CR)值进行改善,而逻辑不一致只有将判断矩阵中所有三阶回路去除才能得到解决。因此,通过对n阶判断矩阵进行基本矩阵分解,得到C³_n个3阶的基本矩阵,其中存在三阶回路的称为基本回路,从而将判断矩阵的一致性修正问题转化为基本回路的一致性修正问题。通过对基本回路的一致性比较,提出了两种确定最不一致元素的方法,即CR和最大法和优化法,并设计了优化模型对最不一致元素进行修正。最后,通过算例分析验证了本文方法的可行性,与已有方法的对比结论证明了本文方法更为有效。     

A geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices

The usefulness of encoding the fuzzy evaluations of alternatives and the importance weights of criteria, in a multiple objective decision problem through binary comparison matrices (or pairwise judgment matrices) is receiving considerable attention. The methodology for identifying the best alternative in a given decision problem involves the computation of the principal eigenvectors of the binary comparison matrices. The eigenvectors transform the fuzzy evaluations of the importance of the criteria and the ratings of the alternatives into a ratio scale. A difficulty that is often experienced in using this approach in practice, is the inconsistency of the binary evaluations. This paper proposes a simple averaging procedure to construct a supertransitive approximation to a binary comparison matrix, where inconsistency is a problem. It is further suggested that such an adjustment might be necessary to more closely reflect the inherent fuzziness of the evaluations contained in a binary comparison matrix. The procedure is illustrated by means of examples.     

Fuzzy analytic hierarchy process: A logarithmic fuzzy preference programming methodology

Fuzzy analytic hierarchy process (AHP) proves to be a very useful methodology for multiple criteria decision-making in fuzzy environments, which has found substantial applications in recent years. The vast majority of the applications use a crisp point estimate method such as the extent analysis or the fuzzy preference programming (FPP) based nonlinear method for fuzzy AHP priority derivation. The extent analysis has been revealed to be invalid and the weights derived by this method do not represent the relative importance of decision criteria or alternatives. The FPP-based nonlinear priority method also turns out to be subject to significant drawbacks, one of which is that it may produce multiple, even conflict priority vectors for a fuzzy pairwise comparison matrix, leading to entirely different conclusions. To address these drawbacks and provide a valid yet practical priority method for fuzzy AHP, this paper proposes a logarithmic fuzzy preference programming (LFPP) based methodology for fuzzy AHP priority derivation, which formulates the priorities of a fuzzy pairwise comparison matrix as a logarithmic nonlinear programming and derives crisp priorities from fuzzy pairwise comparison matrices. Numerical examples are tested to show the advantages of the proposed methodology and its potential applications in fuzzy AHP decision-making.     

An integrated approach for deriving priorities in analytic network process

A multiple objective programming approach for the analytic network process (ANP) is proposed to obtain all local priorities for crisp or interval judgments at one time, even in an inconsistent situation. The weakness of the ANP and fuzzy ANP (FANP) is that the complexity of generating priorities is equal to the number of comparison matrices. In the proposed approach, all sets of crisp priorities for each pairwise comparison matrix can be obtained directly. Moreover, from the outcomes of three examples, the power to reach a limiting supermatrix is less than or equal to the power of the FANP. Thus, the proposed approach can be regarded as an efficient alternative of the fuzzy ANP.     

A simple formula to find the closest consistent matrix to a reciprocal matrix

Achieving consistency in pair-wise comparisons between decision elements given by experts or stakeholders is of paramount importance in decision-making based on the AHP methodology. Several alternatives to improve consistency have been proposed in the literature. The linearization method (Benítez et al., 2011 [10]), derives a consistent matrix based on an original matrix of comparisons through a suitable orthogonal projection expressed in terms of a Fourier-like expansion. We propose a formula that provides in a very simple manner the consistent matrix closest to a reciprocal (inconsistent) matrix. In addition, this formula is computationally efficient since it only uses sums to perform the calculations. A corollary of the main result shows that the normalized vector of the vector, whose components are the geometric means of the rows of a comparison matrix, gives the priority vector only for consistent matrices.     

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.     

Deriving weights from pairwise comparison ratio matrices: An axiomatic approach

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