Constructing a quasi-concave quadratic objective function from interviewing a decision maker

A model for constructing quadratic objective functions (=utility functions) from interviewing a decision maker is considered. The interview is designed to guarantee a unique non-trivial output of the model and to enable estimating both cardinal and ordinal utility, depending on interview scenarios. The model is provided with operational restrictions for the monotonicity of the objective function (=either only growth, or only decrease in every variable) and its quasi-concavity (=convexity of the associated preference). Thereby constructing a monotonic quasi-concave quadratic objective function is reduced to a problem of non-linear programming. To support interactive editing of a quadratic objective function, the stability of the model (the continuous dependence of the output ordinal preference on the input data) is proved. In illustration, we construct a quadratic objective function of ski station customers. Then it is used to adjust prices of 10 ski stations in the south of Stuttgart.