Characterising ranking stability in interval pairwise comparison matrices

Abstract

Relative pairwise comparisons represent the cornerstone for several decision-making methods. Such approaches aim to support complex decision-making situations with multiple alternatives and are essential in order to provide an overall absolute evaluation of the alternatives despite the presence of experts and/or decision-makers with conflicting opinions. Moreover, when decision-maker's opinions are affected by uncertainty, there is the need to analyse the effect of such uncertain measures on the result of the decision-making process. We propose an approach based on a multi-objective optimisation problem, able to identify the presence of rank reversal issues in order to evaluate the stability of the final outcome of the decision-making process and a metric able to support experts in evaluating the effects of their uncertainty. We characterise the robustness of the ranking with respect to rank reversal by identifying a perturbation that is as small as possible, while causing the maximum number of ordinal swaps.

Keywords:

• Decision-making
• pairwise comparison matrix
• rank reversal

Acknowledgment

The research of S. Bozóki was supported in part by the TKP2021-NKTA-01 NRDIO grant.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Also, in this case, we operate the substitution $\mathbf{z}=\ln (\mathbf{x})$; we only report the equivalent problem formulation with respect to the variable $\mathit{z}$ for the sake of brevity.

2 It can be easily shown that the Hessian matrix of $\kappa (\cdot )$ is the Laplacian matrix $\mathcal{L}$, which is positive semidefinite, thus implying convexity of $\kappa (\cdot )$.

Additional information

Funding

This work was supported by the Italian National Project ‘DRIVERS: Approccio combinato data-driven ed experience-driven all'analisi del rischio sistemico’ funded by National Institute for Insurance Against Accidents at Work (Inail) [grant number BRIC2021-ID03].

Notes on contributors

L. Faramondi

L. Faramondi received the Laurea degree in Computer Science and Automation (2013) and the PhD degree on Computer Science and Automation (2017) from the University Roma Tre of Rome. He is currently assistant professor at Complex Systems & Security Laboratory at the University Campus Bio-Medico of Rome. He is involved in several national and European projects about the Critical Infrastructure and Indoor Localisation. His research interests include the identification of network vulnerabilities, cyber physical systems, multi-criteria decision support systems, and optimisation at large.

G. Oliva

G. Oliva received the Laurea degree and the Ph.D in Computer Science and Automation Engineering in 2008 and 2012, respectively, both at University Roma Tre of Rome, Italy. He is currently assistant professor in Automatic Control at the University Campus Bio-Medico of Rome, Italy. His main research interests include distributed systems, distributed optimisation, and applications of graph theory in technological and biological systems.

R. Setola

R. Setola received the Laurea degree in Electronic Engineering and the Ph.D. degree in Control Engineering from the University of Naples Federico II, in 1992 and 1996, respectively. He was responsible for the Italian Government Working Group on Critical Information Infrastructure Protection (CIIP) and a member of the G8 Senior Experts' Group for CIIP. He is currently a Full Professor with the University Campus Bio-Medico of Rome, where he directs the Automation Research Unit and the Master Program in homeland security. He has been the coordinator of several EU projects. He has authored nine books and more than 250 scientific papers. His main research interests include simulation, modelling and control of complex systems, and critical infrastructure protection.

S. Bozóki

S. Bozóki obtained his MSc degree in Applied Mathematics from Eötvös Lorand University, and PhD degree in economics from Corvinus University of Budapest, Hungary. He is a senior research fellow at the Research Group of Operations Research and Decision Systems, Laboratory on Engineering and Management Intelligence, Institute for Computer Science and Control (SZTAKI). He is an associate professor at the Department of Operations Research and Actuarial Sciences, Corvinus University of Budapest. His research interests include multi-attribute decision-making, pairwise comparison matrices, preference modelling, global optimisation and multivariate polynomial systems.