# A Modified Analytic Hierarchy Process Suitable for Online Survey Preference Elicitation
## 1 Introduction  
The estimation of objective or criteria importance weights is an integral component of most multi-criteria decision analysis studies [1]. A range of methods has been applied in the literature to assess objective weights, each with advantages and disadvantages [2, 3, 4, 5, 6, 7, 8, 9]. Comparative studies of these methods suggest in some cases that the objective weights may vary considerably between methods [10], although others have found higher correlations between the results of the different methods [11]. The Analytic Hierarchy Process (AHP) [12, 13] is one commonly applied approach for deriving such weights. The AHP has been used in a number of applications covering business (e.g., [14]), manufacturing (e.g., [15]), transport (e.g., [16]), healthcare (e.g., [17]), and a wide range of applied environmental and natural resource management studies [18, 19, 20, 21, 22, 23, 24]. It has become one of the most common approaches used for preference elicitation for multi-criteria decision analysis [25]. The method estimates the relative importance of each criterion being assessed to different stakeholders or decision-makers through a series of pairwise comparisons. The derived weights subsequently help determine which set of outcomes, given a range of different options, may be overall most preferable to different stakeholder groups and/or decision-makers based on their preference sets. A key challenge facing the use of the AHP is the issue of inconsistencies in the responses by those whose preferences are being elicited. Preference weightings are highlysubjective, and inconsistency between responses to different combinations of comparisons is a common problem facing the AHP, particularly when decision-makers are confronted with many sets of comparisons [26]. Respondents do not necessarily cross-check their responses, and even if they do, ensuring a perfectly consistent set of responses when many sub-components are compared is difficult. The discrete nature of the 1-9 scale in the traditional AHP approach can also contribute to inconsistency, as a perfectly consistent response may require a fractional preference score [27]. Inconsistency can also arise through errors in entering judgments, lack of concentration, and inappropriate use of extremes [24; 28; 29]. The pairwise comparisons underpinning preference choice in the AHP are believed to make the process of assigning importance weights relatively easy for respondents, as only two sub-components are being compared at a time [1]. However, as more criteria are compared, the required number of pairwise comparisons increases at an exponential rate, requiring approximately \((n^{2}-n)/2\) comparisons, where \(n\) is the number of objectives or criteria to be compared. For example, comparing two criteria requires only one pairwise comparison, three criteria require three pairwise comparisons, four criteria require six pairwise comparisons, and so on. The propensity for respondents to be inconsistent in their responses also correspondingly increases when they are confronted with many sets of comparisons [26; 30; 31]. Such inconsistency in the comparison matrix may result in incorrect priority weighting being associated with each of the attributes being considered [28]. Inconsistency is generally considered to affect the reliability of the resultant set of preference weights derived from the AHP, with reliability inversely related to the level of inconsistency [32; 33; 34]. The increased development of online survey platforms provides both opportunities and challenges for preference elicitation using the AHP. Foremost, the number of stakeholders that can be captured in the analysis can be greatly increased, providing greater information to decision-makers around the distribution of preferences across different stakeholder groups.