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**The analytic hierarchy process. Planning, priority setting, resource allocation.**
*(English)*
Zbl 0587.90002

New York etc.: McGraw-Hill International Book Company. XIII, 287 p. DM 181.70 (1980).

The book gives a broad presentation of a decision model which the author has developed and applied extensively over the last 15 years. He first gives an informal description of the model, which should have some intuitive appeal to the non-mathematical reader, and in a later chapter a rigorous mathematical presentation. At every stage the author gives examples and applications from economics and social sciences, drawn from a score of his earlier publications.

The basic idea behind the model is the following: In a pairwise comparison between elements i and j, the former is judged to be \(a_{ij}\) times as important as the latter, and \(a_{ij}a_{ji}=1\). Pairwise comparisons lead to a ”consistent” ordering (hierarchy) of all elements if \(a_{ij}a_{jk}=a_{ik}\). In this case the \(n\times n\) matrix \(\{a_{ij}\}\) has one eigenvalue equal to n, and n-1 equal to zero. In practice pairwise comparisons do not always lead to a consistent hierarchy. This may in fact be impossible if the comparisons are made by majority voting of a group of decision makers. If \(\lambda_{\max}\) stands for the largest eigenvalue, the author takes \((\lambda_{\max}- n)/(n-1)\) as an index of deviation from consistency. From the eigenvectors corresponding to \(\lambda_{\max}\) the author determines the weights which the decision maker should give to the different elements. He thus offers a solution of a problem to which more orthodox decision theory can only suggest that the decision maker should make an effort to reach a consistent preference ordering.

The author argues that for practical purposes it is sufficient to take \(a_{ij}\) as one of the integers 1, 3, 5, 7 or 9, and their inverses.

The basic idea behind the model is the following: In a pairwise comparison between elements i and j, the former is judged to be \(a_{ij}\) times as important as the latter, and \(a_{ij}a_{ji}=1\). Pairwise comparisons lead to a ”consistent” ordering (hierarchy) of all elements if \(a_{ij}a_{jk}=a_{ik}\). In this case the \(n\times n\) matrix \(\{a_{ij}\}\) has one eigenvalue equal to n, and n-1 equal to zero. In practice pairwise comparisons do not always lead to a consistent hierarchy. This may in fact be impossible if the comparisons are made by majority voting of a group of decision makers. If \(\lambda_{\max}\) stands for the largest eigenvalue, the author takes \((\lambda_{\max}- n)/(n-1)\) as an index of deviation from consistency. From the eigenvectors corresponding to \(\lambda_{\max}\) the author determines the weights which the decision maker should give to the different elements. He thus offers a solution of a problem to which more orthodox decision theory can only suggest that the decision maker should make an effort to reach a consistent preference ordering.

The author argues that for practical purposes it is sufficient to take \(a_{ij}\) as one of the integers 1, 3, 5, 7 or 9, and their inverses.

Reviewer: K.Borch

### MSC:

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

91B06 | Decision theory |

90B50 | Management decision making, including multiple objectives |