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Journal Article•DOI•

A new consensus ranking approach for correlated ordinal information based on Mahalanobis distance

Teresa González-Arteaga1, José Carlos R. Alcantud2, R. de Andrés Calle2•
University of Valladolid1, University of Salamanca2
01 Dec 2016-Information Sciences (Elsevier)-Vol. 372, pp 546-564
TL;DR: A new procedure to codify ordinal information is provided and characterized and a new measurement of the degree of dissensus among individual preferences based on the Mahalanobis distance is defined.
About: This article is published in Information Sciences.The article was published on 2016-12-01 and is currently open access. It has received 25 citations till now. The article focuses on the topics: Ranking & Mahalanobis distance.

Summary (2 min read)

Jump to: [1. Introduction] – [2. Ordinal information] – [3. A new dissensus measure for ordinal information: The class of Mahalanobis dissensus measures] – [4. Reaching a social consensus solution based on Mahalanobis distance] and [5. Concluding remarks]

1. Introduction

  • A considerable amount of literature has contributed to the research issue of obtaining consensus in group decision making problems.
  • Generally speaking, experts can express their opinions by means of ordinal or cardinal information, the former being more extensively used in the research issue addressed in this work.
  • The characterization of the new codification procedure is a key point because it ensures consistency of their approach and its use in any methodology.
  • Then the authors exploit these measures in order to propose a consensus solution especially designed for profiles of preferences on possibly correlated alternatives and to overcome the drawbacks of the aforementioned distance-based methodologies.
  • That solution aggregates individual opinions into a social preference on the alternatives by minimizing dissensus with respect to the original profile of preferences.

2. Ordinal information

  • Most group decision making problems can usually manage different types of information.
  • In the specialized literature it is possible to find several approaches or procedures to codify ordinal information into numerical values (see [8], [7], [14] and [25], among others).
  • Moreover, the authors characterize the new codification procedure to associate every profile of complete preorders with a unique matrix.
  • The t vector is also called the ties vector of KR.

3. A new dissensus measure for ordinal information: The class of Mahalanobis dissensus measures

  • A considerable amount of the most cited contributions on consensus measurement have addressed this topic considering functions that assign to every ranking profile a real number from the unit interval.
  • This distance is a common tool in multivariate statistical analysis, e.g., in regression models.
  • This relation verifies the property of reflexivity, antisymmetry and transitivity.
  • In order to analize the properties of the Mahalanobis dissensus measures, it seems reasonable that the authors initially explore if these measures satisfy anonymity and neutrality, that is, if the Mahalanobis dissensus measures are normal dissensus measures and then the rest of their properties.
  • Note compatibility refers to the behavior of the ranking of the profiles previously provided in Definition 6.

4. Reaching a social consensus solution based on Mahalanobis distance

  • The problem of reaching a social consensus solution intends to determine the ranking of alternatives that best agrees with individual preferences, or in other words, the ranking that minimizes the disagreement among individuals.
  • The authors now proceed to define and prove the main properties of the Mahalanobis social consensus solution.
  • In their example this means that all destinations are equally treated.
  • About graphical interpretation, on the left of Figure 2 the elements of the feasible set F are displayed like dots using a color scale.
  • Finally, the authors examine the case of a non-diagonal matrix, which allows to incorporate the interdependence of the alternatives because the role of Σ in the Mahalanobis distance.

5. Concluding remarks

  • This study is aimed at proposing a new approach to obtain a group consensus solution under the assumption of ordinal information.
  • A new procedure based on an optimization model has been developed, obtaining a social 8The element ij of the corelation matrix Corr is Σij√ Σii √ Σjj , where Σij is the element ij of variance-covariance matrix Σ. consensus solution based on the Mahalanobis distance.
  • To accomplish such target two new contributions have been developed in addition of the main result: the characterization of a codification procedure for ordinal information, namely, the canonical codification and the definition and analysis of a new dissensus measure, namely, the Mahalanobis dissensus measure.
  • Moreover, the operational character and intuitive interpretation of their approaches have been illustrated by an explanatory example.

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Citations
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Journal Article•DOI•
An agglomerative hierarchical clustering algorithm for linear ordinal rankings

[...]

Nana Liu1, Nana Liu2, Zeshui Xu2, Xiao-Jun Zeng1, Peijia Ren3, Peijia Ren1 •
University of Manchester1, Sichuan University2, South China University of Technology3
01 May 2021-Information Sciences
TL;DR: In this article, a new method for clustering linear ordinal ranking (LOR) information by agglomerative hierarchical clustering (AHC) algorithm is proposed, and the corresponding new distance measure is proposed starting from the perspective of utilizing the rankings' position information and relationship information together.

22 citations

Journal Article•DOI•
A limited cost consensus approach with fairness concern and its application

[...]

Junliang Du1, Sifeng Liu1, Yong Liu2•
Nanjing University of Aeronautics and Astronautics1, Jiangnan University2
25 Jun 2021-European Journal of Operational Research
TL;DR: A limited cost consensus approach with fairness concern of decision makers is proposed, which can obtain a stable and balanced consensus and can provide significant managerial references for real-world economic and management activities.

