Improving consistency in AHP decision-making processes

Abstract

Decision making in engineering is becoming increasingly complex due to the large number of alternatives and multiple conflicting goals. Powerful decisionsupport expert systems powered by suitable software are increasingly necessary. In this paper, the multiple attribute decision method known as analytical hierarchy process (AHP), which uses pairwise comparisons with numerical judgments, is considered. Since judgments may lack a minimum level of consistency, mechanisms to improve consistency are necessary. A method to achieve consistency through optimisation is described in this paper. This method has the major advantage of depending on just n decision variables -the number of compared elements -and so is less computationally expensive than other optimisation methods, and can be easily implemented in virtually any existing computer environment. The proposed approach is exemplified by considering a simplified version of one of the most important problems faced by water supply managers, namely, the minimisation of water loss.

Consistent completion of incomplete judgments in decision making using AHP

“…This task is within a project for assessing the best leakage control alternative [14] that was specifically developed by the authors with Valencia's main water utility. Many of the matrices obtained in the course of this project have already been used by the authors in previous works [3,4,5,6,13]. The matrix in Table 1 was not considered in the main decision-making process as it was incomplete.…”

“…This matrix cannot be completed (assigning values to the asterisks) consistently since its rank is greater than or equal to 3 (there is a 3 × 3 nonsingular submatrix) taking into account that, as proven in [6], a consistent matrix is necessarily of rank 1.…”

“…Consequently, it is predictable -even reasonable-that the issued global opinion body lacks a minimum consistency, which is essential for the prioritization to be meaningful, reliable, and consequently, conducive to a sound decision. The literature contains numerous mechanisms to improve consistency, [11,21,8,1,20,9,13,3,4,6] among many others, and any attempt to improve consistency results in a better quality decision [18]. Among the methods to improve consistency, we present in Section 3 the linearization method introduced by the authors [4] on which the main contribution of this article is built.…”

A simple formula to find the closest consistent matrix to a reciprocal matrix

“…Achieving consistency in AHP has become an important issue and different methods have been proposed in the literature [1,2,3,6,7,8,9,10,11,12,13,14]. In this paper, we focus on the linearization process [10] not as method to directly obtain the priority vector, but as a method that provides a closed form for achieving complete consistency.…”

“…It is also well known that any consistent matrix has rank one [3], and as a consequence, any of its normalized rows and, in particular, the normalized vector of the geometric means of the rows, also provides the priority vector. Taking into account the (natural lack of) consistency of human thinking, some degree of inconsistency is expected and, as a result, in general A is not consistent.…”