Optimization of the second order logarithmic machining economics problem by extended geometric programming

Abstract

The theory of geometric programming is extended to include a new function with logarithmic exponents. The function, defined as a Quadratic Posylognomial (QPL) is a series of nonlinear product terms with positive coefficients and positive variables. A QPL may be created by adding a linear function of the logarithms of the variables to the constant exponents of posynomial. The logarithm of each nonlinear term is a quadratic form in the logarithm of the primal variables. The primal-dual relationships and necessary conditions are developed and special cases where sufficient conditions exist are derived. The constrained Machining Economics Problem (MEP) using second order logarithmic tool life equations are characterized and formulated using the above theory. An algorithm to solve special cases that meet the sufficient conditions with single term posynomial constraints is developed. A peripheral end milling example problem, based on experimental tool life data, was solved using the algorithm. The QPL formulation using second order tool life models was compared to the posynomial formulation using extended Taylor tool life models.