On the Proper Interpretation and Impact of the AHP Axiom of Independence in Single-Criterion and Multiple-Criteria Problems

Some authors have proposed that the analytic hierarchy process (AHP) axiom of independence be relaxed to accommodate observations drawn from (a) examples of pairwise comparisons of alternatives in clusters in single-criterion AHP problems and (b) examples of problems in which the criteria have the same underlying measurement and both the achievement of the goal and the alternatives are measured objectively. We show that the illustrations given are actually single-criterion problems according to the AHP. Thus the AHP axiom of independence is inapplicable in both situations and therefore not violated. We also consider the consequence of failure to distinguish between a criterion as an attribute of alternatives and cluster of alternatives, the two being different in the hierarchic structure. Finally, we discuss transformable problems, which look like multicriteria problems but are actually single-criterion problems, and how failure to recognize this fact may lead to incorrect syntheses and false conclusions.     

On the measurement of preferences in the analytic hierarchy process

In this paper we apply multiattribute value theory as a framework for examining the use of pairwise comparisons in the analytic hierarchy process (AHP). On one hand our analysis indicates that pairwise comparisons should be understood in terms of preference differences between pairs of alternatives. On the other hand it points out undesirable effects caused by the upper bound and the discretization of any given ratio scale. Both these observations apply equally well to the SMART procedure which also uses estimates of weight ratios. Furthermore, we demonstrate that the AHP can be modified so as to produce results similar to those of multiattribute value measurement; we also propose new balanced scales to improve the sensitivity of the AHP ratio scales. Finally we show that the so-called supermatrix technique does not eliminate the rank reversal phenomenon which can be attributed to the normalizations in the AHP. © 1997 John Wiley & Sons, Ltd.     

A Statistical Hypothesis Testing Method for the Rank Ordering of the Priorities of the Alternatives in the Analytic Hierarchy Process

In analytic hierarchy process (AHP), a ratio scale (π₁, π₂, ⋯, π_t) for the priorities of the alternatives {T₁, T₂, ⋯, T_t} is used for a decision problem in which π_i/π_j is used to quantify the ratio of the priority of T_i to that of T_j. In a group decision‐making setup, the subjective estimates of π_i/π_j are obtained as entries of a pairwise comparison matrix for each member of the group. On the basis of these pairwise comparison matrices, one of the topics of interest in some situation is the total rank ordering of the priorities of the alternatives. In this article, a statistical method is proposed for testing a specific total rank ordering of the priorities of the alternatives. The method developed is then illustrated using numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Euclid: Strategic alternative assessment matrix

The vast amount of information that must be considered to solve inherently ill‐structured and complex strategic problems creates a need for tools to help decision makers (DMs) recognize the complexity of this process and develop a rational model for strategy evaluation. Over the last several decades, a philosophy and a body of intuitive and analytical methods have been developed to assist DMs in the evaluation of strategic alternatives. However, the intuitive methods lack a structured framework for the systematic evaluation of strategic alternatives while the analytical methods are not intended to capture intuitive preferences. Euclid is a simple and yet sophisticated multiobjective value analysis model that attempts to uncover some of the complexities inherent in the evaluation of strategic alternatives. The proposed model uses a series of intuitive and analytical methods including environmental scanning, the analytic hierarchy process (AHP), subjective probabilities, and the theory of displaced ideal, to plot strategic alternatives on a matrix based on their Euclidean distance from the ideal alternative. Euclid is further compared to the quantitative strategic planning matrix (QSPM) in a real world application. The information provided by the users shows that Euclid can significantly enhance decision quality and the DM's confidence. Euclid is not intended to replace the DMs, rather, it provides a systematic approach to support, supplement, and ensure the internal consistency of their judgments through a series of logically sound techniques. Copyright © 2003 John Wiley & Sons, Ltd.     

A proposal for handling missing data

A method for dealing with the problem of missing observations in multivariate data is developed and evaluated. The method uses a transformation of the principal components of the data to estimate missing entries. The properties of this method and four alternative methods are investigated by means of a Monte Carlo study of 42 computer-generated data matrices. The methods are compared with respect to their ability to predict correlation matrices as well as missing entries. The results indicate that whenever there exists modest intercorrelations among the variables (i.e., average off diagonal correlation above .2) the proposed method is at least as good as the best alternative (a regression method) while being considerably faster and simpler computationally. Models for determining the best alternative based upon easily calculated characteristics of the matrix are given. The generality of these models is demonstrated using the previously published results of Timm.     

