Fuzzy analytic hierarchy process: Fallacy of the popular methods

The fuzzy Analytic Hierarchy Process (fuzzy AHP) is a very popular decision making method and literally thousands of papers have been published about it. However, we find the basic logic of this approach has problems. From its methodology, the definition and operational rules of fuzzy numbers not only oppose the main logic of fuzzy set theory, but also oppose the basic principles of the AHP. In dealing with the outcomes, fuzzy AHP does not give a generally accepted method to rank fuzzy numbers and a way to check the validity of the results. Besides, we discuss the validity of the Analytic Hierarchy/Network Process (AHP/ANP) in complex and uncertain environments and find that fuzzy ANP is a false proposition because there is no fuzzy priority in the super matrix which provides the basis for the ANP. Although fuzzy AHP has been applied in many cases and cited hundreds of times, we hoped that those who use fuzzy AHP would understand the problems associated with this method.     

Shortening the comparison process in the AHP

The Analytic Hierarchy Process (AHP) has become a popular decision aid since its development by Thomas L. Saaty in the 1970's. However, the number of pairwise comparisons which must be made during the course of this method often become prohibitive. This paper summarizes a method to reduce the number of comparisons by using the derivatives of the right Perron vector and the graph-theoretic interpretation of a positive reciprocal matrix.     

A quality control approach to consistency paradoxes in AHP

The Analytic Hierarchy Process (AHP) requires a specific consistency check of the pairwise comparisons in order to ensure that the decision maker is being neither inconsistent nor random in his or her pairwise comparisons. However, there are many situations where the decision maker has been reasonable, logical and non-random in making the pairwise comparison and yet will fail the consistency check. This paper argues against the use of the standard consistency check. If a consistency test is to be done, a quality control approach is recommended.     

ANP法在区域企业技术创新能力评价中的应用

在应用网络层次分析法(ANP)进行综合评价中存在两个问题:一是网络层次结构中子系统内部的依存关系往往缺乏实质性的阐述和研究,而这是网络层次结构的关键;二是没有通过实例比较网络层次法和层次分析法,证明网络层次分析法的实用价值.以区域企业技术创新能力的评价为背景,分别应用网络层次分析法(ANP)和层次分析法(AHP)建立评价指标体系,并利用各省数据评价了区域企业技术创新能力的现状,给出了政策建议,以期为制定区域企业技术创新战略、规划等提供了定量参考依据.     

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.     

The categorical data analysis approach for ratio model of pairwise comparisons

In the Analytic Hierarchy Process (AHP), pairwise comparison values are elicited by a ratio scale representing a set of categories. In this article, we propose a method of statistical inference in AHP which pertains to cardinal representations of categorical variables. Relevant estimation and hypothesis testing procedures are addressed in this perspective.     

Incomplete pairwise comparisons in the analytic hierarchy process

The Analytic Hierarchy Process is a decision-analysis tool which was developed by T.L. Saaty in the 1970s and which has been applied to many different decision problems in corporate, governmental and other institutional settings. The most successful applications have come about in group decisionmaking sessions, where the group structures the problem in a hierarchical framework and pairwise comparisons are elicited from the group for each level of the hierarchy. However, the number of pairwise comparison necessary in a real problem often becomes overwhelming. For example, with 9 alternatives and 5 criteria, the group must answer 190 questions. This paper explores various methods for reducing the complexity of the preference eliciting process. The theory of a method based upon the graph-theoretic structure of the pairwise comparison matrix and the gradient of the right Perron vector is developed, and simulations of a series of random matrices are used to illustrate the properties of this approach.     

Mixed valuation methods: a combined AHP-GP procedure for individual and group multicriteria agricultural valuation

This paper introduces a new assessment method classification, in which a third procedure, mixed valuation, is jointly included with the traditional economic and non-economic methodologies. The paper considers a case of multiple actors (from a previous work by the same authors—Aznar et al. (Estudios de Economía Aplicada, 25(2):389–409, 2007), in which a new technique for multicriteria agriculture valuation (MAVAM) was proposed. The method is specifically designed for situations in which scarce information about the elements being compared (quantified or not) is available. It works in individual and group decision making contexts and attempts to both obtain and incorporate the objective information associated with the tangible aspects of the problem and the subjective knowledge associated with the human factor into the valuation process. It combines two of the most extended multicriteria decision making techniques: the Analytic Hierarchy Process (AHP) and Goal Programming (GP). The first of these enables tangible and intangible information stemming from known elements to be collected by using pairwise comparisons; the second allows the scarce information available and the personal approach to the valuation to be included in the valuation process. The proposed methodology is illustrated by means of its application to a case of individual and group valuation of an agricultural asset in the La Ribera district, Valencia (Spain).     

An action learning evaluation procedure for multiple criteria decision making problems

In this paper we focus on an extension of the Analytic Hierarchy Process (AHP) that accommodates ambiguity on the part of the decision maker (DM), and facilitates the exploration of the decision domain. We propose a systematic action learning process that builds confidence as it converges from numeric interval estimates to numeric point estimates. Our Multiple Criteria Decision Making (MCDM) problem procedure structures the problem as a hierarchy, evaluates all objects using pairwise comparisons that accommodate vagueness and ambiguity, uses interval prioritization techniques, and does synthesis using the linear additive value function. This action learning process facilitates the understanding of key stakeholders, which is imperative for the successful implementation of the subsequent decision.     

A probabilistic interpretation of the final rankings in AHP

The Analytic Hierarchy Process (AHP) has become a popular and practical tool for dealing with complex decision problems. It provides a ranking for the decision alternatives. This article recommends treating the pairwise comparison input data as random variables. This will allow the determination of whether the differences between alternatives are statistically significant.     

