An enhanced goal programming method for generating priority vectors

Generating priority vectors from pairwise comparison matrices is an essential part of the analytical hierarchy process. Besides the well-known right eigenvector method and the logarithmic least squares method, Bryson proposed a goal programming method (GPM) for achieving the above. Although the GPM always obtains an optimal solution, this study shows the existence of concealed alternative optimal solutions that might lead to different priority vectors. This study proposed a GPM for overcoming the problem of alternative optimal solution. In contrast with the GPM, the proposed method obtains a unique optimal solution and performs better in various aspects.     

Inferring consensus weights from pairwise comparison matrices without suitable properties

Pairwise comparison is a popular method for establishing the relative importance of n objects. Its main purpose is to get a set of weights (priority vector) associated with the objects. When the information gathered from the decision maker does not verify some rational properties, it is not easy to search the priority vector. Goal programming is a flexible tool for addressing this type of problem. In this paper, we focus on a group decision-making scenario. Thus, we analyze different methodologies for getting a collective priority vector. The first method is to aggregate general pairwise comparison matrices (i.e., matrices without suitable properties) and then get the priority vector from the consensus matrix. The second method proposes to get the collective priority vector by formulating an optimization problem without determining the consensus pairwise comparison matrix beforehand.     

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.     

A parametric GP model dealing with incomplete information for group decision-making

We consider a group decision-making problem where preferences given by the experts are articulated into the form of pairwise comparison matrices. In many cases, experts are not able to efficiently provide their preferences on some aspects of the problem because of a large number of alternatives, limited expertise related to some problem domain, unavailable data, etc., resulting in incomplete pairwise comparison matrices. Our goal is to develop a computational method to retrieve a group priority vector of the considered alternatives dealing with incomplete information. For that purpose, we have established an optimization problem in which a similarity function and a parametric compromise function are defined. Associated to this problem, a logarithmic goal programming formulation is considered to provide an effective procedure to compute the solution. Moreover, the parameters involved in the method have a clear meaning in the context of group problems.     

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 fuzzy programming method for deriving priorities in the analytic hierarchy process

The estimation of the priorities from pairwise comparison matrices is the major constituent of the Analytic Hierarchy Process (AHP). The priority vector can be derived from these matrices using different techniques, as the most commonly used are the Eigenvector Method (EVM) and the Logarithmic Least Squares Method (LLSM). In this paper a new Fuzzy Programming Method (FPM) is proposed, based on geometrical representation of the prioritisation process. This method transforms the prioritisation problem into a fuzzy programming problem that can easily be solved as a standard linear programme. The FPM is compared with the main existing prioritisation methods in order to evaluate its performance. It is shown that it possesses some attractive properties and could be used as an alternative to the known prioritisation methods, especially when the preferences of the decision-maker are strongly inconsistent.     

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.     

Pairwise opposite matrix and its cognitive prioritization operators: comparisons with pairwise reciprocal matrix and analytic prioritization operators

The pairwise reciprocal matrix (PRM) of the analytic hierarchy/network process has been investigated by many scholars. However, there are significant queries about the appropriateness of using the PRM to represent the pairwise comparison. This research proposes a pairwise opposite matrix (POM) as the ideal alternative with respect to the human linguistic cognition of the rating scale of the paired comparison. Several cognitive prioritization operators (CPOs) are proposed to derive the individual utility vector (or priority vector) of the POM. Not only are the rigorous mathematical proofs of the new models demonstrated, but solutions of the CPOs are also illustrated by the presentation of graph theory. The comprehensive numerical analyses show how the POM performs better than the PRM. POM and CPOs, which correct the fallacy of the PRM associated with its prioritization operators, should be the ideal solutions for multi-criteria decision-making problems in various fields.     

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.     

Building a set of additive value functions representing a reference preorder and intensities of preference: GRIP method

We present a method called Generalized Regression with Intensities of Preference (GRIP) for ranking a finite set of actions evaluated on multiple criteria. GRIP builds a set of additive value functions compatible with preference information composed of a partial preorder and required intensities of preference on a subset of actions, called reference actions. It constructs not only the preference relation in the considered set of actions, but it also gives information about intensities of preference for pairs of actions from this set for a given decision maker (DM). Distinguishing necessary and possible consequences of preference information on the considered set of actions, GRIP answers questions of robustness analysis. The proposed methodology can be seen as an extension of the UTA method based on ordinal regression. GRIP can also be compared to the AHP method, which requires pairwise comparison of all actions and criteria, and yields a priority ranking of actions. As for the preference information being used, GRIP can be compared, moreover, to the MACBETH method which also takes into account a preference order of actions and intensity of preference for pairs of actions. The preference information used in GRIP does not need, however, to be complete: the DM is asked to provide comparisons of only those pairs of reference actions on particular criteria for which his/her judgment is sufficiently certain. This is an important advantage comparing to methods which, instead, require comparison of all possible pairs of actions on all the considered criteria. Moreover, GRIP works with a set of general additive value functions compatible with the preference information, while other methods use a single and less general value function, such as the weighted-sum.     

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.     

A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations

Decision makers (DMs)’ preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations. This paper proposes a chi-square method (CSM) for obtaining a priority vector from multiplicative and fuzzy preference relations. The proposed CSM can be used to obtain a priority vector from either a multiplicative preference relation (i.e. a pairwise comparison matrix) or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures. Theorems and algorithm about the CSM are developed. Three numerical examples are examined to illustrate the applications of the CSM and its advantages.     

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.     

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.     

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.     

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.     

Representing the strengths and directions of pairwise comparisons

We consider the problem of finding weights that well represent a set of pairwise multiplicative comparisons of a set of objects (as in the AHP and other methods). Our main contribution is a method for deriving such weights that takes into consideration not only the strengths of the pairwise comparisons, but also their directions. For example, if the comparison directions satisfy transitivity, then the weights produced by our method also satisfy transitivity (this is not always true for other methods). We also present a set of reasonable axioms for which our method is the (essentially) unique solution. Our method and axioms are closely related to those of Cook and Kress [Eur. J. Oper. Res. 37 (1988) 335]. Our method, like theirs, reduces to solving a linear program (hence it is different from the approach used in the AHP). For the special case that the comparison directions satisfy transitivity, our method is quite simple and reduces to performing a forward pass as in the critical path method.     

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.     

GPM(1,1,m)模型及其在中国能源消费预测中的应用

客观准确地预测能源消费,可以为政府制定社会经济发展政策提供重要参考.利用矩阵分析的思想研究了灰色预测模型的建模机理,提出了基于时间多项式的可拓形式GPM(1,1,m)模型,并分析了其理论意义.在此基础上,通过研究了时间多项式对模型参数和预测值的影响,推导了它们之间的定量关系,设计了实际建模中的优化方法和参数估计的一般形...     

Interval estimation of priorities in the AHP

This paper extends and modifies the Analytic Hierarchy Process (AHP) and the Synthetic Hierarchy Method (SHM) of priority estimation to accommodate random data in the pairwise comparison matrices. It employs a Cauchy distribution to describe the pairwise comparison of alternatives in Saaty matrices, and shows how to modify these matrices in order to handle random data. The use of random data yields Saaty matrices that are not reciprocally symmetrical. Several variants of the AHP are then modified (i) to accommodate reciprocally asymmetric matrices, and (ii) to allow each priority estimate to be expressed on an interval of possible values, rather than as a single discrete point. The merits of interval estimation are illustrated by an example.