Generating ranking groups in the analytical hierarchy process

The paper introduces a convenient procedure of ranking N alternatives through direct comparisons in AHP. The alternatives are divided into groups in such a way that dominant relationship exists between the groups but not among the alternatives within each group. This method is suitable for situations where the strict ranking in a sequence for all alternatives is not reliable or not necessary. Two procedures are proposed to construct the AHP ranking groups. The proposed grouping procedures can be used in conjunction with the traditional approaches.     

A statistical approach to the analytic hierarchy process with interval judgements. (I). Distributions on feasible regions

This paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the Analytic Hierarchy Process (AHP). The method of Arbel for extracting such preferences, which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives, is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed. In particular it is proposed that a distribution for the weights on the feasible region, which is both tractable and meaningful, be adopted. The mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest, such as probabilities and marginal distributions, can immediately be quantified. Two specific distributions on the feasible region, the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings, are examined in some detail and the ideas which emerge are illustrated by means of selected examples.     

AHP model for competitiveness analysis of selected countries

Competitiveness analysis of countries and regions stands in the foreground in recent years. Different methods as well as indicators are used to assess competitiveness, but no single procedure is considered to be the main one and it can hardly be stated, which of the measurement approaches is the most proper. The IMD World Competitiveness Online database represents one of such tools, containing the Overall Competitiveness ranking that evaluates 59 world countries by more than 300 individual indicators in 2012. The paper proposes an original Analytic Hierarchy Process (AHP) model for competitiveness ranking of selected countries. The model is based, due to the high number of indicators and countries, on absolute measurement and expert evaluation. The results given by the AHP model are compared to IMD competitiveness ranking. Differences of both results are analysed and discussed.     

A fuzzy ranking method with range reduction techniques

For ranking alternatives based on pairwise comparisons, current analytic hierarchy process (AHP) methods are difficult to use to generate useful information to assist decision makers in specifying their preferences. This study proposes a novel method incorporating fuzzy preferences and range reduction techniques. Modified from the concept of data envelopment analysis (DEA), the proposed approach is not only capable of treating incomplete preference matrices but also provides reasonable ranges to help decision makers to rank decision alternatives confidently.     

一个模糊层次分析法在方案排序中的应用

给出了一个模糊层次分析法(FAHP).该方法的决策矩阵的元素为三角模糊数.结合三角模糊数比较的可能度理论,提出了一个基于模糊层次分析法的有限方案决策方法,最后的实例说明方法的有效性和合理性.     

On Maximizing Discrimination in Multiple Criteria Decision Making

We examine an LP/DEA-based technique for establishing an overall ranking of alternatives that are ranked on multiple criteria, which themselves are ranked. This two-stage process involves one LP in the first stage, and N LPs in the second stage to rank N alternatives. We find that the information from N + 1 LPs can be obtained by solving two LPs. In many cases, the solution of one LP, which can be done by inspection, is almost as informative as the two-stage procedure. We also indicate when the second stage would be redundant. If maximum technical discrimination between the alternatives is sought, we consider how this might be achieved by aggressive cross-evaluation via N LPs. We also show how to identify a subset of the alternatives that would be ranked in the first place under any ordering of the criteria, and thus play an important role in the evaluation procedure.     

Interval cross efficiency for fully ranking decision making units using DEA/AHP approach

Data envelopment analysis (DEA) is a popular technique for measuring the relative efficiency of a set of decision making units (DMUs). Fully ranking DMUs is a traditional and important topic in DEA. In various types of ranking methods, cross efficiency method receives much attention from researchers because it evaluates DMUs by using self and peer evaluation. However, cross efficiency score is usual nonuniqueness. This paper combines the DEA and analytic hierarchy process (AHP) to fully rank the DMUs that considers all possible cross efficiencies of a DMU with respect to all the other DMUs. We firstly measure the interval cross efficiency of each DMU. Based on the interval cross efficiency, relative efficiency pairwise comparison between each pair of DMUs is used to construct interval multiplicative preference relations (IMPRs). To obtain the consistency ranking order, a method to derive consistent IMPRs is developed. After that, the full ranking order of DMUs from completely consistent IMPRs is derived. It is worth noting that our DEA/AHP approach not only avoids overestimation of DMUs’ efficiency by only self-evaluation, but also eliminates the subjectivity of pairwise comparison between DMUs in AHP. Finally, a real example is offered to illustrate the feasibility and practicality of the proposed procedure.     

