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.     

On the extent analysis method for fuzzy AHP and its applications

In a paper by Chang [D.Y. Chang, Applications of the extent analysis method on fuzzy AHP, European Journal of Operational Research 95 (1996) 649–655], an extent analysis method on fuzzy AHP was proposed to obtain a crisp priority vector from a triangular fuzzy comparison matrix. It is found that the extent analysis method cannot estimate the true weights from a fuzzy comparison matrix and has led to quite a number of misapplications in the literature. In this paper, we show by examples that the priority vectors determined by the extent analysis method do not represent the relative importance of decision criteria or alternatives and that the misapplication of the extent analysis method to fuzzy AHP problems may lead to a wrong decision to be made and some useful decision information such as decision criteria and fuzzy comparison matrices not to be considered. We show these problems to avoid any possible misapplications in the future.     

On pairwise comparison matrices that can be made consistent by the modification of a few elements

Pairwise comparison matrices are often used in Multi-attribute Decision Making for weighting the attributes or for the evaluation of the alternatives with respect to a criteria. Matrices provided by the decision makers are rarely consistent and it is important to index the degree of inconsistency. In the paper, the minimal number of matrix elements by the modification of which the pairwise comparison matrix can be made consistent is examined. From practical point of view, the modification of 1, 2, or, for larger matrices, 3 elements seems to be relevant. These cases are characterized by using the graph representation of the matrices. Empirical examples illustrate that pairwise comparison matrices that can be made consistent by the modification of a few elements are present in the applications.     

Analysis of pairwise comparison matrices: an empirical research

Pairwise comparison (PC) matrices are used in multi-attribute decision problems (MADM) in order to express the preferences of the decision maker. Our research focused on testing various characteristics of PC matrices. In a controlled experiment with university students (N=227) we have obtained 454 PC matrices. The cases have been divided into 18 subgroups according to the key factors to be analyzed. Our team conducted experiments with matrices of different size given from different types of MADM problems. Additionally, the matrix elements have been obtained by different questioning procedures differing in the order of the questions. Results are organized to answer five research questions. Three of them are directly connected to the inconsistency of a PC matrix. Various types of inconsistency indices have been applied. We have found that the type of the problem and the size of the matrix had impact on the inconsistency of the PC matrix. However, we have not found any impact of the questioning order. Incomplete PC matrices played an important role in our research. The decision makers behavioral consistency was as well analyzed in case of incomplete matrices using indicators measuring the deviation from the final order of alternatives and from the final score vector.     

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.     

A comparison of the Saaty's AHP and modified AHP for right and left eigenvector inconsistency

Over the past two decades, Saaty's Analytic Hierarchy Process (AHP) has been developed to solve decision problems in various fields by prioritization of alternatives using eigenvectors and manipulations in matrix algebra. However, a fundamental problem called “Right and Left Eigenvector Inconsistency” has been observed which may yield inconsistent results using the right and the left eigenvectors. A new method known as the Modified AHP has been recently devised by H.A. Donegan, F.J. Dodd, T.B.M. McMaster, The Statistician 41 (1992) 295–302 who claimed that the inconsistency problem can be effectively reduced. This work is an attempt to compare the Saaty's AHP (SAHP) and the Modified AHP (MAHP) using 42 models comprising 294 reciprocal matrices. It was discovered that the Modified AHP is no better than the Saaty's AHP.     

Rank reversals in multi-criteria decision analysis with statistical modelling of ratio-scale pairwise comparisons

In multi-criteria decision analysis, the overall performance of decision alternatives is evaluated with respect to several, generally conflicting decision criteria. One approach to perform the multi-criteria decision analysis is to use ratio-scale pairwise comparisons concerning the performance of decision alternatives and the importance of decision criteria. In this approach, a classical problem has been the phenomenon of rank reversals. In particular, when a new decision alternative is added to a decision problem, and while the assessments concerning the original decision alternatives remain unchanged, the new alternative may cause rank reversals between the utility estimates of the original decision alternatives. This paper studies the connections between rank reversals and the potential inconsistency of the utility assessments in the case of ratio-scale pairwise comparisons data. The analysis was carried out by recently developed statistical modelling techniques so that the inconsistency of the assessments was measured according to statistical estimation theory. Several type of decision problems were analysed and the results showed that rank reversals caused by inconsistency are natural and acceptable. On the other hand, rank reversals caused by the traditional arithmetic-mean aggregation rule are not in line with the ratio-scale measurement of utilities, whereas geometric-mean aggregation does not cause undesired rank reversals.     

