# A Simplified Algorithm for Dealing with Inconsistencies Using the Analytic Hierarchy Process
## 1 Introduction
In natural resource management in particular, management agencies are turning to online surveys to understand the priorities of a wide range of stakeholders to better support policy development and management decision making, with many of these surveys using AHP approaches, e.g., [14; 15; 16; 17; 18]. The use of online surveys to elicit preferences is not unique to natural resource management, with a range of other AHP studies implemented through online surveys, e.g., [19; 20]. A key advantage of the use of online surveys is that allows access to relevant stakeholders who may be geographically dispersed, even if not large in absolute numbers. For example, [PERSON] et al. [21] employed an online AHP survey to assess satisfaction with the working environment within a large real estate firm with offices spread across the UK, while [PERSON] et al. [18] used an online AHP survey to elicit the relative importance of different information types to decision making by fisheries managers across Australia. This lack of direct interaction with the respondents creates additional challenges for deriving priorities through approaches such as AHP. Direct interactions with the individual respondents are not generally feasible, and in many cases responses are anonymous. At the same time, high levels of inconsistency are relatively common in online surveys. For example, [PERSON], et al. [22] found only 26% of respondents satisfied a relaxed threshold consistency ratio of 0.3 (compared to the standard threshold of 0.1) in their online survey; [PERSON], et al. [23] found 67% of respondents satisfied a relaxed threshold consistency ratio of 0.2; [PERSON], et al. [18] found 64% of the general public and 72% of resource managers provided consistent responses, while [PERSON] and [PERSON] [24] found only 25% or respondents satisfied the standard threshold consistency ratio of 0.1. Most previous online-based AHP studies have tended to exclude responses that have a high level of inconsistency, resulting in a substantially reduced, and potentially unrepresentative, sample e.g., [22; 23; 18; 24]. Given this, there is a growing need to develop methods for adjusting inconsistencies in AHP studies where direct interactions are not possible[PERSON] [PERSON] and [PERSON] [25] found (through stochastic simulation) that ex post adjustment of pairwise comparison matrix to improve consistency also improved the reliability of the preferences scores[PERSON] Several algorithms have since been developed to adjust the preference matrix in order to reduce inconsistency, although these are not widely applied. Some involve iterative approachesthat aim to reduce inconsistencies while maintaining the relative relationship between the weights as much as possible, utilizing the principal eigenvector [26; 27; 28][PERSON] [PERSON] and [PERSON] [29] combined iterative approaches with optimization models to identify and adjust scores, while [PERSON] and [PERSON] [30] used a non-linear programming approach to find a consistent set of scores that minimized the changes in the derived weights. Other approaches involve the use of evolutionary optimization procedures to adjust the values of the comparison matrices to minimize changes while improving consistency [31; 32; 33]. These implicitly assume that the relative weights derived from the inconsistent matrix are still largely representative of the relative preferences of the respondent. Most of these above methods aim to minimize the change in the derived set of weights. However, given that these weights are potentially distorted by the inconsistencies, aiming to minimize divergence from these may be misleading. In contrast, [PERSON] et al. [34] and [PERSON] et al. [35] used orthogonal projection to modify the entire comparison matrix, deriving a completely new matrix that was close to the original but consistent. [PERSON] et al. [36] used deviations from an expected score value (based on other scores) to identify problematic elements within the matrix, which are subsequently replaced with a modified estimate.