17 citations

Journal Article•DOI•
A decision support system for detecting and handling biased decision-makers in multi criteria group decision-making problems

[...]

Meysam Rabiee1, Babak Aslani2, Jafar Rezaei3•
University of Oregon1, University of Oklahoma2, Delft University of Technology3
01 Jun 2021-Expert Systems With Applications
TL;DR: In this article, an anti-biased statistical approach, including extreme, moderate, and soft versions, was developed as a decision support system for group decision-making (GDM) to detect and handle the bias.
Abstract: Detecting and handling biased decision-makers in the group decision-making process is overlooked in the literature. This paper aims to develop an anti-biased statistical approach, including extreme, moderate, and soft versions, as a decision support system for group decision-making (GDM) to detect and handle the bias. The extreme version starts with eliminating the biased decision-makers (DMs). For this purpose, the DMs with a lower Biasedness Index value than a predefined threshold are removed from the process. Next, it continues with a procedure to mitigate the effect of partially biased DMs by assigning different weights to DMs with respect to their biasedness level. To do so, two ratios for the remaining DMs are calculated: (i) Overlap Ratio, which shows the relative value of overlap between the confidence interval (CI) of each DM and the maximum possible overlap value. (ii) Relative confidence interval CI which reflects the relative value of CI for each DM compared to the confidence interval CI of all DMs. The final step is assigning weight to each DM, considering the two values Overlap Ratio and Relative confidence interval. DMs with closer opinions to the aggregated opinion of all DMs, or those with an adequate level of discrimination in their judgments gain more weight. The framework addresses and prescribes possible actions for all possible cases in GDM including without outliers, cases with partial outliers, and extreme cases with complete disagreement among DMs, or when none of the DMs show an adequate level of discrimination power. The moderate version preassigns a minimum weight to all unbiased DMs and then follows the weighting step for the remaining total weight. However, the soft version follows the preassignmnet of weights to all DMs in the initial pool, meaning there is no elimination in this setting. The proposed approach is tested for several scenarios with different sizes. Four performance measures are introduced to evaluate the effectiveness of the proposed method. The resulted performance measures show the reliability of the outcomes.

16 citations

Journal Article•DOI•
A limited cost consensus approach with fairness concern and its application

[...]

01 Apr 2022-European Journal of Operational Research
TL;DR: Wang et al. as mentioned in this paper proposed a limited cost consensus approach with fairness concern of decision makers to obtain a stable and balanced consensus, which can provide significant managerial references for real-world economic and management activities.

11 citations

Journal Article•DOI•
Renyi’s-Tsallis fuzzy divergence measure and its applications to pattern recognition and fault detection

[...]

Ratika Kadian1, Satish Kumar1•
Maharishi Markandeshwar University, Mullana1
01 Jan 2020-Journal of Intelligent and Fuzzy Systems

10 citations

References
More filters
Book•
Rank correlation methods

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Maurice G. Kendall
01 Jan 1948
TL;DR: The measurement of rank correlation was introduced in this paper, and rank correlation tied ranks tests of significance were applied to the problem of m ranking, and variate values were used to measure rank correlation.
Abstract: The measurement of rank correlation introduction to the general theory of rank correlation tied ranks tests of significance proof of the results of chapter 4 the problem of m ranking proof of the result of chapter 6 partial rank correlation ranks and variate values proof of the result of chapter 9 paired comparisons proof of the results of chapter 11 some further applications.

