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.     

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.     

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.     

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.     

Deriving crisp and consistent priorities for fuzzy AHP-based multicriteria systems using non-linear constrained optimization

Fuzzy optimization models are used to derive crisp weights (priority vectors) for the fuzzy analytic hierarchy process (AHP) based multicriteria decision making systems. These optimization models deal with the imprecise judgements of decision makers by formulating the optimization problem as the system of constrained non linear equations. Firstly, a Genetic Algorithm based heuristic solution for this optimization problem is implemented in this paper. It has been found that the crisp weights derived from this solution for fuzzy-AHP system, sometimes lead to less consistent or inconsistent solutions. To deal with this problem, we have proposed a consistency based constraint for the optimization models. A decision maker can set the consistency threshold value and if the solution exists for that threshold value then crisp weights can be derived, otherwise it can be concluded that the fuzzy comparison matrix for AHP is not consistent for the given threshold. Three examples are considered to demonstrate the effectiveness of the proposed method. Results with the proposed constraint based fuzzy optimization model are more consistent than the existing optimization models.     

The extended linear assignment method for multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets

The theory of interval-valued intuitionistic fuzzy sets is useful and beneficial for handling uncertainty and imprecision in multiple criteria decision analysis. In addition, the theory allows for convenient quantification of the equivocal nature of human subjective assessments. In this paper, by extending the traditional linear assignment method, we propose a useful method for solving multiple criteria evaluation problems in the interval-valued intuitionistic fuzzy context. A ranking procedure consisting of score functions, accuracy functions, membership uncertainty indices, and hesitation uncertainty indices is presented to determine a criterion-wise preference of the alternatives. An extended linear assignment model is then constructed using a modified weighted-rank frequency matrix to determine the priority order of various alternatives. The feasibility and applicability of the proposed method are illustrated with a multiple criteria decision-making problem involving the selection of a bridge construction method. Additionally, a comparative analysis with other methods, including the approach with weighted aggregation operators, the closeness coefficient-based method, and the auxiliary nonlinear programming models, has been conducted for solving the investment company selection problem to validate the effectiveness of the extended linear assignment method.     

模糊判断矩阵排序向量的确定方法研究

首先给出模糊判断矩阵的两种一致性的定义。然后分析现有确定模糊判断矩阵排序向量的方法的特点及存在的问题，在此基础上，系统研究确定模糊判断矩阵排序向量的两类方法，第一类方法是先将模糊判断矩阵转化为AHP判断矩阵，然后将后者的排序向量作为前者的排序向量；另一类方法是直接由一致性或具有满意一致性的模糊判断矩阵计算排序向量。最后用算例说明所提出方法的应用。     

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.     

Fuzzy AHP中的一种加权群体决策方法

考虑在一些群体决策问题中,由于决策者个人的经验、才知、权力等因素的不同,因而拥有不同决策权重的情况,提出Fuzzy环境中层次分析法的一种加权群体决策模型,同时还给出由Fuzzy判断矩阵计算被比较元素的Fuzzy排序权重的几种方法,并给出一个应用该群体决策模型的例子。     

Research on AHP with interval-valued intuitionistic fuzzy sets and its application in multi-criteria decision making problems

This paper investigates an approach for multi-criterion decision making (MCDM) problems with interval-valued intuitionistic fuzzy preference relations (IVIFPRs). Based on the novel interval score function, some extended concepts associated with IVIFPRs are defined, including the score matrix, the approximate optimal transfer matrix and the possibility degree matrix. By using these new matrixes, a prioritization method for IVIFPRs is proposed. Then, we investigate an interval-valued intuitionistic fuzzy AHP method for multi-criteria decision making (MCDM) problems. In the end, a numerical example is provided to illustrate the application of the proposed approach.     

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.     

基于模糊判断矩阵满意一致性的方案排序研究

应用模糊判断矩阵的完全一致性进行多属性方案排序因其条件较苛刻，有时会存在与专家原始判断意见偏离较大的缺陷。为此本文提出了一种基于满意一致性的排序新方法。首先提出了顺序模糊判断矩阵的概念，证明了任何满足满意一致性的模糊判断矩阵均存在顺序模糊判断矩阵。然后给出了顺序模糊判断矩阵的影子矩阵所具有的性质，并且根据这些性质对满足满意一致性的模糊判断矩阵提出了方案排序算法，最后进行了算例分析。从分析可知：这种基于满意一致性进行排序的算法不仅简便、实用，而且更符合专家的原始判断。     

An analysis of distributions of priority values from alternative comparison scales within AHP

The analytic hierarchy process (AHP) introduced by T.L. Saaty is a well known and popular method of multi-criteria decision making. Central to this method are the pairwise comparisons between criteria (and decision alternatives) made using a 9-unit scale. The appropriateness of Saaty's original one-to-nine (1–9) scale has been the subject of much debate and cause for concern. This paper contrasts the appropriateness of the 1–9 scale with other alternative 9-unit scales also used in AHP, by looking at the probability distributions of the associated priority values. For large problems, estimated probability distributions are found for the priority values through using the method of Parzen Windows.     

毕达哥拉斯群决策方法及其在绿色供应商选取中的应用

针对专家权重未知且属性值为毕达哥拉斯模糊数的多属性群决策问题,基于证据理论和混合加权毕达哥拉斯MSM算子,提出了一种群决策方法。 首先,由决策信息矩阵获取专家的模糊测度,并赋予其相应的权重;其次,基于新构造的混合加权毕达哥拉斯MSM算子对专家所提供的属性信息分别进行集结,得到各个专家的综合评价信息;再次,利用证据合成方法,对专家综合评价信息进行融合,获得候选方案的综合证据信息,进而可知备选方案的信任区间,并据此对候选方案进行优选决策;最后,绿色供应商选取案例的分析与对比验证了方法的可行性与合理性。     

2-order additive fuzzy measure identification method based on diamond pairwise comparison and maximum entropy principle

Fuzzy measures can flexibly describe the relative importance of decision criterion as well as their interactions in multicriteria decision making. Based on the diamond pairwise comparison, a new identification method of 2-order additive fuzzy measure is proposed. The relative weight and the interaction degree can be obtained simultaneously for every pair of criteria in the diamond pairwise comparison. The Choquet integral-based equivalent alternative curve can help the decision maker estimate the interaction degrees between criteria. The overall importance of each criterion is obtained by the maximum eigenvector method of AHP. According to the maximum fuzzy measure entropy principal, a nonlinear programming is constructed to identify the interaction indices among criteria. Finally, an illustrative example shows the feasibility and validity of the proposed identification method.     

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     

A cosine maximization method for the priority vector derivation in AHP

The derivation of a priority vector from a pair-wise comparison matrix (PCM) is an important issue in the Analytic Hierarchy Process (AHP). The existing methods for the priority vector derivation from PCM include eigenvector method (EV), weighted least squares method (WLS), additive normalization method (AN), logarithmic least squares method (LLS), etc. The derived priority vector should be as similar to each column vector of the PCM as possible if a pair-wise comparison matrix (PCM) is not perfectly consistent. Therefore, a cosine maximization method (CM) based on similarity measure is proposed, which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a PCM. An optimization model for the CM is proposed to derive the reliable priority vector. Using three numerical examples, the CM is compared with the other prioritization methods based on two performance evaluation criteria: Euclidean distance and minimum violation. The results show that the CM is flexible and efficient.