Weighting by pairwise comparison

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# Weighting by pairwise comparison

Another method for weighting several criteria is the pairwise comparison. It stems from the Analytic Hierarchy Process (AHP), a famous decision-making framework developed by the American Professor of mathematics (1980).
The following three steps lead to the result:

• Completion of the pairwise comparison matrix: Step 1 – two criteria are evaluated at a time in terms of their relative importance. Index values from 1 to 9 are used. If criterion A is exactly as important as criterion B, this pair receives an index of 1. If A is much more important than B, the index is 9. All gradations are possible in between. For a "less important" relationship, the fractions 1/1 to 1/9 are available: if A is much less important than B, the rating is 1/9. The values are entered row by row into a cross-matrix. The diagonal of the matrix contains only values of 1. First, the right upper half of the matrix is filled until each criterion has been compared to every other one. If A to B was rated with the relative importance of n, B to A has to be rated with 1/n. If the vegetation cover is a little more important than slope (index 3), the slope is a little less important than vegetation cover (index 1/3). For reasons of consistency, the lower left half of the matrix can thus be filled with the corresponding fractions.
• Calculating the criteria weights: Step 2 – the weights of the individual criteria are calculated. First, a normalized comparison matrix is created: each value in the matrix is divided by the sum of its column. To get the weights of the individual criteria, the mean of each row of this second matrix is determined. These weights are already normalized; their sum is 1.
• Assessment of the consistency matrix: Step 3 – a statistically reliable estimate of the consistency of the resulting weights is made (1999). This method however, goes beyond this course and is not discussed any further.

This method allows a concentration on the comparison of only two criteria at a time. Thereby, the effort required to compare each criteria with every other one is increasing rapidly when handling many classes (to be exact: with n criteria there are n(n-1)/2 comparisons). The integration of the method into a digital environment is easy and can be mastered with a spreadsheet or a GIS. A successful example of the latter is the use of this method in the environmental SDSS in "IDRISI". You can find further information about IDRISI at Clark Labs.

This animation allows you to try out the AHP with your own weightings for the case study of the wolf habitat. As an expert on wolves you are asked to evaluate all five criteria in their relative importance. How the standard values can be read: starting in the first row, vegetation was evaluated as "equally to slightly more important" than slope; this cell gets an index of 2. The value 0.5 in the last cell of this first line indicates that slope is "equally or slightly less important" than vegetation. The whole upper right half of the matrix is filled in this manner. The lower left half contains the corresponding fractions of the evaluations. Replace the standard values with your own weightings and study the change of weights.

Pairwise comparison (Saaty's AHP)
Definition Index Definition Index
Equally important 1 Equally important 1/1
Equally or slightly more important 2 Equally or slightly less important 1/2
Slightly more important 3 Slightly less important 1/3
Slightly to much more important 4 Slightly to way less important 1/4
Much more important 5 Way less important 1/5
Much to far more important 6 Way to far less important 1/6
Far more important 7 Far less important 1/7
Far more important to extremely more important 8 Far less important to extremely less important 1/8
Extremely more important 9 Extremely less important 1/9
Weights