Optimization of the Second Order Logarithmic Machining Economics Problem by Extended Geometric Programming Part I—Unconstrained

Abstract

Abstract An extended theory of geometric programming is applied to the machining economics problem utilizing a more accurate tool-life equation that is quadratic in the logarithms of the machining variables. The tool-life equation and corresponding unit cost or productivity functions are classified as quadratic posylognomials (QPL). By comparison, the extended Taylor tool-life equation and corresponding objective functions are classified as posynomials. Necessary conditions for a minimum of the unconstrained cost or productivity functions are given and are related to the R-T (Removal Rate-Tool Life) characteristic curve. Sufficient conditions for a minimum are also given and the problem is characterized in terms of the matrix of the second-order coefficients in the logarithmic tool-life equation. The computational aspects are illustrated with a peripheral end-milling example constructed from an experimentally derived tool-life equation from the literature. The theory and solution techniques for the unconstrained problem in Part I form a basis for solution of the machining economics problem with posynomial constraints in Part II (to appear in the September issue).