A Comparison of Verbal and Numerical Judgments in the Analytic Hierarchy Process

In the Analytic Hierarchy Process (AHP), decision makers make pairwise comparisons of alternatives and criteria. The AHP allows to make these pairwise comparisons verbally or numerically. Although verbal statements are intuitively attractive for preference elicitation, there is overwhelming evidence that people have very different numerical interpretations of the same verbal expressions. This study explores the consequences of these differences for the quality of the AHP analysis. The results of the laboratory study with 180 participants confirm that the 1-to-9 conversion table, as is often used in the AHP, tends to overestimate differences in preference. Concerning the outcome of the AHP analysis the numerical mode shows slightly better results (not significant). Given the preference of many people for the verbal mode we conclude that if accuracy is not of the highest importance, the ease and comfort of verbal expressions may be worth the small loss in decision quality.     

A Comprehensive Literature Review of the Rank Reversal Phenomenon in the Analytic Hierarchy Process

During the last few decades, several multi‐criteria decision analysis methods have been proposed to help in selecting the best compromise alternatives. Among them, analytic hierarchy process (AHP) and its applications have attracted much attention from academics and practitioners. However, since the early 1980s, critics have raised questions regarding its proper use. One of them concerns the unacceptable changes in the ranks of the alternatives, called rank reversal, upon changing the structure of the decision. Several modifications were suggested to preserve ranks. In this paper, a classification scheme and a comprehensive literature review are presented in order to uncover, classify and interpret the current research on AHP methodologies and rank reversals. On the basis of the scheme, 61 scholarly papers from 18 journals are categorized into specific areas. The specific areas include the papers on the topics of adding/deleting alternatives and the papers published in adding/deleting criteria. The scholarly papers are also classified by (1) year of publication, (2) journal of publication, (3) authors' geographic location and (4) using the AHP in association with other methods. It is hoped that the paper can meet the needs of researchers and practitioners for convenient references of AHP methodologies and rank reversals and hence promote the future of rank reversal research. Copyright © 2012 John Wiley & Sons, Ltd.     

A Comment on ‘Rank Disagreement: A Comparison of Multi-Criteria Methodologies’

In a recent paper in this journal, Buede and Maxwell found reasonable agreement between MAVT and conventional AHP most preferred alternatives in a large-scale simulation. We show that MAVT would have had total agreement on preferred alternatives with two modified forms of AHP, referenced AHP and linking pin AHP, under simulation. © 1997 John Wiley & Sons, Ltd.     

Item Response Modeling of Paired Comparison and Ranking Data

The comparative format used in ranking and paired comparisons tasks can significantly reduce the impact of uniform response biases typically associated with rating scales. Thurstone's (1927, 1931) model provides a powerful framework for modeling comparative data such as paired comparisons and rankings. Although Thurstonian models are generally presented as scaling models, that is, stimuli-centered models, they can also be used as person-centered models. In this article, we discuss how Thurstone's model for comparative data can be formulated as item response theory models so that respondents' scores on underlying dimensions can be estimated. Item parameters and latent trait scores can be readily estimated using a widely used statistical modeling program. Simulation studies show that item characteristic curves can be accurately estimated with as few as 200 observations and that latent trait scores can be recovered to a high precision. Empirical examples are given to illustrate how the model may be applied in practice and to recommend guidelines for designing ranking and paired comparisons tasks in the future.     

Measurement scales and scale independence in the analytic hierarchy process

One approach to evaluate the relative performance of decision alternatives with respect to multiple criteria is provided by the analytic hierarchy process. The method is based on pairwise comparisons between attributes, and several numerical measurement scales for the ratio statements have been proposed. The choice of measurement scale is re‐examined, and new arguments supporting the measurement scale of geometric progression are derived. Separately from the measurement scale considerations, the effects of the scale parameter in geometric measurement scale are also studied. By using a regression model for pairwise comparisons data, it is shown that the statistical inference does not depend on the value of the scale parameter in the case of a single pairwise comparison matrix. It is also shown when the scale independence of statistical inference can be achieved in a decision hierarchy. This requires the use of the geometric‐mean aggregation rule instead of the traditional arithmetic‐mean aggregation. The results of the case study demonstrate that the measurement scale and the aggregation rule have potentially large impacts on decision support. Copyright © 2000 John Wiley & Sons, Ltd.     

