О некорректности метода анализа иерархий
The paper considers the basic statement of the analytic hierarchy process (AHP), that the priorities of decision variants on individual criteria are compared on the ratio scales that are not linked to each other and are also independent of the priorities of criteria. According to the mathematical theory of measurement this approach is incorrect. To demonstrate its potential consequences a simple example in which the use of the AHP procedure leads to a clearly erroneous result is provided.
In authors' previous paper published in 2011 in «Control Sciences» journal one example of a bi-criterion decision analysis problem demonstrating that the use of Analytic Hierarchy Process (AHP) may lead to a clearly erroneous result is given. However, the author of another paper published in 2012 in the same journal suggested that he found an error in our use of AHP and, consequently, our criticism of AHP is unsubstantiated. In this new paper the authors show that there was no mistake in the use of AHP in their original counter-example, and provide two further counter-examples that support their original conclusion.
Influence of the assumption about the existence of quantitative coefficients of criteria importance, consistent with the criteria importance order (according to the definitions in the criteria importance theory), on the preference relation, generated by this information, is investigated.
This abstract offers a method for ranking alternatives in a decison making problem. It determines importance of the criteria with help of factor analysis. Though the alternatives are evaluated by each of the criteria by a group of experts, the weights for the criteria are to be found with the help of factor analysis.
The algorithm of the method is as follows:
1. Under the constraint that the problem handles several evaluation criteria, several items to compare (alternatives) and several experts to give their evaluation.
2. Find the principal components that replace the input criteria implicitly.
3. To find the final mark for each of the alternatives the marks given by experts are multiplied with the regression coefficients, found in the step 2.
4. The final marks are represented in axes „crieria“ and „mark“ so that each alternative is described with a curve (trajectory). These curves represent the map of graded alternatives. Depending on the problem to be solved (min or max,) a record for each main criteria is to be found.
5. With help of special deviation measure procedures (Minkowski, Chebyshev e.t.s) a matrix of deviations from ideal solution is to be built.
6. The alternatives are to be rated in accordance to the deviation from the ideal trajectory.
To prove the effectiveness of the method it was applied to a problem for 5 alternatives, 3 experts and 38 evaluation criteria. The problem was also solved with the help of most popular method of Weighted Sum Model (WSM) and TOPSIS method. The problem was also being solved by finding the geometric mean for each alternative. The results for approaches were compared and the method, offered in this abstrat, proved itself as a feasible one.
The influence of the assumption about the existence of cardinal coefficients of criteria importance, consistent with the importance ordering of criteria (according to the definitions in the criteria importance theory), on the preference relation generated by this information, is investigated.