6,404 citations

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4,255 citations

Book•
The Proof and Measurement of Association Between Two Things

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C. Spearman
05 Sep 2011
TL;DR: The present article is a commencement at attempting to remedy this deficiency of scientific correlation, and the meaning and working of the various formulæ have been explained sufficiently, it is hoped, to render them readily usable even by those whose knowledge of mathematics is elementary.
Abstract: All knowledge—beyond that of bare isolated occurrence—deals with uniformities. Of the latter, some few have a claim to be considered absolute, such as mathematical implications and mechanical laws. But the vast majority are only partial; medicine does not teach that smallpox is inevitably escaped by vaccination, but that it is so generally; biology has not shown that all animals require organic food, but that nearly all do so; in daily life, a dark sky is no proof that it will rain, but merely a warning; even in morality, the sole categorical imperative alleged by Kant was the sinfulness of telling a lie, and few thinkers since have admitted so much as this to be valid universally. In psychology, more perhaps than in any other science, it is hard to find absolutely inflexible coincidences; occasionally, indeed, there appear uniformities sufficiently regular to be practically treated as laws, but infinitely the greater part of the observations hitherto recorded concern only more or less pronounced tendencies of one event or attribute to accompany another. Under these circumstances, one might well have expected that the evidential evaluation and precise mensuration of tendencies had long been the subject of exhaustive investigation and now formed one of the earliest sections in a beginner’s psychological course. Instead, we find only a general naı̈ve ignorance that there is anything about it requiring to be learnt. One after another, laborious series of experiments are executed and published with the purpose of demonstrating some connection between two events, wherein the otherwise learned psychologist reveals that his art of proving and measuring correspondence has not advanced beyond that of lay persons. The consequence has been that the significance of the experiments is not at all rightly understood, nor have any definite facts been elicited that may be either confirmed or refuted. The present article is a commencement at attempting to remedy this deficiency of scientific correlation. With this view, it will be strictly confined to the needs of practical workers, and all theoretical mathematical demonstrations will be omitted; it may, however, be said that the relations stated have already received a large amount of empirical verification. Great thanks are due from me to Professor Haussdorff and to Dr. G. Lipps, each of whom have supplied a useful theorem in polynomial probability; the former has also very kindly given valuable advice concerning the proof of the important formulæ for elimination of ‘‘systematic deviations.’’ At the same time, and for the same reason, the meaning and working of the various formulæ have been explained sufficiently, it is hoped, to render them readily usable even by those whose knowledge of mathematics is elementary. The fundamental procedure is accompanied by simple imaginary examples, while the more advanced parts are illustrated by cases that have actually occurred in my personal experience. For more abundant and positive exemplification, the reader is requested to refer to the under cited research, which is entirely built upon the principles and mathematical relations here laid down. In conclusion, the general value of the methodics recommended is emphasized by a brief criticism of the best correlational work hitherto made public, and also the important question is discussed as to the number of ‘‘cases’’ required for an experimental series.

3,687 citations

Additional excerpts

  • ..., [55], [33], [14], [42], [21] and [27])....

    [...]

Book Chapter•DOI•
Rank Correlation Methods

[...]

Ron N. Forthofer, Robert G. Lehnen
01 Jan 1981
TL;DR: RankRank correlation coefficients as mentioned in this paper are statistical indices that measure the degree of association between two variables having ordered categories, and are defined such that a coefficient of zero means "no association" between the variables and a value of +1.0 or -1.
Abstract: Rank correlation coefficients are statistical indices that measure the degree of association between two variables having ordered categories. Some well-known rank correlation coefficients are those proposed by Goodman and Kruskal (1954, 1959), Kendall (1955), and Somers (1962). Rank correlation methods share several common features. They are based on counts and are defined such that a coefficient of zero means “no association” between the variables and a value of +1.0 or -1.0 means “perfect agreement” or “perfect inverse agreement,” respectively.

3,475 citations

Additional excerpts

  • ..., [55], [33], [14], [42], [21] and [27])....

    [...]

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Frequently Asked Questions (11)
Q1. What are the contributions mentioned in the paper "A new consensus ranking approach for correlated ordinal information based on mahalanobis distance" ?

The authors investigate from a global point of view the existence of cohesiveness among experts ’ opinions. Accordingly, the authors propose and characterize a new procedure to codify ordinal information. Finally, the authors investigate a procedure to obtain a social consensus solution that also includes the possibility of alternatives that are correlated. In addition, the authors examine the main traits of the dissensus measurement as well as the social solution proposed. The operational character and intuitive interpretation of their approaches are illustrated by an explanatory example. 

Let R ∈ W (X) be a complete preorder on X, a codified complete preorder is a real-valued vector MR = (m1, . . . ,mk) where mj represents the codification value corresponding to alternative xj. 

Whatever the statement of the minimization problem, the objective function is restricted to feasible codified vectors, which emphasizes the importance of their characterization for the canonical codification procedure. 

In the literature several procedures to codify linear and complete preorders into numerical values can be found (see [8], [7], [14] and [25], among others), Borda [8] being the first author to manage ordinal preferences in such way. 

Due to the importance of the choice of the codification procedure to accomplish any methodology over ordinal information, it should be relevant to dispose of a consistent codification procedure. 

In order to analize the properties of the Mahalanobis dissensus measures, it seems reasonable that the authors initially explore if these measures satisfy anonymity and neutrality, that is, if the Mahalanobis dissensus measures are normal dissensus measures and then the rest of their properties. 

Consider the spectral decomposition of the matrix Σ = ΓtDλΓ where Γ and Dλ contain eigenvectors (by columns) and the corresponding eigenvalues of Σ as diagonal elements, respectively. 

In order to simplify the notation and due to the equivalence among the set of complete preorders and the set of their permutations, the authors can write πMR for some MR. 

Cook and Seiford [14] formalized the so-called monotone non-decreasing property for the case of the Minimum Variance method in order to realize the potential of the alignment between the average point and the ranking that minimizes the Euclidean distance. 

Since Σ matrix can be considered as a variance-covarince matrix in the Mahalanobis distance, it is easy to compute the corresponding correlation matrix Corr, that is, the correlation among the alternatives. 

Definition 1. The canonical codified complete preorder associated with R ∈ W (X) is defined by the numerical vector KR = (c1, ..., ck) ∈ ({1, . . . , k})k where cj = |{q : xj <R xq}| and therefore cj accounts for the number of alternatives that are graded at most as good as xj.