Linear programming techniques for multidimensional analysis of preferences

This paper offers a new methodology for analyzing individual differences in preference judgments with regard to a set of stimuli prespecified in a multidimensional attribute space. The individual is modelled as possessing an ideal point denoting his most preferred stimulus location in this space and a set of weights which reveal the relative saliences of the attributes. He prefers those stimuli which are closer to his ideal point (in terms of a weighted Euclidean distance measure). A linear programming model is proposed for external analysisi.e., estimation of the coordinates of his ideal point and the weights (involved in the Euclidean distance measure) by analyzing his paired comparison preference judgments on a set of stimuli, prespecified by their coordinate locations in the multidimensional space. A measure of poorness of fit is developed and the linear programming model minimizes this measure overall possible solutions. The approach is fully nonmetric, extremely flexible, and uses paired comparison judgments directly. The weights can either be constrained nonnegative or left unconstrained. Generalizations of the model to consider ordinal or interval preference data and to allow an orthogonal transformation of the attribute space are discussed. The methodology is extended to perform internal analysis,i.e., to determine the stimuli locations in addition to weights and ideal points by analyzing the preference judgments of all subjects simultaneously. Computational results show that the methodology for external analysis is unbiased—i.e., on an average it recovers the true ideal point and weights. These studies also indicate that the technique performs satisfactorily even when about 20 percent of the paired comparison judgments are incorrectly specified.The authors wish to express their deep appreciation to J. Douglas Carroll and Joseph B. Kruskal for their most valuable comments on an earlier version of this paper.