Homotopy Continuation Method

Abstract One of the most important steps in solving Multi-Attribute Decision Making (MADM) problems is to derive the weights of importance of the attributes. The decision maker is requested to compare the importance for each pair of attributes. The result expressed in numbers is written in a pairwise comparison matrix. The aim is to determine a weight vector w=(w1, w2,, wn), which reflects the preferences of the decision maker, in the positive orthant of the n-dimensional Euclidean space.

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In this paper, an efficient iterative method of arbitrary positive integer order of convergence ≥ 2 will be established for the two-body universal initial value problem. The method is of dynamic nature in the sense that, on going from one iterative scheme to the subsequent one, only additional instruction is needed. Moreover, which is the most important, the method does not need any a priori knowledge of the initial guess. A property which avoids the critical situations between divergent to very slow convergent solutions, that may exist in other numerical methods which depend on initial guess. Some applications of the method are also given.