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The Analytic Hierarchy Process The Analytic Hierarchy Process (AHP), introduced by Thomas Saaty (1980), is an effective tool for dealing with complex decision making, and may aid the decision maker to set priorities and make the best decision. By reducing complex decisions to a series of pairwise comparisons, and then synthesizing the results, the AHP helps to capture both subjective and objective aspects of a decision. In addition, the AHP incorporates a useful technique for checking the consistency of the decision maker’s evaluations, thus reducing the bias in the decision making process. 1 How the AHP works The AHP considers a set of evaluation criteria, and a set of alternative options among which the best decision is to be made. It is important to note that, since some of the criteria could be contrasting, it is not true in general that the best option is the one which optimizes each single criterion, rather the one which achieves the most suitable trade-off among the different criteria.      The AHP generates a weight for each evaluation criterion according to the decision maker’s pairwise comparisons of the criteria. The higher the weight, the more important the corresponding criterion. Next, for a fixed criterion, the AHP assigns a score to each option according to the decision maker’s pairwise comparisons of the options based on that criterion. The higher the score, the better the performance of the option with respect to the considered criterion. Finally, the AHP combines the criteria weights and the options scores, thus determining a global score for each option, and a consequent ranking. The global score for a given option is a weighted sum of the scores it obtained with respect to all the criteria. 2 Features of the AHP The AHP is a very flexible and powerful tool because the scores, and therefore the final ranking, are obtained on the basis of the pairwise relative evaluations of both the criteria and the options provided by the user. The computations made by the AHP are always guided by the decision maker’s experience, and the AHP can thus be considered as a tool that is able to translate the evaluations (both qualitative and quantitative) made by the decision maker into a multicriteria ranking. In addition, the AHP is simple because there is no need of building a complex expert system with the decision maker’s knowledge embedded in it. On the other hand, the AHP may require a large number of evaluations by the user, especially for problems with many criteria and options. Although every single evaluation is very simple, since it only requires the decision maker to express how two options or criteria compare to each other, the load of the evaluation task may become unreasonable. In fact the number of pairwise comparisons grows quadratically with the number of criteria and options. For instance, when comparing 10 alternatives on 4 criteria, 4·3/2=6 comparisons are requested to build the weight vector, and 4·(10·9/2)=180 pairwise comparisons are needed to build the score matrix. However, in order to reduce the decision maker’s workload the AHP can be completely or partially automated by specifying suitable thresholds for automatically deciding some pairwise comparisons.  3 Implementation of the AHP The AHP can be implemented in three simple consecutive steps: 1) Computing the vector of criteria weights. 2) Computing the matrix of option scores. 3) Ranking the options. Each step will be described in detail in the following. It is assumed that m evaluation criteria are considered, and n options are to be evaluated. A useful technique for checking the reliability of the results will be also introduced.  3.1 Computing the vector of criteria weights In order to compute the weights for the different criteria, the AHP starts creating a pairwise comparison matrix A. The matrix A is a m×m real matrix, where m is the number of evaluation criteria considered. Each entry ajk of the matrix A represents1  the importance of the jth criterion relative to the kth criterion. If ajk > 1, then the jth criterion is more important than the kth criterion, while if ajk < 1, then the jth criterion is less important than the kth criterion. If two criteria have the same importance, then the entry ajk is 1. The entries ajk and akj satisfy the following constraint: (1) .1=? kjjk aa Obviously, ajj = 1 for all j. The relative importance between two criteria is measured according to a numerical scale from 1 to 9, as shown in Table 1, where it is assumed that the jth criterion is equally or more important than the kth criterion. The phrases in the "Interpretation" column of Table 1 are only suggestive, and may be used to translate the decision maker's qualitative evaluations of the relative importance between two criteria into numbers. It is also possible to assign intermediate values which do not correspond to a precise interpretation. The values in the matrix A are by construction pairwise consistent, see (1). On the other hand, the ratings may in general show slight inconsistencies. However these do not cause serious difficulties for the AHP.   Value of ajk Interpretation 1 j and k are equally important 3 j is slightly more important than k 5 j is more important than k 7 j is strongly more important than k 9 j is absolutely more important than k Table 1. Table of relative scores. Once the matrix A is built, it is possible to derive from A the normalized pairwise comparison matrix Anorm by making equal to 1 the sum of the entries on each column, i.e. each entry jk a of the matrix Anorm is computed as .1 ? = = m llkjkjkaa a (2) Finally, the criteria weight vector w (that is an m-dimensional column vector) is built by averaging  the entries on each row of Anorm, i.e. .1 m awmljlj? ==(3)                                                  1 For a matrix A, aij denotes the entry in the ith row and the jth column of A. For a vector v, vi denotes the ith element of v.   3.2 Computing the matrix of option scores  The matrix of option scores is a n×m real matrix S. Each entry sij of S represents the score of the ith option with respect to the jth criterion. In order to derive such scores, a pairwise comparison matrix  is first built for each of the m criteria, j=1,...,m. The matrix  is a n×n real matrix, where n is the number of options evaluated. Each entry  of the matrix  represents the evaluation of the ith option compared to the hth option with respect to the jth criterion. If , then the ith option is better than the hth option, while if , then the ith option is worse than the hth option. If two options are evaluated as equivalent with respect to the jth criterion, then the entry  is 1. The entries  and  satisfy the following constraint:  )( jB ) ( jB )( j ihb ) ( jB 1)( >jihb 1)(