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.     

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.     

Some applications of the analytic hierarchy process

In this paper we describe our experience using the Analytic Hierarchy Process in the evaluation of drug effectiveness in medicine, wine tasting and tea production, selection of team members in sports, forecasting in the industrial sector and service and trade. Each of the applications presented involves the quantification of qualitative information.     

A new macroeconomic forecasting and policy evaluation method using the analytic hierarchy process

The Analytic Hierarchy Process is used to show how forecasts can be made of the effects of monetarist, Keynesian and supply-side macroeconomic policies and to determine their impact on important variables such as unemployment, inflation and GNP growth.     

Hilbert's metric and the analytic hierarchy process

This paper explores some of the properties of Hilbert's projective metric as a measure of closeness between two ratio scales in the context of the Analytic Hierarchy Process. Smallperturbation arguments are used to contrast the sensitivity and the distributional behavior of this metric with the more traditional Euclidean distance function, in situations where the paired comparison of alternatives is subject to random perturbations, and priorities are estimated either by Saaty's eigenvalue method or by the logarithmic least squares principle. A pivotal property of Hilbert's metric has surfaced which allows for the construction of confidence regions for an underlying priority vector. These regions are seen to enjoy good coverage properties.     

The analytic hierarchy process in medical and health care decision making: A literature review

This paper presents a literature review of the application of the analytic hierarchy process (AHP) to important problems in medical and health care decision making. The literature is classified by year of publication, health care category, journal, method of analyzing alternatives, participants, and application type. Very few articles were published prior to 1988 and the level of activity has increased to about three articles per year since 1997. The 50 articles reviewed were classified in seven categories: diagnosis, patient participation, therapy/treatment, organ transplantation, project and technology evaluation and selection, human resource planning, and health care evaluation and policy. The largest number of articles was found in the project and technology evaluation and selection category (14) with substantial activity in patient participation (9), therapy/treatment (8), and health care evaluation and policy (8). The AHP appears to be a promising support tool for shared decision making between patient and doctor, evaluation and selection of therapies and treatments, and the evaluation of health care technologies and policies. We expect that AHP research will continue to be an important component of health care and medical research.     

Perturbation of analytic hermitian matrix functions

The behaviour of real eigenvalues of selfadjoint analytic matrix valued functions under small selfadjoint analytic perturbations is studied. Attention is paid mainly to the case when the perturbation is definite (or semidefi-nite). Earlier results of the authors concerning matrix polynomials of first degree are extended to the case of analytic functions.     

On modeling dynamic priorities in the analytic hierarchy process using compositional data analysis

In a rapidly changing environment, the priorities derived using the analytic hierarchy process (AHP) approach at one point in time might very likely change in the near future. Thus, in order to adapt to such ever-changing environment, it is of primary importance to be able to follow the change over time as to enable the system to respond differently and continuously over time of its operation. This paper proposes the use of a time-based compositional forecasting method, which is based on the idea of exponential smoothing, to deal with the AHP priority dynamics. The proposed method is particularly useful when there is a limited number of historical data, and might be considered to be more effective and time-efficient compared to that of multivariate time series method. It was also shown that the proposed method provides much greater adaptability in modeling the AHP priorities change over time compared to that of recently developed methods in compositional data research field. The shortcoming of Saaty’s dynamic judgment approach and some limitations of the other existing methods will be discussed. Finally, to substantiate the validity of the proposed method and to give some practical insights, an illustrative case study is provided.     

A utility approach to the analytic hierarchy process

This paper attempts to examine the utility foundation of the Analytic Hierarchy Process (AHP). It identifies the conditions under which the selection of an alternative is consistent with the maximization of an underlying utility function, or more precisely, the conditions under which the AHP-recommended choice corresponds with the solution attained from maximizing the respondent's utility function.     

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.     

Perturbation of the generating operator of an analytic semigroup

Voronezh Military Aviation Engineering Academy. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 1, pp. 62–63, January–March, 1988.     

