Optimization of the Second-Order Logarithmic Machining Economics Problem by Extended Geometric Programming Part II: Posynomial Constraints

Abstract

An algorithm is presented for solution of the machining economics problem with a Quadratic Posylognomial (QPL) objective function and single term posynomial constraints, meeting certain sufficient conditions. The algorithm applies to minimum cost or maximum productivity when the tool-life equation is a single term QPL and the removal rate is a single-term posynomial. A peripheral end-milling example, using the same tool-life equation and cost parameters as Part I, with the addition of experimentally derived constraints, is solved to illustrate the computational aspects of the algorithm. The QPL and posynomial (Taylor) formulations of the constrained machining problem are compared using the same experimental tool-life data. The QPL formulation is based on a quadratic logarithmic model whereas the posynomial formulation is based on a linear logarithmic tool-life model. An optimum without active constraints is possible using the QPL formulation in several independent machining variables, such as feed, speed, and depth, whereas the posynomial optimum, by nature, requires active constraints for more than one independent variable. This is tantamount to having additional “degrees of freedom” for optimization, as illustrated by the example problem.