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.     

An intelligent decomposition of pairwise comparison matrices for large-scale decisions

A Pairwise Comparison Matrix (PCM) has been used to compute for relative priorities of elements and are integral components in widely applied decision making tools: the Analytic Hierarchy Process (AHP) and its generalized form, the Analytic Network Process (ANP). However, PCMs suffer from several issues limiting their applications to large-scale decision problems. These limitations can be attributed to the curse of dimensionality, that is, a large number of pairwise comparisons need to be elicited from a decision maker. This issue results to inconsistent preferences due to the limited cognitive powers of decision makers. To address these limitations, this research proposes a PCM decomposition methodology that reduces the elicited pairwise comparisons. A binary integer program is proposed to intelligently decompose a PCM into several smaller subsets using interdependence scores among elements. Since the subsets are disjoint, the most independent pivot element is identified to connect all subsets to derive the global weights of the elements from the original PCM. As a result, the number of pairwise comparison is reduced and consistency is of the comparisons is improved. The proposed decomposition methodology is applied to both AHP and ANP to demonstrate its advantages.     

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.     

A comparative study of the numerical scales and the prioritization methods in AHP

In the analytic hierarchy process (AHP), a decision maker first gives linguistic pairwise comparisons, then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them, and finally derives a priority vector from the numerical pairwise comparisons. In particular, the validity of this decision-making tool relies on the choice of numerical scale and the design of prioritization method. By introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices (LPCMs), and by defining the deviation measures of LPCMs, we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods. Using these performance measure algorithms, we compare the most common numerical scales (the Saaty scale, the geometrical scale, the Ma–Zheng scale and the Salo–Hämäläinen scale) and the prioritization methods (the eigenvalue method and the logarithmic least squares method). In addition, we also discuss the parameter of the geometrical scale, develop a new prioritization method, and construct an optimization model to select the appropriate numerical scales for the AHP decision makers. The findings in this paper can help the AHP decision makers select suitable numerical scales and prioritization methods.     

On the extraction of weights from pairwise comparison matrices

We study properties of weight extraction methods for pairwise comparison matrices that minimize suitable measures of inconsistency, ‘average error gravity’ measures, including one that leads to the geometric row means. The measures share essential global properties with the AHP inconsistency measure. By embedding the geometric mean in a larger class of methods we shed light on the choice between it and its traditional AHP competitor, the principal right eigenvector. We also suggest how to assess the extent of inconsistency by developing an alternative to the Random Consistency Index, which is not based on random comparison matrices, but based on judgemental error distributions. We define and discuss natural invariance requirements and show that the minimizers of average error gravity generally satisfy them, except a requirement regarding the order in which matrices and weights are synthesized. Only the geometric row mean satisfies this requirement also. For weight extraction we recommend the geometric mean.     

Scale transitivity in the AHP

Transitivity is important in multicriteria decision-making. The analytic hierarchy process (AHP), as one of the widely used decision analysis tools, is criticized since it suffers from scale intransitivity. This paper first reviews and compares different scales from different aspects, then discusses the transitivity of AHP scales and derives a scale based on the transitivity, so it is naturally transitive. Besides, two approaches are provided to determine the scale parameter for the derived transitive scale. In order to deal with the transitivity problem, the AHP provides a consistency index for testing pairwise comparison consistency. So, finally, this paper proposes a consistency measure to reflect the judgmental inconsistency.     

Priority theory and utility theory

The first axiom of the Analytic Hierarchy Process (AHP) states that the pairwise comparison judgements are reciprocal. Here we explore the relationships between the reciprocal property and preference relations with and without the axiom of transitivity and point out some important distinctions between utility theory and the AHP.     

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.     

Fractional Minmax Goal Programming: A Unified Approach to Priority Estimation and Preference Analysis in MCDM

A special minmax goal programming model with fractional goals is formulated and then used as the basis for developing two specific MCDM methods: (1) a method to derive priorities for decision elements from pairwise comparison matrices in the AHP framework; (2) a method to assess an additive value function by analysing some preference information expressed over the set of alternatives on a ratio scale. For both methods a common iterative solution procedure is proposed by using a linear programming formulation.     

