A Systemic Rebuttal to the Criticism of Using the Eigenvector for Priority Assessment in the Analytic Hierarchy Process for Decision Making

Una Refutación Sistémica a la Crítica de Usar el Vector Propio para Calcular Prioridades en el Proceso Analítico Jerárquico para Toma de Decisiones

Claudio Garuti Anderlini¹, Valério Pamplona Salomon² and Isabel Spencer González¹

¹ Fulcrum Ingeniería Luis Thayer Ojeda 0180 Of. 1004, Providencia, Santiago, Chile e–mail: claudiogaruti@fulcrum.cl

² São Paulo State University Av. Dr. A. P. Cunha 333, 12.516–410 Guaratinguetá, Brazil e–mail:, salomon@feg.unesp.br

Article received on March 06, 2008
Accepted on August 12, 2008

Abstract

Arguments have been provided against the use of the eigenvector as the operator that derives priorities. A highlight of the arguments is that the eigenvector solution does not always respect the condition of ordinal preference (COP) based on the decision maker's judgments. While this condition may be reasonable when dealing with measurable concepts (such as distance or time) that lead to consistent matrices, it is questionable whether it is to be expected in all situations, particularly when the information provided by the decision maker is not fully consistent. The judgments that lead to inconsistency may also contain valuable information that must be considered in the priority assessment process as well. By the other hand, the analytic hierarchy process (AHP) use the eigenvector operator to derive the priorities that represent the cardinal decision maker preferences from a pairwise comparison matrix, which do not always respect the COP condition. The AHP and still deeper the ANP (the mathematical generalization of AHP) start from concepts of ordinal metric of dominance and system theory, which is well supported by graph theory and ordinal topology with the Cesaro sum as its fundamental pillar to build metric of dominance. These mathematic concepts has no relation with COP preservation moreover, this two way of thinking are in a course of collision since the second (COP) inhibit the first (Cesaro sum).

Systems theory claims that the whole is more than its standalone components, and that internal relationships provide additional information as well. Given that the pairwise comparison matrix is an interrelated system and not just a collection of standalone judgments, we plan to show that the eigenvector, because it is a systemic operator, is the most suitable to represent and capture the behavior of the whole system and its emerging properties.

Keywords: AHP/ANP, Eigenvector, Systems, Condition of Order Preservation, Ordinal Topology and Metric of Dominance.

Resumen

Se han entregado argumentos en la literatura contra el uso del vector propio para obtener prioridades. Uno de los principales argumentos dice que el vector propio no respeta la condición de ordinalidad de preferencia (COP) obtenida del decisor. Si bien, esta condición suena razonable cuando tratamos con conceptos clásicos de medida como distancia o tiempo, que conllevan intrínsicamente niveles de consistencia completa, es cuestionable que este comportamiento deba ser esperado en todo tipo de situaciones y variables, particularmente cuando la información entregada por el decisor no es completamente consistente. Los juicios que conllevan inconsistencia, normalmente contienen información valiosa, la que debe ser considerada en el proceso de evaluación. Por otro lado, el AHP usa el vector propio para derivar las prioridades cardinales que representan las preferencias del decisor a partir de una matriz de comparaciones a pares, la que no siempre respeta la condición COP. El AHP y con mayor fuerza aún el ANP, parten de los conceptos de métrica ordinal de dominancia y de la teoría de sistemas, las que son bien sustentadas por teoría de grafos y topología de ordinales, a través de la suma de Cesaro como su pilar fundamental para la construcción de esta métrica de dominancia. Estos conceptos matemáticos no guardan ninguna relación con la preservación de COP, mas aún, estas dos formas de pensamiento se hallan en curso de colisión, ya que la segunda (COP) coarta a la primera (suma de Cesaro).

Uno de los principales pilares de la teoría de sistemas corresponde al hecho indiscutible que el todo es más importante que la suma de sus partes aisladas, y que las relaciones internas del sistema, proveen información adicional relevante. Dado que la matriz de comparaciones a pares es un sistema interrelacionado y no una colección de juicios sueltos, nosotros planteamos mostrar que el vector propio, como un operador eminentemente sistémico, es el más adecuado para capturar y representar el comportamiento del sistema como un todo, incluyendo sus propiedades emergentes.

Palabras Clave: AHP/ANP, vector propio, Sistemas, Condición de Preservación de Orden (COP), Topología Ordinal y Métricas de Dominancia.

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