5 - Hidden Nonlinearities in Linear Goal Programming Models

Abstract

This chapter discusses hidden nonlinearities in linear goal programming models. When goals are linear and the relationship between deviational variables in the objective function is also linear, then the weighted goal programming (WGP) problems can be solved by the traditional simplex method. In the same linear context, the lexicographic goal programming (LGP) problems can be solved with very little difficulty by using the Simplex in a sequential way or by resorting to some extensions of the simplex, such as the modified or multiphase simplex among other algorithmic approaches. However, when the goals are not linear and/or the relationship between deviational variables in the objective function of the WGP Model or in any of the components of the achievement function of the LGP models is not linear, then the mathematical complications in their solution increase drastically. Although the direct linearization of fractional goals can lead to erroneous results, in some cases, it is a legitimate practice. By taking into account the computational advantages of this kind of straightforward linearization, it can be very useful to have available an operational test capable of detecting when this transformation is legitimate.