Modellistica e Gestione dei Sistemi Ambientali A tool for multicriteria analysis: The Analytic Hierarchy Process Chiara Mocenni University of Siena

Modellistica e Gestione dei Sistemi Ambientali A general framework (1/3) Decision problem n alternatives A 1,…,A n m decision criteria C 1,…,C m expressed by indicators I 1,…,I m Select the alternative that achieves the best trade-off among the different criteria We need: Weights w 1,…,w m for the criteria The weights reflect the importance given to the criteria Normalize the indicators Transform the indicators I 1,…,I m into the scores s 1,…,s m

Modellistica e Gestione dei Sistemi Ambientali A general framework (2/3) Let: w=(w 1,…,w m ) the vector of criteria weights I=(I 1,…,I m ) the vector of indicators I (i) the instance of I under the ith alternative s=(s 1,…,s m ) the vector of scores s (i) the instance of s under the ith alternative v (i) the global score assigned to the ith alternative

Modellistica e Gestione dei Sistemi Ambientali A general framework (3/3) For each alternative A i, i=1,…,n, repeat: Rank the scores v (1),…, v (n) indicators normalization I (i) s (i) =f(I (i) ) v (i) =w T s (i) v (i) w s (i) scores weights weighted sum global score

Modellistica e Gestione dei Sistemi Ambientali The Analytic Hierarchy Process I (i) s (i) =f(I (i) ) v (i) =w T s (i) v (i) w s (i) Multicriteria decision technique for weighting the criteria computing the scores A weight is associated to each criterion based on the pairwise comparisons of the criteria Scores are associated to each alternative based on the pairwise comparisons of the alternatives according to the different criteria Weights and scores are combined to determine a global score for each alternative

Modellistica e Gestione dei Sistemi Ambientali Weighting the criteria The criteria pairwise comparison matrix A is such that: Each entry a jk of A represents the importance of the jth criterion relative to the kth criterion If a jk >1 then the jth criterion is more important than the kth criterion The larger a jk (on a scale from 1 to 9), the more important is the jth criterion compared to the kth criterion The entries a jk and a kj satisfy a jk a kj =1 Clearly, a jj =1

Modellistica e Gestione dei Sistemi Ambientali Example j is absolutely more important than k 9 j is strongly more important than k 7 j is more important than k 5 j is slightly more important than k 3 j and k are equally important 1 InterpretationValue of a jk Remark: also intermediate values can be used

Modellistica e Gestione dei Sistemi Ambientali Weighting the criteria (contd) The weights vector w is computed by means of simple calculations on the rows and the columns of A: Divide each entry of A by the sum of the entries in the same column Average the entries on each row In the previous example, the following weights correspond to the criteria pairwise comparison matrix A: w=(0.633, 0.261, 0.106)

Modellistica e Gestione dei Sistemi Ambientali Computing the scores A pairwise comparison matrix B (j) of the alternatives is built according to each criterion, j=1,…,m The matrix B (j) contains the pairwise evaluations of the alternatives according to the jth criterion The matrix B (j) is built in a consistent way by following the same rules described for the matrix A The pairwise comparisons are carried out on the basis of the values assumed by the indicator I j on each alternative Scores for the alternatives according to the jth criterion are obtained by means of simple calculations on the rows and the columns of B (j), as described for the matrix A

Modellistica e Gestione dei Sistemi Ambientali Automating the pairwise comparisons Let the indicator I j lie in the interval [I j,max, I j,min ] Assume the larger I j, the better the system performance according to the jth criterion If, then: The relative evaluation of the ith and hth alternatives is a linear function of the difference More sophisticated functions can be designed by exploiting specific knowledge and/or experience