Time complexity and consistency issues in using the AHP for making group decisions

Two different ways of using the AHP in making group decisions are compared and evaluated. The first method combines different experts’ opinions before applying an eigenvalue method to obtain final weights for decision alternatives. The second, in contrast, derives each expert's rating for the decision alternatives before combining them. Both methods take into account the relative importance of different experts in making decisions. Comparison and evaluation of these two methods are made by using two criteria: time complexity and consistency indices. A numerical example is provided to illustrate the use of these two methods, and results of a mathematical simulation are presented for comparing the time complexity in different-sized problems. © 1998 John Wiley & Sons, Ltd.     

Integration of Analytic Hierarchy Process with Regression Analysis to Identify Attractive Locations for Market Expansion

Selection of new geographies in which to expand is a key decision for businesses aspiring to go beyond the opportunities in the existing markets. The conventional approaches of market selection can only provide a set of systematic steps for problem solving without considering the relationships between the decision factors. Decision models based on statistical techniques are able to examine the relationship between decision factors but are unable to effectively assist decision makers in identifying the most promising market, particularly in terms of prioritizing across decision factors. Analytic Hierarchy Process (AHP) is a commonly used approach for choosing alternatives by prioritizing across multiple decision factors. The typical AHP modelling requires knowledge of criteria and/or alternatives along with their relative weights, generally elicited from field experts. Quite often, firms encounter situations where decision makers are aware of only the overall objective and a set of earmarked geographies for setting up market locations while being relatively unaware of decision criteria and relative weights. This precludes using AHP to identify promising market locations. This paper conceptualizes a market selection decision model that integrates AHP with statistical modelling techniques to identify the attractive market locations for the purpose of expansion. The model first uses principal component analysis and multiple regression to determine significant decision criteria and their weights. Thereafter, it applies AHP to prioritize the market locations across the decision criteria. This integrative approach is illustrated for identifying the attractive locations in rural markets for a steel firm in India. The major advantage of this approach is that unlike the existing models, it works in situations when firms have not enough knowledge about factors for evaluating alternative market locations. Another key advantage of the proposed model is that of economizing resources for data collection on variables representing decision factors. Copyright © 2012 John Wiley & Sons, Ltd.     

MCDM methods in strategic planning of forestry on state‐owned lands in Finland: applications and experiences

The forests in Finland have been under intensive planning for decades. Currently, mathematical programming is widely used in planning of wood production. Today's multi‐functional forestry, however, calls for more flexible decision support methods. MCDM tools have been used in responding to fresh planning challenges. For example, the Finnish Forest and Park Service, entrusted with the care of the vast majority of state‐owned natural resources in Finland, endeavours to produce large‐scale natural resource plans satisfying the needs of both economic, social, and ecological sustainability. Participatory approach is applied in the process. Several forestry applications of MCDM methods, particularly those making use of the AHP or the HIPRE program, have been presented. Also, the outranking methods ELECTRE and PROMETHEE have been tested. Due to the nature of forestry applications, statistical techniques for analysing uncertainties in pairwise comparisons and for utilizing interval judgement data have been developed to improve the usability of the AHP. Recently, a hybrid method called A'WOT, making use of the AHP and SWOT, was also introduced into strategic forest planning. This paper summarizes the experiences gained in applying a MAVT and two outranking methods in connection with a participatory natural resource planning process in Finland. In addition, some results of the method development work related to application needs are briefly presented. The details of the planning cases reviewed here have previously been presented in forestry journals. The purpose of this paper is not only to show how MCDM methods have been applied in forestry, but also to discuss the usability and usefulness of MCDM methods from the viewpoint of supporting forestry decision making—and how they might further be improved. Also, some perspectives for the future development work of MCDM applications in the field of natural resource management are focused on. As a conclusion, the use of more than just one MCDM method in a single planning process is seen usually recommendable. In addition, developing hybrid MCDM methods is regarded as a potential direction for future research. Also, closer co‐operation between method developers and appliers is called for to produce more useful applications. Copyright © 2002 John Wiley & Sons, Ltd.     

Identification and Small Sample Estimation of Thurstone's Unrestricted Model for Paired Comparisons Data

The interpretation of a Thurstonian model for paired comparisons where the utilities' covariance matrix is unrestricted proved to be difficult due to the comparative nature of the data. We show that under a suitable constraint the utilities' correlation matrix can be estimated, yielding a readily interpretable solution. This set of identification constraints can recover any true utilities' covariance matrix, but it is not unique. Indeed, we show how to transform the estimated correlation matrix into alternative correlation matrices that are equally consistent with the data but may be more consistent with substantive theory. Also, we show how researchers can investigate the sample size needed to estimate a particular model by exploiting the simulation capabilities of a popular structural equation modeling statistical package.     

