# A Modified PCA Approach for Solving MADM Problems with Dependent Criteria

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Abstract

A Basic assumption of MADM methods such as SAW and TOPSIS is the statistical independence of criteria. Violating this assumption would result in erroneous decisions. However, there are many cases in which criteria are dependent and based on this situation decision has to be made. In this paper, an approach based on the property of principal component analysis (PCA) in producing independent variables is developed. In addition, weights of criteria are considered for decision making considering the fact that PCA does not take them into account. Since each of the new components obtained from PCA is a linear combination of the initial criteria therefore one cannot say whether the new component is a larger the better variable or a smaller the better one so the decision making is difficult. To determine the direction of the new components, we developed a new method based on the variation mode chart. This method uses the weights of the initial criteria as well as their coefficients in the new components and considering the direction of the initial criteria it determines the direction of the new components. After that it becomes obvious which component is a larger the better or a smaller the better variable. Meanwhile, the basic assumption of criteria independency is met. To show the efficiency of the proposed method, a real case in a military decision making problem as well as a numerical example is presented. In the military decision making problem, the Mann-Whitney non parametric test shows that the ranking of alternatives obtained from proposed method has more accordance with the ranking derived from expert's choice in comparison with the other methods ignoring the effect of dependencies. For more assurance about the performance of the proposed method, an example with a large number of alternatives and attributes (20 alternatives and 10 criteria) is presented. Using the properties of PCA, the initial criteria are replaced by four independent components. After determining the direction of new components using TOPSIS and SAW two different rankings are produced to choose from. For the validation of the proposed method this problem was also solved by ignoring dependencies of alternatives. The final results showed that using the proposed method will result in similar ranking under implementation of different MADM methods, herein Topsis and SAW. Ignoring the dependency would cause very different and sometimes conflicting ranking under implementation of different MADM techniques. In other words, the proposed method results in more robust ranking compared to the methods that ignore the dependency of criteria.

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