Heuristic Rating Estimation Approach to The Pairwise Comparisons Method

Kułakowski, Konrad

doi:10.3233/fi-2014-1081

Cited by 20 publications

(23 citation statements)

References 41 publications

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“…This result, however, is in contradiction with (23), hence, it must be w(a i ) ÷ w(a j ) > w(a k ) ÷ w(a l ), therefore, condition POIP is satisfied. Remark 3.…”

mentioning

confidence: 88%

“…Despite the relative maturity of the area, it still invites further exploration. Examples of new exploration are the Rough Set approach [13], voting systems [9], fuzzy PC relation handling [28,38,29], incomplete PC relation [4,11], non-numerical rankings [19,18], nonreciprocal PC relation properties [12], rankings with the reference set of alternatives [23,26], applications to software correctness and cybersecurity [33] and others. Further references: [34,16,22].…”

Section: Introductionmentioning

confidence: 99%

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When is the condition of order preservation met?

Kułakowski

Mazurek

Ramík

et al. 2019

European Journal of Operational Research

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This article explores a relationship between inconsistency in the pairwise comparisons method and conditions of order preservation. A pairwise comparisons matrix with elements from an alo-group is investigated. This approach allows for a generalization of previous results. Sufficient conditions for order preservation based on the properties of elements of pairwise comparisons matrix are derived. A numerical example is presented.

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“…This result, however, is in contradiction with (23), hence, it must be w(a i ) ÷ w(a j ) > w(a k ) ÷ w(a l ), therefore, condition POIP is satisfied. Remark 3.…”

mentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

When is the condition of order preservation met?

Kułakowski

Mazurek

Ramík

et al. 2019

European Journal of Operational Research

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“…Despite its long existence the area still prompts researchers to further exploration. Examples of such exploration are the Rough Set approach [10], voting systems [5], fuzzy PC relation handling [22,32], incomplete PC relation [2,7], nonnumerical rankings [14], nonreciprocal PC relation properties [9], rankings with the reference set of alternatives [18,19] and others. Further references can be found in [28,12].…”

Section: Introductionmentioning

confidence: 99%

On the Properties of the Priority Deriving Procedure in the Pairwise Comparisons Method

Kułakowski

2015

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The pairwise comparisons method is a convenient tool used when the relative order of preferences among different concepts (alternatives) needs to be determined. There are several popular implementations of this method, including the Eigenvector Method, the Least Squares Method, the Chi Squares Method and others. Each of the above methods comes with one or more inconsistency indices that help to decide whether the consistency of input guarantees obtaining a reliable output, thus taking the optimal decision. This article explores the relationship between inconsistency of input and discrepancy of output. A global ranking discrepancy describes to what extent the obtained results correspond to the single expert's assessments. On the basis of the inconsistency and discrepancy indices, two properties of the weight deriving procedure are formulated. These properties are proven for Eigenvector Method and Koczkodaj's Inconsistency Index. Several estimates using Koczkodaj's Inconsistency Index for a principal eigenvalue, Saaty's inconsistency index and the Condition of Order Preservation are also provided. IntroductionThe first documented uses of comparisons in pairs date back to the thirteenth century [4]. Later, the method was developed by Fechner [6], Thurstone [30] and Saaty [25]. The latter proposed the Analytic Hierarchy Process (AHP) extension to the pairwise comparisons (herein abbreviated as PC) theory, the framework allowing dealing with a large number of criteria.At the beginning of the twentieth century the method was used in psychometrics and psychophysics [30]. Now the method is considered part of decision theory [26]. Its utility has been confirmed in numerous examples [31,11,21,29]. Despite its long existence the area still prompts researchers to further exploration. Examples of such exploration are the Rough Set approach [10], voting systems [5], fuzzy PC relation handling [22,32], incomplete PC relation [2,7], nonnumerical rankings [14], nonreciprocal PC relation properties [9], rankings with the reference set of alternatives [18,19] and others. Further references can be found in [28,12].The aim of the PC method is to determine the weights of objects, so that the more important object is, the higher (or the lower) weight it has. The method provides the user with a number of specific techniques for deriving weights from the PC matrix M [3]. With every specific technique an appropriate inconsistency index is associated that describes to what extent M is inconsistent. The value of an inconsistency index is perceived as a kind of quality determinant for M -the input data to the deriving weight procedure. Following the popular adage "garbage in, garbage out" one could say that when the inconsistency is high (consistency is low) the result must be poor. Indeed, for various inconsistency indices there exist thresholds of acceptability, above which the obtained results are considered to be unreliable.However, poor and unreliable results may be also due to the deriving method itself. One way to learn something abou...

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“…Currently, the Heuristic Rating Estimation (HRE) method which enables the user to explicitly define the reference set of concepts, for which the ranking values are a priori known, is being developed [11,12]. The base heuristics used in HRE proposes to determine the relative values of a single non-reference concept as a weighted arithmetic mean of all the other concepts.…”

Section: Introductionmentioning

confidence: 99%

“…The presented work is a follow-up of research initiated in [11,12]. It redefines the main heuristics of HRE and the method of calculating the solution.…”

Section: Introductionmentioning

confidence: 99%

Heuristic rating estimation: geometric approach

2014

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Heuristic rating estimation is a newly proposed method that supports decisions analysis based on the use of pairwise comparisons. It allows the ranking values of some alternatives (herein referred to as concepts) to be initially known, whilst ranks for other concepts have yet to be estimated. To calculate the missing ranks it is assumed that the priority of every single concept can be determined as the weighted arithmetic mean of the priorities of all the other concepts. It has been shown that the problem has an admissible solution if the inconsistency of the pairwise comparisons is not too high. The proposed approach adopts heuristics according to which a weighted geometric mean is used to determine the missing priorities. In this approach, despite increased complexity, a solution always exists and its existence does not depend on the inconsistency or reciprocity of the input matrix. Thus, the presented approach might be appropriate for a larger number of problems than previous methods. Moreover, it turns out that the geometric approach, as proposed in the article, can be optimal. The optimality condition is presented in the form of a corresponding theorem. A formal definition of the proposed geometric heuristics is accompanied by two numerical examples.

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Heuristic Rating Estimation Approach to The Pairwise Comparisons Method

Cited by 20 publications

References 41 publications

When is the condition of order preservation met?

When is the condition of order preservation met?

On the Properties of the Priority Deriving Procedure in the Pairwise Comparisons Method

Heuristic rating estimation: geometric approach

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