# A Simplified Algorithm for Dealing with Inconsistencies Using the Analytic Hierarchy Process
## 1 Introduction  
The estimation of objective or criteria importance weights is an integral component of most multi-criteria decision analysis studies [1]. The analytic hierarchy process (AHP) [2, 3] is a commonly applied approach for deriving such weights, e.g., [4, 5, 6, 7]. The method estimates the relative importance of each criterion being assessed to different stakeholders or decision makers. The derived weights subsequently help determine which set of outcomes given a range of different management options may be overall most preferable to the different groups based on their preference sets. AHP is based upon the construction of a series of comparison matrices, with each element of the matrix representing the relative importance of each criterion relative to another. From these comparison matrices, the set of criteria importance weights can be determined using a number of different approaches [8], as detailed in the following sections. The elements themselves are derived through stakeholders being required to compare and prioritize only two criteria at any one time. The pair-wise comparison used in the AHP method makes the process of assigning importance weights relatively easy for stakeholder respondents as only two sub-components are being compared at a time [1]. However, preference or importance weightings are highly subjective, and inconsistency is a common problem facing AHP, particularly when decision makers are confronted with many sets of comparisons [9]. For example, asked to consider three criteria through pairwise comparison, a stakeholder may rate A > B, B > C but C > A. This is an extreme outcome, known as a circular triad, but can and does occur in some cases. More often, we might find that the \"implicit\" relationship between, say C and A, in terms of the preference strength given the other prioritizations (i.e., A and B, B and C) differs from the stated relationship. The propensity for these inconsistencies to emerge increases with the numberof criteria to be compared. The presence of inconsistency in the comparison matrix may result in issues such as the incorrect priority weighting associated with each of the attributes being considered [10]. When applied in a multi-criteria decision-making framework, these issues may result in incorrect ordering of management options, and the selection of a less-than-optimal option. Inconsistency may arise for several reasons. Respondents do not necessarily cross check their responses, and even if they do, ensuring a perfectly consistent set of responses when many attributes are compared is difficult. The process of choosing a particular trade-off value can also depend on respondent's attention, mood, feelings and mental efficiency [11], and inconsistencies may arise through essentially random errors created by variations in these respondent's state of mind. The discrete nature of the 1-9 scale applied in AHP can also contribute to inconsistency, as a perfectly consistent response may require a fractional preference score [12]. [PERSON], et al. [13] found that excessive use of extreme values by respondents trying to highlight their strong preferences was a main factor contributing to inconsistency in their study. [PERSON] [4] suggested that inconsistency can also arise through respondents entering their judgments incorrectly, lack of concentration, as well as inappropriate use of extremes. [PERSON] and [PERSON] [10] termed these \"Unusual and False Observations\" (UFO) that appear because of inaccurate data entry and other lapses of judgement. Ideally, UFOs would be identified and respondents would be requested to reconsider their choices in the light of these identified inconsistencies. However, this is not always practical. The ease of implementing online surveys, the larger number of potential respondents these can reach and the relatively low cost per response has made online surveys widely attractive for gaining preference information from the general community as well as special interest groups.