A linear programming approach to fitting an upper quadratic boundary line to natural rubber data

Abstract

In the context of biology, the problem of finding the maximum or minimum response in a cause and effect relationship is recognized as boundary line fitting. Frequently used methods involve fitting curves through the boundary points using the least square principle. However, identification of boundary points especially when there are not any multiple responses at values of the explanatory variable is ad hoc. Existing methods involve dividing the range of explanatory variable into bins and considering points with the maximum response or the response above some value in each bin. However, the results depend heavily on the way of dividing into bins and the number of bins. There is no agreement on the best way of dividing or the number of bins. Furthermore, the least square line is not consistent with the theory of limiting response because it goes through the points rather than going above all the points or below all the points, representing the boundary. This paper presents a new method that avoids all the above drawbacks. It involves the theory of linear programming. The proposed method has been compared with commonly used methods by using simulated data and shown to perform better. The method is illustrated by applying to experimental data on the response of latex yield of natural rubber to leaf nutrient concentrations.