Combining compound linguistic ordinal scale and cognitive pairwise comparison in the rectified fuzzy TOPSIS method for group decision making

Group decision making is the process to explore the best choice among the screened alternatives under predefined criteria with corresponding weights from assessment of a group of decision makers. The Fuzzy TOPSIS taking an evaluated fuzzy decision matrix as input is a popular tool to analyze the ideal alternative. This research, however, finds that the classical fuzzy TOPSIS produces a misleading result due to some inappropriate definitions, and proposes the rectified fuzzy TOPSIS addressing two technical problems. As the decision accuracy also depends on the evaluation quality of the fuzzy decision matrix comprising rating scores and weights, this research applies compound linguistic ordinal scale as the fuzzy rating scale for expert judgments, and cognitive pairwise comparison for determining the fuzzy weights. The numerical case of a robot selection problem demonstrates the hybrid approach leading to the much reliable result for decision making, comparing with the conventional fuzzy Analytic Hierarchy Process and TOPSIS.     

A fuzzy outranking relation in multicriteria decision making

It frequently happens that a decision maker must establish a ranking within a finite set of alternatives with respect to multiple criteria. The subjective evaluation of each alternative according to each criterion is expressed in the form of a distributive evaluation. To capture the preferences of one alternative over another, a concept of fuzzy outranking relation can be used. This fuzzy outranking relation is characterized by a degree of credibility which is computed from two indices: a confidence index and a doubt index. Each of these indices is calculated from the distributive evaluations over the various criteria. In this paper, such a fuzzy outranking relation (fuzzy binary relation) is constructed and an application is presented.     

基于二元联系数距离的不确定语言多属性决策模型

研究了属性权重范围已知,方案主观偏好值为语言变量,决策信息为不确定语言决策矩阵的多属性决策问题.在给出不确定语言变量转换为二元联系数的公式以及二元联系数距离公式的基础上,将方案主观偏好语言评价值转换为二元联系数,将不确定语言决策矩阵转换为二元联系数决策矩阵,从而得到方案的二元联系数综合属性值,通过最小化方案的二元联系数综合属性值和主观偏好值之间距离,建立多目标优化模型,并将其转换为一个单目标规划模型计算出属性权重.然后,通过对方案的二元联系数综合属性值进行不确定性分析,得到各方案的排序总数,利用排序总数对方案进行排序择优.应用实例表明该决策方法可行有效.     

Multiple criteria group decision-making with generalized interval-valued fuzzy numbers based on signed distances and incomplete weights

Decision-making information provided by decision makers is often imprecise or uncertain, due to lack of data, time pressure, or the decision makers’ limited attention and information-processing capabilities. Interval-valued fuzzy sets are associated with greater imprecision and more ambiguity than are ordinary fuzzy sets. For these reasons, this paper presents a signed distance-based method for handling fuzzy multiple-criteria group decision-making problems in which individual assessments are provided as generalized interval-valued trapezoidal fuzzy numbers, and the information about criterion weights are not precisely but partially known. First, concerning the relative importance of decision makers and the group consensus of fuzzy opinions, all individual decision opinions were aggregated into group opinions using a hybrid average with weighted averaging and signed distance-based ordered weighted averaging operations. Next, considering a decision situation with incomplete weight information of criteria, an integrated programming model was developed to estimate criterion weights and to order the priorities of various alternatives based on signed distances. In addition, several deviation variables were introduced to mitigate the effect of inconsistent evaluations on the importance of criteria. Finally, the feasibility of the proposed method is illustrated by a numerical example of a multi-criteria supplier selection problem. Furthermore, a comparative analysis with other methods was conducted to validate the effectiveness and applicability of the proposed methodology.     

Multi-person multi-attribute decision making models under intuitionistic fuzzy environment

Intuitionistic fuzzy numbers, each of which is characterized by the degree of membership and the degree of non-membership of an element, are a very useful means to depict the decision information in the process of decision making. In this article, we investigate the group decision making problems in which all the information provided by the decision makers is expressed as intuitionistic fuzzy decision matrices where each of the elements is characterized by intuitionistic fuzzy number, and the information about attribute weights is partially known, which may be constructed by various forms. We first use the intuitionistic fuzzy hybrid geometric (IFHG) operator to aggregate all individual intuitionistic fuzzy decision matrices provided by the decision makers into the collective intuitionistic fuzzy decision matrix, then we utilize the score function to calculate the score of each attribute value and construct the score matrix of the collective intuitionistic fuzzy decision matrix. Based on the score matrix and the given attribute weight information, we establish some optimization models to determine the weights of attributes. Furthermore, we utilize the obtained attribute weights and the intuitionistic fuzzy weighted geometric (IFWG) operator to fuse the intuitionistic fuzzy information in the collective intuitionistic fuzzy decision matrix to get the overall intuitionistic fuzzy values of alternatives by which the ranking of all the given alternatives can be found. Finally, we give an illustrative example.     

Interactive procedure for a multiobjective stochastic discrete dynamic problem

Multiple objectives and dynamics characterize many sequential decision problems. In the paper we consider returns in partially ordered criteria space as a way of generalization of single criterion dynamic programming models to multiobjective case. In our problem evaluations of alternatives with respect to criteria are represented by distribution functions. Thus, the overall comparison of two alternatives is equivalent to the comparison of two vectors of probability distributions. We assume that the decision maker tries to find a solution preferred to all other solutions (the most preferred solution). In the paper a new interactive procedure for stochastic, dynamic multiple criteria decision making problem is proposed. The procedure consists of two steps. First, the Bellman principle is used to identify the set of efficient solutions. Next interactive approach is employed to find the most preferred solution. A numerical example and a real-world application are presented to illustrate the applicability of the proposed technique.     

Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems

In this paper, we investigate the group decision making problems in which all the information provided by the decision-makers is presented as interval-valued intuitionistic fuzzy decision matrices where each of the elements is characterized by interval-valued intuitionistic fuzzy number (IVIFN), and the information about attribute weights is partially known. First, we use the interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator to aggregate all individual interval-valued intuitionistic fuzzy decision matrices provided by the decision-makers into the collective interval-valued intuitionistic fuzzy decision matrix, and then we use the score function to calculate the score of each attribute value and construct the score matrix of the collective interval-valued intuitionistic fuzzy decision matrix. From the score matrix and the given attribute weight information, we establish an optimization model to determine the weights of attributes, and then we use the obtained attribute weights and the interval-valued intuitionistic fuzzy weighted geometric (IIFWG) operator to fuse the interval-valued intuitionistic fuzzy information in the collective interval-valued intuitionistic fuzzy decision matrix to get the overall interval-valued intuitionistic fuzzy values of alternatives, and then rank the alternatives according to the correlation coefficients between IVIFNs and select the most desirable one(s). Finally, a numerical example is used to illustrate the applicability of the proposed approach.     

Fuzzy multi-criteria decision-making based on positive and negative extreme solutions

To encompass decision data vagueness, many researchers generalized multi-criteria decision-making (MCDM) methods in certain environment into fuzzy multi-criteria decision-making (FMCDM) methods under fuzzy environment. In these FMCDM methods, ranking fuzzy numbers based on fuzzy pair-wise comparison is normally essential, but the comparison is a complexity work. To avoid fuzzy pair-wise comparison, we propose a FMCDM method based on positive and negative extreme solutions of alternatives. In the proposed method, two extreme solutions of alternatives are obtained by MAX and MIN operations of fuzzy TOPSIS. Then weakness and strength matrices between alternatives and extreme solutions are derived by a difference function revised from fuzzy preference relation of Lee, and multiplied with weight matrix to be weighted weakness and strength indices. The two weighted indices are respectively transferred into positive and negative indices, and then the two indices integrated into a total performance index. Finally, alternatives can be sorted according to their related performance indices, and FMCDM problems are easily solved, not by fuzzy pair-wise comparison.     

Multi-criteria decision making with fuzzy linguistic preference relations

Although the analytic hierarchy process (AHP) and the extent analysis method (EAM) of fuzzy AHP are extensively adopted in diverse fields, inconsistency increases as hierarchies of criteria or alternatives increase because AHP and EAM require rather complicated pairwise comparisons amongst elements (attributes or alternatives). Additionally, decision makers normally find that assigning linguistic variables to judgments is simpler and more intuitive than to fixed value judgments. Hence, Wang and Chen proposed fuzzy linguistic preference relations (Fuzzy LinPreRa) to address the above problem. This study adopts Fuzzy LinPreRa to re-examine three numerical examples. The re-examination is intended to compare our results with those obtained in earlier works and to demonstrate the advantages of Fuzzy LinPreRa. This study demonstrates that, in addition to reducing the number of pairwise comparisons, Fuzzy LinPreRa also increases decision making efficiency and accuracy.     

Ranking multicriteria alternatives: The method ZAPROS III

The new version of the method for the construction of partial order on the set of multicriteria alternatives is presented. This method belongs to the family of verbal decision analysis (VDA) methods and gives a more efficient means of problem solution. The method is based on psychologically valid operations for information elicitation from a decision maker: comparisons of two distances between the evaluations on the ordinal scales of two criteria. The information received from a decision maker is used for the construction of a binary relation between a pair of alternatives which yields preference, indifference and incomparability relations. The method allows construction of a partial order on the set of given alternatives as well as on the set of all possible alternatives. The illustrative example is given.     

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.     

Extensions of the multicriteria analysis with pairwise comparison under a fuzzy environment

Multicriteria decision-making (MCDM) problems often involve a complex decision process in which multiple requirements and fuzzy conditions have to be taken into consideration simultaneously. The existing approaches for solving this problem in a fuzzy environment are complex. Combining the concepts of grey relation and pairwise comparison, a new fuzzy MCDM method is proposed. First, the fuzzy analytic hierarchy process (AHP) is used to construct fuzzy weights of all criteria. Then, linguistic terms characterized by L–R triangular fuzzy numbers are used to denote the evaluation values of all alternatives versus subjective and objective criteria. Finally, the aggregation fuzzy assessments of different alternatives are ranked to determine the best selection. Furthermore, this paper uses a numerical example of location selection to demonstrate the applicability of the proposed method. The study results show that this method is an effective means for tackling MCDM problems in a fuzzy environment.