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.     

Another potential shortcoming of AHP

In the last twenty years many features of Saaty’s Analytical Hierarchy Process (AHP) have been criticised, especially the additive hierarchical composition of conventional AHP, which leads to the possibility of occurrence of the Rank Reversal phenomenon (adding an irrelevant alternative may cause a reversal in the ranking at the top). In this paper we show another feature of AHP which may be, and in many application contexts will inneed be, an even stronger shortcoming of the method. It consists in the fact that the addition of indifferent criteria (for which all alternatives perform equally) causes a significant alteration of the aggregated priorities of alternatives, with important consequences. In hierarchies with four or more levels, rank reversal may happen. Since in almost all applications of AHP the set of criteria is not fixed ex-ante but is variable and is constructed in accordance with reasons of relevance and simplicity, almost all applications of AHP are potentially flawed.     

Group decision making in a multiple criteria environment: A case using the AHP in software selection

This paper reports the results of a case study where the analytic hierarchy process (AHP) technique was employed to support the selection of a multi-media authorizing system (MAS) in a group decision environment. Three MAS products were identified and ultimately ranked using the AHP. Six software engineers, who are technically competent and experienced, participated in our study. These engineers were trained to use the AHP and asked to apply this technique to select the most appropriate MAS product for adoption. A post-study survey and interview were conducted with all the engineers to collect further feedback on the use of the AHP, as compared to their frequently used Delphi technique, in supporting group decisions. The experiment results and survey findings indicated that the AHP is preferable to Delphi as the AHP helps group members center a discussion around objectives, rather than alternatives. We also found the AHP to be more conducive to consensus building in group decision settings.     

The synthetic hierarchy method: An optimizing approach to obtaining priorities in the AHP

A new method of synthesizing local and criteria priorities into global priorities is suggested. This approach is a development of the Analytic Hierarchy Process enabling the united consideration of all horizontal and vertical connections of a hierarchical system in a single optimizing objective function based on statistical models of the synthesis process. The solution can be reduced to a linear system or to an eigenproblem of a special matrix constructed as a combination of Kronecker's sums and products of pairwise judgement matrices. A numerical example shows that the optimizing approach produces a ranking of global priorities that may be different from the ranking produced by the classical AHP.     

Comparison of Prioritization Techniques Using Interhierarchy Mappings

This paper discusses a method of comparing different prioritization techniques that entails mapping the input data from one system to another. The ranking vectors output by the methods should then be comparable. As an illustration, a method that uses pairwise comparison, e.g. the Analytic Hierarchy Process (AHP) of Saaty, itself viewed as a mapping, is compared with a method using cross impact analysis, the Edinburgh Approach. The problem centres on an appropriate inverse of the process mapping. Although the discussion is restricted to simple bilevel hierarchies, useful insight is provided into the methodologies in general and, in particular, the problem of inverse inconsistency associated with rank reversals. A way in which the theory of AHP might be incorporated into a Differential Hierarchy Process is suggested.     

A probabilistic interpretation of the final rankings in AHP

The Analytic Hierarchy Process (AHP) has become a popular and practical tool for dealing with complex decision problems. It provides a ranking for the decision alternatives. This article recommends treating the pairwise comparison input data as random variables. This will allow the determination of whether the differences between alternatives are statistically significant.     

