Positive Mathematical Programming for Farm Planning: Review

In the field of farm management and in related multidisciplinary fields such as bio-economic farm modeling and hydro-economic regional modeling, much attention has recently been paid to positive mathematical programming (PM P), primarily because of its ability to exactly reproduce an observed set of endogenous input variables in the model (e.g., an observed land-use pattern for crop productions) as the result of optimization. In mathematical terms, PMP is an inverse problem of quadratic programming (QP), where the objective function is calibrated on the basis of the Kuhn-Tucker conditions for the optimization of the QP model and of a linear programming model that is prepared for the calibration of parameters in the QP model. The two types of optimum conditions derived from the models are combined to obtain linear equations for the calibration of the QP model. However, as is often the case with inverse problems, the equations for the calibration are indefinite because the number of parameters to be calibrated surpasses the number of equations. As a result, various methods have been proposed to solve this so-called ill-posed problem. The main objectives of the present paper are to examine how the calibration methods developed in previous PM P models are related to one another and to propose practical procedures for determining which calibration method is the most appropriate from the viewpoint of sensitivity analyses. A simple conceptual framework is proposed to relate the previously developed calibration methods, and it is then applied to exemplify criteria for selecting a calibration method from the viewpoint of simulation results. A new direction in PMP-based farm modeling in which more feasible simulation results can be derived is also discussed.