The analytic hierarchy process with stochastic judgements

The analytic hierarchy process (AHP) is a widely-used method for multicriteria decision support based on the hierarchical decomposition of objectives, evaluation of preferences through pairwise comparisons, and a subsequent aggregation into global evaluations. The current paper integrates the AHP with stochastic multicriteria acceptability analysis (SMAA), an inverse-preference method, to allow the pairwise comparisons to be uncertain. A simulation experiment is used to assess how the consistency of judgements and the ability of the SMAA-AHP model to discern the best alternative deteriorates as uncertainty increases. Across a range of simulated problems results indicate that, according to conventional benchmarks, judgements are likely to remain consistent unless uncertainty is severe, but that the presence of uncertainty in almost any degree is sufficient to make the choice of best alternative unclear.     

The approach to consistency in the analytic hierarchy process

In his first book on the Analytic Hierarchy Process, T. L. Saaty left open several mathematical questions about the structure of the set of positive reciprocal matrices. In this paper we consider three of these questions: Given an eigenvector and all matrices which give rise to it, can one go from one of them to any order by making small perturbations in the entries? Given two positive column vectors v and w is there a perturbation which carries the set of all positive reciprocal matrices with principal right eigenvector v to the set of positive reciprocal matrices with principal right eigenvector w? Does the set of positive reciprocal n×n matrices whose left and right principal eigenvectors are reciprocals coincide with the set of consistent matrices for n⩾4?     

How to handle dependence with the analytic hierarchy process

Hierarchic and network systems are discussed as basic frameworks of unstructured problems modeled by the Analytic Hierarchy Process. A hierarchy represents a linear chain of interactions, whereas a network allows for feedback in the form of cycles and loops. A theory is provided for the priorities of a network system of which those of a hierarchy are shown to be a special case. Practical applications are illustrated.     

Using the analytic hierarchy process in a dynamic environment

The Analytic Hierarchy Process (AHP) is a decision analysis technique that uses judgements from a group of relevant decision makers along with hierarchical decomposition to derive a set of ratioscaled measures for decision alternatives. This paper addresses implementation issues for the AHP when the alternatives become available to the decision maker sequentially rather than simultaneously. Uncertainty about the value of future alternatives and the number of alternatives is included. We present a technique similar to the classic “secretary problem” of operations research and describe some sample results of using this technique. The procedure involves prioritizing criteria of possible alternatives before the alternatives became available, scoring the alternatives and then comparing the score of an alternative with an easily computed (through a dynamic programming recursion) critical value.     

The harmonic consistency index for the analytic hierarchy process

A new consistency measure, the harmonic consistency index, is obtained for any positive reciprocal matrix in the analytic hierarchy process. We show how this index varies with changes in any matrix element. A tight upper bound is provided for this new consistency measure when the entries of matrix are at most 9, as is often recommended. Using simulation, the harmonic consistency index is shown to give numerical values similar to the standard consistency index but it is easier to compute and interpret. In addition, new properties of the column sums of reciprocal matrices are obtained.     

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.     

On the use of information criteria in analytic hierarchy process

In this article, we discuss how the model-selection procedures such as Akaike's information criteria (AIC) can be used for selecting the most appropriate one out of several existing statistical models in the literature for the judgment data used in analytic hierarchy process (AHP). Furthermore, once the appropriate model is selected, a procedure is proposed on the basis of AIC for statistical ranking of the alternatives. This ranking procedure does not suffer from the problem of intransitivity and can be based on non-normal distribution. It enables one to obtain the detailed pattern for the ordered priorities of the alternatives in the decision process involving AHP.     

Characteristics of positive reciprocal matrices in the analytic hierarchy process

Positive reciprocal matrices (PRMs) are the basic elements used by the Analytic Hierarchy Process (AHP) for resolving an important class of multi-criteria decision problems. A PRM, A=(a _ij ), is square with all a _ij >0 and a _ji =1/a _ij . We discuss characteristics of such matrices based on an analysis of both real-world and randomly generated sets.