Fuzzy hierarchical analysis

This paper extends hierarchical analysis to the case where the participants are allowed to employ fuzzy ratios in place of exact ratios. If a person considers alternative A more important than alternative B, then the ratio used might be approximately 3 to 1, or between 2 to 1, and 4 to 1, or at most 5 to 1. The pairwise comparison of the issues and the criteria in the hierarchy produce fuzzy positive reciprocal matrices. The geometric mean method is employed to calculate the fuzzy weights for each fuzzy matrix, and these are combined in the usual manner to determine the final fuzzy weights for the alternatives. The final fuzzy weights are used to rank the alternatives from highest to lowest. The highest ranking contains all the undominated issues. The procedure easily extends to the situation where many experts are utilized in the ranking process, or to the case of missing data. Two examples are presented showing the final fuzzy weights and the final ranking.     

Ranking residential properties by a multicriteria single price model

This study describes an application of the multicriteria single price model (Ballestero) to the ranking of alternatives. By a generalization of the original model, the equilibrium set of alternatives can be characterized from the viewpoints, respectively, of the demander and the supplier, and from that the efficiency index can be calculated. We demonstrate how, in a state of equilibrium, the two viewpoints result inevitably in inverse orders of ranking. In contrast with other proposals for full ranking of alternatives, the method used in the present study (i) assumes a moderate attitude on the part of the decision maker towards risk, with a robust axiomatic basis; (ii) assigns weights to the criteria independently of which alternative is being evaluated and the attitude (optimistic or pessimistic) of the decision maker; (iii) produces a cardinal hierarchy of the alternatives and not just an ordinal one. The model is illustrated by a sample of residential properties in the city of Valencia, Spain.     

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.     

Does AHP help us make a choice? An experimental evaluation

In this paper, we use experimental economics methods to test how well Analytic Hierarchy Process (AHP) fares as a choice support system in a real decision problem. AHP provides a ranking that we statistically compare with three additional rankings given by the subjects in the experiment: one at the beginning, one after providing AHP with the necessary pair-wise comparisons and one after learning the ranking provided by AHP. While the rankings vary widely across subjects, we observe that for each individual all four rankings are similar. Hence, subjects are consistent and AHP is, for the most part, able to replicate their rankings. Furthermore, while the rankings are similar, we do find that the AHP ranking helps the decision makers reformulate their choices by taking into account suggestions made by AHP.     

The Analytic Hierarchy Process in an uncertain environment: A simulation approach

Traditionally, decision makers were forced to converge ambiguous judgments to a single point estimate in order to describe a pairwise relationship between alternatives relative to some criterion for use in the Analytic Hierarchy Process (AHP). Since many circumstances exist which make such a convergence difficult, confidence in the outcome of an ensuing AHP synthesis may be reduced. Likewise, when a group of decision makers cannot arrive at a consensus regarding a judgment, some members of the group may simply lose confidence in the overall synthesis if they lack faith in some of the judgments. The AHP utilizes point estimates in order to derive the relative weights of criteria, sub-criteria, and alternatives which govern a decision problem. However, when point estimates are difficult to determine, distributions describing feasible judgments may be more appropriate. Using simulation, we will demonstrate that levels of confidence can be developed, expected weights can be calculated and expected ranks can be determined. It will also be shown that the simulation approach is far more revealing than traditional sensitivity analysis.     

Evidential reasoning based preference programming for multiple attribute decision analysis under uncertainty

Multiple attribute decision analysis (MADA) problems having both quantitative and qualitative attributes under uncertainty can be modelled and analysed using the evidential reasoning (ER) approach. Several types of uncertainty such as ignorance and fuzziness can be consistently modelled in the ER framework. In this paper, both interval weight assignments and interval belief degrees are considered, which could be incurred in many decision situations such as group decision making. Based on the existing ER algorithm, several pairs of preference programming models are constructed to support global sensitivity analysis based on the interval values and to generate the upper and lower bounds of the combined belief degrees for distributed assessment and also the expected values for ranking of alternatives. A post-optimisation procedure is developed to identify non-dominated solutions, examine the robustness of the partial ranking orders generated, and provide guidance for the elicitation of additional information for generating more desirable assessment results. A car evaluation problem is examined to show the implementation process of the proposed approach.