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.     

Note on group consistency in analytic hierarchy process

We study the paper of Xu [Z. Xu, On consistency of the weighted geometric mean complex judgement matrix in AHP, European Journal of Operational Research 126 (2000) 683–687] for the group consistency in analytic hierarchy process of multicriteria decision-making. The purpose of this note is threefold. First, we point out the questionable results in this paper. Second, for three by three comparison matrices, we provide a patchwork for his method. Third, we constructed a counter example to show that in general his method is wrong. Numerical examples are provided to illustrate our findings. If there are four or more alternatives, then we may advise researchers to ignore his results to avoid questionable estimation of group consistency.     

A consistency improving method in the analytic hierarchy process

In this paper, we propose a method to modify a given comparison matrix, by which the consistency ratio (CR) value of the modified matrix is less than that of the original one, and give an algorithm to derive a positive reciprocal matrix with acceptable consistency (i.e., CR < 0.1), then the convergence theorem for the given algorithm is established and its practicality is shown by some examples.     

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.     

Acceptable consistency of aggregated comparison matrices in analytic hierarchy process

The analytic hierarchy process is a method for solving multiple criteria decision problems, as well as group decision making. The weighted geometric mean method is appropriate when aggregation of individual judgements is used. This paper presents a new proof which confirms the property that if the comparison matrices of all decision makers are of acceptable consistency, then the weighted geometric mean complex judgement matrix (WGMCJM) also is of acceptable consistency. This property was presented and first proved by Xu (2000), but Lin et al. (2008) rejected the proof. We also discuss under what conditions the WGMCJM is of acceptable consistency when not all comparison matrices of decision makers are of acceptable consistency. For this case we determine the sufficient condition for the WGMCJM to be of acceptable consistency and provide numerical examples. For a special case of two decision makers with 3 × 3 comparison matrices we find out some additional conditions for the WGMCJM to be of acceptable consistency.     

Reducing inconsistency measured by the geometric consistency index in the analytic hierarchy process

This paper presents a theoretical framework and a procedure for revising the judgements and improving the inconsistency of an Analytic Hierarchy Process (AHP) pairwise comparison matrix when the Row Geometric Mean (RGM) is used as the prioritisation procedure and the Geometric Consistency Index (GCI) is the inconsistency measure. Inconsistency is improved by slightly modifying the judgements that further reduce the GCI. Both the judgements and the derived priority vector will be close to the initial values. A simulation study is utilised to analyse the performance of the algorithm. The proposed framework allows the specification of the procedure to particular interests. A numerical example illustrates the proposed procedure.     

A simulation approach for handling uncertainty in the analytic hierarchy process

Uncertainty considerations are introduced into the analytic hierarchy process (AHP). The rank order of decision alternatives depends on two types of related uncertainties: (1) uncertainty regarding the future characteristics of the decision making environment described by a set of scenarios, and (2) uncertainty associated with the decision making judgment regarding each pairwise comparison. A simulation approach for handling both types of related uncertainties in the AHP is described. The example introduced by Saaty and Kearns (1985) is extended here to include uncertainty considerations.     

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.     

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.     

Modeling behavior in competition: The analytic hierarchy process

This paper offers an approach for dealing with prediction of the outcome of World Chess Championship matches based on players experience and attitude towards the game. The paper deals with both the overall outcome and the sequence of game by game outcomes. A method for predicting the overall outcome is advanced and illustrated. Methods for predicting game by game outcomes are examined and compared according to strengths and weaknesses. The analysis is supported by the data on World Championship matches since their beginning 125 years ago.     

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.     

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.     

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.     

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.     

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.     

A Thurstonian view of the analytic hierarchy process

The application of deterministic decision models in situations characterized by noise and uncertainty is likely to produce results of questionable value. In this paper, some very simple probabilistic models are developed and substituted for the deterministic scales used in the Analytic Hierarchy Process (AHP). It is shown that the use of these probabilistic models can extend the domain of AHP to situations, such as consensual or group decision making, that possess significant amounts of uncertainty. In addition, explicit measures of the variation present in the evaluation of decision alternatives and attributes are obtained.     

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.     

Applications of the analytic hierarchy process to long range planning processes

This paper considers the long range planning process from the point of view both of projecting forward likely or desired changes from the current position to define a possible future state, and of identifying desirable future states and working backwards from those to consider ways in which they could be reached. This process is repeated until consistency between the forward and backward looking processes is achieved. A theory of analytic hierarchies is used to give a formal structure of the process. An application to an actual corporate plan is briefly discussed.     

The specification of a reference direction using the analytic hierarchy process

In this paper we deal with the use of the Analytic Hierarchy Process (AHP) for specifying a reference direction, which is used to find a search direction in the visual interactive method developed by Korhonen and Laakso for multiple criteria problems. The reference direction describes how the decision maker would like to improve the values of multiple objectives and we show that the AHP is a convenient way to structure requisite preference information.