Incomplete analytic hierarchy process with minimum weighted ordinal violations

ABSTRACT

Incomplete pairwise comparison matrices offer a natural way of expressing preferences in decision-making processes. Although ordinal information is crucial, there is a bias in the literature: cardinal models dominate. Ordinal models usually yield nonunique solutions; therefore, an approach blending ordinal and cardinal information is needed. In this work, we consider two cascading problems: first, we compute ordinal preferences, maximizing an index that combines ordinal and cardinal information; then, we obtain a cardinal ranking by enforcing ordinal constraints. Notably, we provide a sufficient condition (that is likely to be satisfied in practical cases) for the first problem to admit a unique solution and we develop a provably polynomial-time algorithm to compute it. The effectiveness of the proposed method is analyzed and compared with respect to other approaches and criteria at the state of the art.

Keywords:

• Pairwise comparison matrix
• incomplete data
• logarithmic least squares
• ordinal constraints
• decision making process

Acknowledgements

The authors would like to express their sincere gratitude to Dr.János Fülöp for his valuable comments on the earlier versions of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We point out that, in the general case of arbitrary convex constraints, additional constraint qualifications might be required (e.g. Slater's Condition); moreover, in the case of nonconvex objective functions or constraints, the KKT conditions hold just as necessary conditions. The interested reader is referred to Zangwill (1969) (and references therein) for a comprehensive overview of the topic.

2 For instance, we have a cycle when i is preferred to j, j is preferred to k and k is preferred to i. In this case, the information available is remarkably inconsistent.

3 We reiterate that we consider only links in $G_d$, i.e. links with associated weights ${\mathcal{A}}_{ij}\ge 1$; hence, it holds ${\mathcal{A}}_{min}\ge 1$.

4 When the objective function is convex and the constraints are linear, in order to guarantee that the KKT first-order criterion is a necessary and sufficient global optimality condition, there is no need to check for constraint qualification conditions such as the Slater's condition, (see, for instance, Zangwill 1969); it is sufficient to show that the set of admissible solutions is nonempty.

5 The example in Csató and Rónyai (2016) is given for generic coefficients $b,1/b$, in this case, we set b = 2.

6 In the simulations we always have ${\mathcal{A}}_{ij}\ne 1$ for $i\ne j$; hence, $\tau =0$.

Additional information

Notes on contributors

L. Faramondi

Luca 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 PostDoc Fellow 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 Localization. His research interests include the identification of network vulnerabilities, cyber physical systems, and optimization at large.

G. Oliva

Gabriele 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 optimization, and applications of graph theory in technological and biological systems.

Sándor Bozóki

Sándor 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 modeling, global optimization and multivariate polynomial systems.