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**Bi-criteria evolution strategy in estimating weights from the AHP ratio-scale matrices.**
*(English)*
Zbl 1229.65070

Summary: The problem of deriving weights from ratio-scale matrices in an analytic hierarchy process (AHP) is addressed by researchers worldwide. There are various ways to solve the problem that are generally grouped into simple matrix and optimization methods. All methods have received criticism regarding the accuracy of derived weights, and different criteria are in use to compare the weights obtained from different methods. Because the set of Pareto non-dominated solutions (weights) is unknown and for inconsistent matrices is indefinite, a bi-criterion optimization approach is proposed for manipulating such matrices.

The problem-specific evolution strategy algorithm is implemented for a robust stochastic search over a feasible indefinite solution space. The fitness function is defined as a scalar vector function composed of the common error measure, i.e. the Euclidean distance and a minimum violation error that accounts for no violation of the rank ordering. The encoding scheme and other components of the search engine are adjusted to preserve the imposed constraints related to the required normalized values of the weights.

The solutions generated by the proposed approach are compared with solutions obtained by five well-known prioritization techniques for three judgment matrices taken from the literature. In these and other test applications, the prioritization method that uses the entitled weights estimation by evolution strategy algorithm appears to be superior to other methods if only two, the most commonly used methods, are applied: the Euclidean distance and minimum violation exclusion criteria.

The problem-specific evolution strategy algorithm is implemented for a robust stochastic search over a feasible indefinite solution space. The fitness function is defined as a scalar vector function composed of the common error measure, i.e. the Euclidean distance and a minimum violation error that accounts for no violation of the rank ordering. The encoding scheme and other components of the search engine are adjusted to preserve the imposed constraints related to the required normalized values of the weights.

The solutions generated by the proposed approach are compared with solutions obtained by five well-known prioritization techniques for three judgment matrices taken from the literature. In these and other test applications, the prioritization method that uses the entitled weights estimation by evolution strategy algorithm appears to be superior to other methods if only two, the most commonly used methods, are applied: the Euclidean distance and minimum violation exclusion criteria.

### MSC:

65F30 | Other matrix algorithms (MSC2010) |

### Keywords:

ratio-scale matrix; prioritization method; evolution strategy; algorithm; numerical examples; analytic hierarchy process; Pareto non-dominated solutions; inconsistent matrices
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\textit{B. Srdjevic} and \textit{Z. Srdjevic}, Appl. Math. Comput. 218, No. 4, 1254--1266 (2011; Zbl 1229.65070)

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