Farmland appraisal based on the analytic network process

This paper presents an application of the Analytic Network Process (ANP) to farmland appraisal. The purpose of this new methodology is to solve some of the drawbacks found in comparative and capitalisation methods, called classical appraisal methods, which cannot deal with contexts where only partial information is available and/or qualitative variables are used. The ANP is a method based on the Multiple Criteria Decision Analysis (MCDA). Previous works have already applied other MCDA techniques to the appraisal context, such as the Analytic Hierarchy Process (AHP), however they have not been able to handle all the complexities of many real world appraisal problems. The ANP provides a more accurate approach for modelling complex environment because it allows the general study of the quantitative and qualitative explanatory variables of the price and the incorporation of feedback and interdependence relationships among variables. The new proposed methodology has been applied to a case study of a farm located in Valencia (Spain) in order to demonstrate its goodness. Both quantitative and qualitative variables, such as the age of the trees, productivity or water quality, have been considered to assess the market value of the farm. Six farms from the same region have been selected as reference assets. The appraisal problem has been solved in three different ways in order to study the influence of each model on the value of the problem farm. In this study it has been proved that the more information is incorporated into the model, the higher accuracy of the solution. From the results of this work we can conclude that the approach proposed stands out as a good alternative to current farmland appraisal approaches, as it has proven to be useful when data are only partially available, qualitative variables are used and influences among the explanatory variables are present.     

模糊广义判断矩阵的一致性检验及合成排序

决策评价过程中往往包含诸多不确定性、随机性和模糊性,广义判断下的ＡＨＰ－ＧＪＡＨＰ是一种广义ＡＨＰ,Ｆｕｚｚｙ 环境下的ＧＪＡＨＰ决策方法是应用集值统计的方法,在区间判断标度基础上确定模糊判断矩阵元素的正模糊数表示,并根据模糊集理论的扩展原理,求得Ｆｕｚｚｙ 环境下的模糊排序权值向量。本文给出模糊广义判断矩阵的一致性定义,讨论了各类判断形式条件下的一致性检验法与Ｆｕｚｚｙ 环境下递阶层次结构中的合成排序问题     

基于DEA-ANP的快递企业绩效评价研究

在建立快递企业绩效评价模型时,提出了利用DEA和ANP相结合对快递企业绩效进行评价的方法.不同于之前研究,文中提出的方法使用ANP替换AHP,充分考虑了物流行业的"效益背反"现象.其解决了AHP确定权重时,无法体现元素之间关联性的问题,单独使用DEA评价时,无法考虑决策者偏好的弊端.首先,方法基于.ANP方法,求出每个指标的权重,再分别对每个因素使用DEA方法求出各系统的相对效率值;然后,将各指标权重和相对效率值结合,求出各快递企业中营业部的整体效率值并进行排序;最后通过实例分析,验证了此方法的实用性、可操作性及优越性.     

A statistical approach to measure the consistency level of the pairwise comparison matrix

The consistency of the pairwise comparison matrix (PCM) has been extensively studied in the Analytic Hierarchy Process. Most of the existing approaches for the consistency test of the PCM are based on the consistency ratio threshold (0.1) introduced by Saaty (1977). To accurately measure the consistency level of the PCM, a statistical approach based on the significance level is proposed in this paper by combining the hypothesis test and the random consistency index. The proposed statistical approach is applied to an education evaluation problem in order to demonstrate the rationality and reliability of it.     

A fuzzy extension of Analytic Hierarchy Process based on the constrained fuzzy arithmetic

The aim of the paper is to highlight the necessity of applying the concept of constrained fuzzy arithmetic instead of the concept of standard fuzzy arithmetic in a fuzzy extension of Analytic Hierarchy Process (AHP). Emphasis is put on preserving the reciprocity of pairwise comparisons during the computations. For deriving fuzzy weights from a fuzzy pairwise comparison matrix, we consider a fuzzy extension of the geometric mean method and simplify the formulas proposed by Enea and Piazza (Fuzzy Optim Decis Mak 3:39–62, 2004). As for the computation of the overall fuzzy weights of alternatives, we reveal the inappropriateness of applying the concept of standard fuzzy arithmetic and propose the proper formulas where the interactions among the fuzzy weights are taken into account. The advantage of our approach is elimination of the false increase of uncertainty of the overall fuzzy weights. Finally, we advocate the validity of the proposed fuzzy extension of AHP; we show by an illustrative example that by neglecting the information about uncertainty of intensity of preferences we lose an important part of knowledge about the decision making problem which can cause the change in ordering of alternatives.     

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.     

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.     

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.     

基于广义判断形式的模糊排序方法

广义判断下的ＡＨＰ（ＧＪＡＨＰ）是一种广义ＡＨＰ。它是在研究不完全信息下的决策排序问题时，通过构造广义判断矩阵的数学模型而建立的一种广义ＡＨＰ。本文应用集值统计的方法，在区间判断标度基础上确定模糊判断矩阵元素的正模糊数表示。给出了基于模糊区间数排序权值向量的特征根算法。讨论了Ｆｕｚｚｙ环境下求解各种判断形式的模糊排序权值向量的方法。     

基于模糊综合评判的航空公司战略联盟合作伙伴选择研究

航空公司战略联盟成功的关键是能否选择出理想的合作伙伴,这是一个复杂的决策问题.提出了一个比较新颖的算法.首先根据AHP方法确定模糊评价矩阵和权重向量,并针对航空联盟的敏捷性要求和特定的市场需求,提出了评价指标两两间的比较标度的修正方法,然后通过模糊运算对目标函数进行排序,实现对合作伙伴的最优选择.最后以实例表明本算法能有效的支持伙伴选择.