Quantitative Techniques for Decision Making in Construction

2.ANALYIC HIERARCHY PROCESS II

aQ Amni: 2.1 Th e Controversies of AHP 2.1.1 Righ t and Left Eigenvector Approaches In Chapter 1 , we have discussed the approach that Saaty proposed, which is called right eigenvector approach. The right eigenvector approach means that the priority vector X is calculated based on the equation AX = ^max X. However, we can also calculate the priority vector using the left eigenvecto r approach. Th e left eigenvector approach means that the priority vector is calculated based on (AX)T = (X X) T max ' ie. X T AT = l X T max Johnson, Beine and Wang (1979), two years after Saat y proposed his AHP theory, discovered that the priority vectors calculated from a same reciprocal matrix using the right eigenvector approach and the left eigenvector approach may have disagreed results. Let us look at the following example. Example 2.1 Find the priority vectors of matrix A using the right eigenvector approac h and the left eigenvector approach. QUANTITATIVE TECHNIQUE FOR DECISION MAKINGIN CONSTRUCTION 1 1 1 / 3 1 / 9 1 / 9 1 1 1 / 5 1 / 8 1 / 5 3 5 1 1 / 9 1 / 5 9 8 9 1 1 9 5 5 1 1 A = Solution Using the right eigenvector approach, we solve AX =A, X . We obtain that X = 0.366 0.389 0.167 0.035 0.042 - 2nd - i s i -3r d -5t h _ At h Using the left eigenvecto r approach , AT = 1 1 3 9 9 1 1 5 8 5 1/3 1/5 1 9 5 1 / 9 1 / 8 1 / 9 1 1 1 / 9 1 / 5 1 / 5 1 1 We solve XT AT = X X T and obtai n tha t max X T = [ 0 . 0 3 9 0.04 3 0.10 5 I I I pt 2 nd 3 rd 0.458 0.355~ | I I CJth Ath In simple words, the right eigenvector approach is based on the pairwise comparison of elements of how one element is better than another, while the left eigenvecto r approach is based on the pairwise comparison o f how one element is worse than another. It can be seen that the ranks of the 1st object and the 2nd object are reversed in the two approaches. Readers should note that, unlike the right eigenvector approach, the smaller the value in the eigenvector, the higher the ranking of the object is when we use the left eigenvector approach. The results show that the two approaches give contradictory rankings. John, Beine and Wang (1979) argued that there is no reason to believe that utilization of a right eigenvector (as proposed by Saaty) yields a better result ANALYTIC HIERARCH Y PROCESS II than the left. They experimented 364 randomly generated reciprocal matrices of size n=6 and there were 195 such ranking reversals between left and right eigenvectors. Three years later, Vargas (1982) reported the finding o f his research that if the consistency inde x of a reciprocal matrix is not larger than 10 % then the result of the right eigenvector approach will be sufficiently reliable . As we have seen in Section 1.2. 3 tha t the further X i s from n for a n n x n max reciprocal matrix, the more the inconsistent the matrix is. The consistency index is indicated by the difference o f X an d n, and is defined as follows: J ma x 7 Consistency index = (^max - n) + (n - 1) Vargas' conclusion is that when a decision is made with the help of Analytic Hierarchy Process, the pairwise comparison reciprocal matrix must be tested first t o see whether or not its consistency index is within 0 to 10 % range, and i f it is, then th e priorty vector obtaine d usin g th e right eigenvecto r approach i s sufficiently reliable . Therefore, i n th e application o f AHP as described i n Chapte r 1 , we need t o first tes t th e consistenc y inde x o f a reciprocal matrix and see if it is within 10%, and then find its priority vector (right eigenvector approach) for any decision making processes using AHP. 2.1.2 Modifie d AHP Proposed by Donegan et al. Later, Donegon, Dodd and McMaster (1992, 1995) in their papers claimed that th e right an d left eigenvecto r...