The mathematical equivalence of the "spanning tree" and row geometric mean preference vectors and its implications for preference analysis

Pairwise comparison is a widely used approach to elicit comparative judgements from a decision maker (DM), and there are a number of methods that can be used to then subsequently derive a consistent preference vector from the DM's judgements. While the most widely used method is the eigenvector method, the row geometric mean approach has gained popularity due to its mathematical properties and its ease of implementation. In this paper, we discuss a spanning tree method and prove the mathematical equivalence of its preference vector to that of the row geometric mean approach. This is an important finding due to the fact that it identifies an approach for generating a preference vector which has the mathematical properties of the row geometric mean preference vector, and yet, in its entirety, the spanning tree method has more to offer than the row geometric mean method, in that, it is inherently applicable to incomplete sets of pairwise comparison judgements, and also facilitates the use of statistical and visual techniques to gain insights into inconsistency in the DM's judgements

Introduction to the Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP) has been one of the foremost mathematical methods for decision making with multiple criteria and has been widely studied in the operations research literature as well as applied to solve countless real-world problems. This book is meant to introduce and strengthen the readers’ knowledge of the AHP, no matter how familiar they may be with the topic. This book provides a concise, yet self-contained, introduction to the AHP that uses a novel and more pedagogical approach. It begins with an introduction to the principles of the AHP, covering the critical points of the method, as well as some of its applications. Next, the book explores further aspects of the method, including the derivation of the priority vector, the estimation of inconsistency, and the use of AHP for group decisions. Each of these is introduced by relaxing initial assumptions. Furthermore, this booklet covers extensions of AHP, which are typically neglected in elementary expositions of the methods. Such extensions concern different numerical representations of preferences and the interval and fuzzy representations of preferences to account for uncertainty. During the whole exposition, an eye is kept on the most recent developments of the method.Peer reviewe