The use of limit theorems in paired and triple comparison model building

The psychological framework proposed by Thompson and Singh (1967) is extended to obtain a statistical interpretation of the stimulus-response phenomenon permitting ties when comparing two stimuli. Using this interpretation, the generalized Bradley-Terry model (Rao and Kupper, 1967) for paired comparisons and the  and  for triple comparisons are derived as the appropriate asymptotic models for experiments permitting tied observations.     

An Interactive Approach for Multicriteria Decision Making Using a Tchebycheff Utility Function Approximation

Multicriteria decision making (MCDM) can provide an efficient mean for considering various and conflicting objectives to reveal the alternative that maximizes the decision maker's (DM) utility. In this paper, we propose a new interactive MCDM method for implicit alternatives to help a DM obtain a most preferred solution. We employ a Tchebycheff function to generate weights for objectives consistent with the DM's responses to pairwise comparisons between alternatives and present a mixed integer linear programming formulation to generate these weights. Thus, we approximate the DM's utility function by a Tchebycheff function and generate weights consistent with the DM's responses. We test our approach with different true utility functions on various sized multiple criteria linear programming problems. The computational results show that even with non‐Tchebycheff true utility functions, our method can generate alternatives very close to the optimal solution with few questions. The comparison of our results with other methods reveals its advantages. Copyright © 2013 John Wiley & Sons, Ltd.     

Performance oriented task-resource mapping and scheduling in fog computing environment

Resource allocation and task scheduling is a complex task in fog computing environment because of the inherent heterogeneity among the fog devices. The proposed work attempts to solve the problem by using the popular multi criteria decision making methods such as AHP and TOPSIS. The goal of this paper is to propose a model for performance oriented task - resource mapping in a fog computing environment. MIPS, RAM & storage, uplink latency, downlink latency, uplink bandwidth, downlink bandwidth, trust, cost per MIPS, cost per memory, cost per storage and cost per bandwidth are the various performance characteristics considered in this work for task – resource mapping. Two different multi-criteria decision making methods are employed in order to assess the performance characteristics of the fog devices. In the first method, Analytic Hierarchy Process (AHP) is used for both priority weight calculation and ranking of fog devices. In the second method, AHP is used for priority weight calculation, based on the weights yielded by AHP, Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) algorithm is executed in order to rank the fog devices. Then the fog devices can be allocated to the tasks based on its rank. Furthermore, a motivational example is also demonstrated to validate the proposed method. Simulation results show that the proposed technique exhibits superior performance over other scheduling algorithms in the fog environment by incorporating performance, security, and cost metrics into scheduling decisions.     

Hierarchical modeling of paired comparison data

The method of paired comparisons belongs to a small group of techniques that provide explicit information about the consistency of individual and aggregated choices. This article investigates the link between the individual- and group-level judgments by extending R. D. Luce's (1959) model, which was originally developed for individual choice behavior, to a mixed-effects paired comparison model. It is shown that standard multilevel software for binary data can be used to estimate the model. The interpretation of the paired comparison parameters and statistical model tests are discussed in detail. An extensive analysis of an experimental study illustrates the usefulness of a hierarchical approach in modeling multiple pairwise judgments.     

An application of a biased version of the Bradley-Terry-Luce model to professional basketball results

Some extensions of the Bradley-Terry-Luce model for paired comparisons are presented and used to assess the existence of a home court advantage for professional basketball teams. It is shown how maximum likelihood estimates for these models can be obtained from a simple iterative scaling algorithm. Won-loss records for the regular season are used to make predictions for playoff series. The models are extended to include possible differences among divisions. These models can be applied in many other situations to incorporate group effects and order effects into analysis of paired comparisons.     

Generating consensus priority interval vectors for group decision‐making in the AHP

Many practical and important decision‐making problems are complicated by at least two factors: (1) the qualitative/subjective nature of some criteria often results in uncertainty in the individual ratings; and (2) group decision‐making is involved and some means of aggregating individual ratings is required. Traditionally, both individual and group priorities have been represented as point estimates, but this approach presents severe limitations for accommodating imprecision in the decision‐making process. This paper examines the group decision‐making problem in the context where priorities are represented as numeric intervals. A set of techniques that could be used at some of the phases of an analytic hierarchy process (AHP)‐based group decision‐making process, which has the objective of generating a ‘consensus’ priority that represents the group's opinion with regards to the relative importance of a set of N objects (e.g. criteria, alternatives), is presented. Copyright © 2000 John Wiley & Sons, Ltd.