A Model for Constructing Monotonic and/or Quasi-Concave Quadratic Objective Functions from Ordinal Data

Abstract

We develop a model for constructing quadratic objective functions in n target variables. At the input, a decision maker is asked a few relatively simple questions about his ordinal preferences (to compare 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. The model is provided with operational restrictions for the monotonicity of the objective function (= either only growth, or only decrease in every variable) and quasi-concavity of the objective function (= convexity of the associated preference). Constructing a monotonic quasi-concave quadratic objective function from ordinal data is reduced to a solvable problem of non-linear programming.As an 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.