Employing linear, Tchebycheff, and hybrid functions to fit prescored discrete alternatives
Ward, Keith J.
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Decisions are a part of everyday life. Decisions can range from the type of job offer we accept, the type of car we purchase, or the kind of home we select. However, in many decisions there are multiple criteria upon which we make our final decision. Mathematical functions can be developed, called utility functions, which model how the decision maker ranks the alternatives under consideration. There are various utility functions; in this thesis we have examined the linear and Tchebycheff utility functions and a hybrid of the two. The purpose of this thesis was to approximate the assumed true utility function (we assumed that the true utility function caused the score and rank of the alternatives in the sample data we obtained from Consumer Reports) by fitting the previously mentioned utility functions to the scored data. We considered three products. The goodness of fit measures we used were the sum of absolute deviations and the sum of squares residual. It was a our goal to show, for the data set we selected, how the linear, Tchebycheff, and the hybrid of the two performed in comparison to one another for each product. The results showed that for a set of alternatives that appeared to have a solely a linear utility function, the hybrid of the linear and Tchebycheff utility functions was able to perform best. In considering all products chosen for this thesis, the true utility function never seemed to be Tchebycheff entirely; yet by adding the linear utility function to it, the goodness of fit results were favorable. In general, the hybrid utility function can be of superior performance compared to the linear and Tchebycheff utility functions independently. The hybrid utility function could provide increased flexibility when used in conjunction with an interactive method. The hybrid utility function is of notable advantage as it can be modified to not only incorporate the combination of linear and Tchebycheff utility functions; it also can be extended to quadratic-Tchebycheff combinations or higher ordered combinations of utility functions and used in union with interactive methods. Keywords: Multicriteria decision making, utility functions, data fitting, discrete alternatives