# A Modified Analytic Hierarchy Process Suitable for Online Survey Preference Elicitation
## 2 Materials and Methods  
### AHP vs. MAHP  
The traditional AHP approach [12] is based on a series of bivariate comparisons. Preferences are expressed on a nine-point scale (Figure 1), with 1 indicating equal preference and 9 indicating an extreme preference for one of the sub-components. Preferences are assumed to be symmetrical, such that if \(A\) against \(B\) has a preference of \(a_{AB}=9\), then \(a_{BA}=1/a_{AB}=1/9\). Figure 1: Example of a bivariate comparison used in the traditional AHP. A separate bivariate comparison is required for each set of alternatives. With the MAHP, we avoid some of these pitfalls by modifying the way in which the data are collected and analyzed, taking into account the symmetry assumption underlying the AHP. In the online survey, respondents were presented with a nine-point importance scale against which they could assess the importance of each objective. A nine-point scale was selected (rather than an \"out of 10\", as might occur with a Likert scale), as it allows five categories to be defined with mid-points between them, consistent with the traditional AHP approach. The options being assessed are presented together as a choice matrix, with the preference score of each being explicitly compared with the preference scores of the other options (Figure 2). A separate value is derived for each objective row (e.g., \(a_{i}=1\) to 9) given the importance level identified in Figure 2 and the relative score between any two objectives \(a_{i,j}\) is derived from the difference between them, such that:  
\[a_{i,j}=\left\{ \begin{array}{l}a_{i}-a_{j}+1\;if\;a_{i}-a_{j}>0\\ \frac{-1}{a_{i}-a_{j}-1}\;if\;a_{i}-a_{j}\leq 0\end{array} \right.\;\;\;\;and \;a_{j,i}=\frac{1}{a_{i,j}} \tag{1}\]  
As with the traditional AHP, preferences are assumed to be symmetrical. The use of the difference rather than the nominal value is to provide a relative preference similar to what is derived from the traditional AHP bivariate comparison. For example, a score of 9 for \(A\) and a score of 7 for \(B\) is equivalent to an element score of \(a_{AB}=3\) and \(a_{BA}=1/3\) in the traditional AHP bivariate comparison. If both are equal (irrespective of the value assigned to both), then the equivalent element score is \(a_{BA}=1\). The use of the difference also reduces the influence of the initial anchor score [51; 52]. For example, a pair of scores (9, 7) and (5, 3) will provide the same element score. Common to both approaches is a comparison matrix of scores (\(A\)), given by:  
\[A= \begin{bmatrix}1&a_{12}&\dots&a_{1n}\\ a_{21}&1&\dots&a_{2n}\\ \dots&\dots&\dots&\dots\\ a_{n1}&a_{n2}&\dots&1\end{bmatrix}  \tag{2}\]  
In the case of the traditional AHP, the values of \(a_{i,j}\) are determined directly from the bivariate comparison in Figure 1. For the MAHP, the values of \(a_{i,j}\) are derived from Equation (1) and based on the responses to the comparisons illustrated in Figure 2.  
Two general approaches used for determining the weights given the comparison matrix are the eigenvalue method (EM) developed by [PERSON] [12] and the geometric mean method (GMM) developed by [PERSON] and [PERSON] [53]. While the former approach has been employed in a wider range of studies, the latter has been found to be less susceptible to the influence of extreme preferences, as well as having better performance around other  
Figure 2: Example of bivariate comparison in the MAHP. Additional alternatives are included as additional rows, and all alternatives are compared in the same question. aspects of theoretical consistency (e.g., less susceptible to rank reversibility if the preference set changes and greater transitivity properties) [54].