Constructing a monotonic quadratic objective function in n variables from a few two-dimensional indifferences

Abstract

We develop a model for constructing quadratic objective functions in n target variables. At the input, a decision maker is asked a few simple questions about his ordinal preferences (comparing two-dimensional alternatives in terms `better', `worse', `indifferent'). At the output, the model mathematically derives a quadratic objective function used to evaluate n-dimensional alternatives. Thus the model deals with some imaginary decisions (criteria aggregates) at the input, and disaggregates the decision maker's preference into partial criteria and their cross-correlations (=a quadratic objective function). Therefore, the model provides an approximation step which is next to the disaggregation of a preference into additively separable linear criteria with weight coefficients. The model is based on least squares fitting a quadratic indifference hypersurface (if n=2, indifference curve) to several alternatives which are supposed to be equivalent in preference. The resulting ordinal preference is independent of the cardinal utility scale used in intermediate computations which implies that the model is ordinal. The monotonicity of the quadratic objective function is implemented by means of a finite number of linear constraints, so that the computational model is reduced to restricted least squares. In illustration, we construct a quadratic objective function of German economic policy in four target variables: inflation, unemployment, GNP growth, and increase in public debt. This objective function is used to evaluate the German economic development in 1980–1994. In another application, we construct a quadratic objective function of ski station customers. Then it is used to adjust prices of 10 ski stations to the South of Stuttgart. In Appendix A we provide an original fast algorithm for restricted least squares and quadratic programming used in the main model.