Interval cross efficiency for fully ranking decision making units using DEA/AHP approach

Data envelopment analysis (DEA) is a popular technique for measuring the relative efficiency of a set of decision making units (DMUs). Fully ranking DMUs is a traditional and important topic in DEA. In various types of ranking methods, cross efficiency method receives much attention from researchers because it evaluates DMUs by using self and peer evaluation. However, cross efficiency score is usual nonuniqueness. This paper combines the DEA and analytic hierarchy process (AHP) to fully rank the DMUs that considers all possible cross efficiencies of a DMU with respect to all the other DMUs. We firstly measure the interval cross efficiency of each DMU. Based on the interval cross efficiency, relative efficiency pairwise comparison between each pair of DMUs is used to construct interval multiplicative preference relations (IMPRs). To obtain the consistency ranking order, a method to derive consistent IMPRs is developed. After that, the full ranking order of DMUs from completely consistent IMPRs is derived. It is worth noting that our DEA/AHP approach not only avoids overestimation of DMUs’ efficiency by only self-evaluation, but also eliminates the subjectivity of pairwise comparison between DMUs in AHP. Finally, a real example is offered to illustrate the feasibility and practicality of the proposed procedure.     

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.     

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.     

Ranking of Sports Teams via the AHP

This paper presents an application of the analytic hierarchy approach (AHP) to predict the ranking of the 16 soccer teams of the Israeli National League. Six criteria (attributes) were used for evaluating each team: the team's facility, the team's trainer, the players' level, the team's fans, the previous season's performance and the current performance. For each criterion a matrix of pairwise comparisons between teams was evaluated by a sports expert. The normalized eigenvector of each matrix scales the teams with regard to that criterion. The overall scaling is constructed by weighting the scales of all the criteria. The weight of the criteria are extracted from a judgemental matrix of pairwise comparisons between every two criteria. The normalized eigenvector of this matrix provides the criteria weights. Some comparisons to simpler ranking methods were made in order to validate the model.     

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.     

一种基于决策矩阵的DST-AHP多属性决策方法

针对层次分析法决策时存在两两判断、一致性检验次数过多和判断矩阵残缺性等问题,本文提出了一种基于决策矩阵的DST-AHP多属性决策方法。该方法结合决策矩阵的特征值,构建DST-AHP方法层次结构模型和判断矩阵,并根据判断矩阵定义不同属性下各焦元的基本概率分配函数;然后利用Dempster合成法则对基本概率分配函数值进行合成,依据合成后值对方案进行排序。最后对AHP和DST-AHP两种方法进行比较分析,说明该方法的有效性。     

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.     

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 fuzzy ranking method with range reduction techniques

For ranking alternatives based on pairwise comparisons, current analytic hierarchy process (AHP) methods are difficult to use to generate useful information to assist decision makers in specifying their preferences. This study proposes a novel method incorporating fuzzy preferences and range reduction techniques. Modified from the concept of data envelopment analysis (DEA), the proposed approach is not only capable of treating incomplete preference matrices but also provides reasonable ranges to help decision makers to rank decision alternatives confidently.     

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.     

Analytic hierarchy process for multi-sensor data fusion based on belief function theory

Multi-sensor data fusion is an evolving technology whereby data from multiple sensor inputs are processed and combined. The data derived from multiple sensors can, however, be uncertain, imperfect, and conflicting. The present study is undertaken to help contribute to the continuous search for viable approaches to overcome the problems associated with data conflict and imperfection. Sensor readings, represented by belief functions, have to be fused according to their corresponding weights. Previous studies have often estimated the weights of sensor readings based on a single criterion. Mono-criteria approaches for the assessment of sensor reading weights are, however, often unreliable and inadequate for the reflection of reality. Accordingly, this work opts for the use of a multi-criteria decision aid. A modified Analytical Hierarchy Process (AHP) that incorporates several criteria is proposed to determine the weights of a sensor reading set. The approach relies on the automation of pairwise comparisons to eliminate subjectivity and reduce inconsistency. It assesses the weight of each sensor reading, and fuses the weighed readings obtained using a modified average combination rule. The efficiency of this approach is evaluated in a target recognition context. Several tests, sensitivity analysis, and comparisons with other approaches available in the literature are described.