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.     

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.     

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 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.     

Implementing the balanced scorecard using the analytic hierarchy process &amp; the analytic network process

The balanced scorecard (BSC) is a multi-attribute evaluation concept that highlights the importance of non-financial attributes. By incorporating a wider set of non-financial attributes into the measurement system of a firm, the BSC captures not only a firm's current performance, but also the drivers of its future performance. Although there is an abundance of literature on the BSC framework, there is a scarcity of literature on how the framework should be properly implemented. In this paper, we use the analytic hierarchy process (AHP) and its variant the analytic network process (ANP) to facilitate the implementation of the BSC. We show that the AHP and the ANP can be tailor-made for specific situations and can be used to overcome some of the traditional problems of BSC implementation, such as the dependency relationship between measures and the use of subjective versus objective measures. Numerical examples are included throughout.     

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.     

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.     

Examining alternatives in the interval analytic hierarchy process using complete enumeration

The analytic hierarchy process as an aid to decision makers has gained in popularity over the years. Recently the procedure has been generalised into the interval analytic hierarchy process. Rather than select a precise ratio for the comparisons between the alternatives offered to the decision maker an interval is proposed. One popular strategy in examining this problem is to adopt a simulation technique. Here two simulation methods are considered and it is demonstrated that total enumeration is a viable alternative.     

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.     

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.     

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.     

Scales and methods for deriving weights in the analytic hierarchy process

Scales and methods for deriving weights of objects being compared used in the analytic hierarchy process (AHP) are theoretically studied. Results are formulated as theorems, which can be applied by a decision maker to choose AHP parameters in solving practical problems.     

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.     

A modified procedure for synthesising ratio judgements in the analytic hierarchy process

The analytic hierarchy process (AHP) has been widely applied to solve problems arising in group decision making, by synthesising different or conflicting judgements. However, directly synthesising conflicting judgements by calculating the geometric mean of preference weights (ratios) in AHP may not reach consensus from all members in a decision making group, especially, when those members represent the stakeholders of the decision making problem. This study proposes a new method that uses the genetic algorithm and utility function to synthesise preference weights to prevent this fallacy occurring when implementing the classical AHP approach. Using the proposed method, the final decision can be achieved with only minimally-adjusted preference weights.     

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.     

Clusters in a group: Decision making in the vector space formulation of the analytic hierarchy process

For a large group that might have a clustered structure, we discuss and implement an algorithm to group individuals into natural clusters using a convenient similarity measure. The cohesiveness of a homogeneous group or cluster is also investigated.     

Delphic hierarchy process (DHP): A methodology for priority setting derived from the Delphi method and analytical hierarchy process

Identifying of criteria and objectives is of paramount importance to a decision-making process and is the basis for a sound decision. A systematic approach, therefore, is needed so that the objectives of the organization not only can be identified but also prioritized so that the resources will be allocated to the relative importance of the objectives and how well the alternatives satisfy them. The methodology proposed here uses the Delphi method and integrates it with the analytic hierarchy process. It assists the decision maker(s) to systematically identify the organizational objectives and then to set priorities among them. The application of the proposed model is illustrated by a case study.     

An analytic framework for the two-dimensional infinite Toda hierarchy associated with a commutative algebra

Starting with the group of operators on a separable Hilbert space that differ from the identity by a trace-class operator, we construct a solution of the two-dimensional infinite Toda hierarchy associated with a maximal commutative subalgebra in complex k×k matrices. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 177–192, April, 2008.