The concept of a cutting rate-tool life (R-T) characteristic curve is extended to the general machining economics problem (MEP) with a quadratic-logarithmic tool life and constraint equations. The R-T characteristic curve presents the general loci of optima, which is useful in selecting optimal parameters for multiple machining conditions. The necessary and sufficient conditions for the global optimum of the unconstrained MEP are presented. These conditions are equivalently applied to the concept of the constrained R-T characteristic curve. In terms of quadratic geometric programming the objective function and constraints of the general MEP are called as quadratic posylognomials (QPL). The QPL problems are classified as convex and nonconvex and the convexity is determined by the second order terms of the tool life model. Nonlinear programming and an exhaustive method are demonstrated to determine the R-T characteristic curve for three cases of posynomial, convex QPL, and non-convex QPL problems.

1.
Beightler, C. S., and Phillips, D. T., 1976 Applied Geometric Programming, John Wiley & Sons, New York.
2.
Colding
B. N.
,
1959
, “
A Three-Dimensional Tool Life Equation—Machining Economics
,”
Transactions of the ASME
, Series B, Vol.
81
, pp.
239
250
.
3.
Ecker
J. G.
, and
Kupferschmid
,
1985
, “
A Computational Comparison of the Ellipsoid Algorithm with Several Nonlinear Programming Algorithms
,”
SIAM J. Control and Optimization
, Vol.
23
, No.
5
, pp.
657
674
.
4.
Friedman
M. Y.
, and
Tipnis
V. A.
,
1976
, “
Cutting Rate-Tool Life Characteristic Functions for Metal Removal Processes—Part I: Theory
,”
ASME Journal of Engineering for Industry
, Vol.
98
, pp.
481
486
.
5.
Hough
C. L.
,
1986
, “
Sufficient Conditions for Cutting Rate-Tool Life Characteristic Functions for Metal Removal Processes
,”
ASME Journal of Engineering for Industry
, Vol.
108
, pp.
235
237
.
6.
Hough
C. L.
, and
Goforth
R. E.
,
1981
, “
Quadratic Posylognomials: An Extension of Posynomial Geometric Programming
,”
AIIE Transactions
, Vol.
13
, No.
1
, pp.
47
54
.
7.
Hough
C. L.
, and
Goforth
R. E.
,
1981
, “
Optimization of the Second Order Logarithmic Machining Economics Problem by Extended Geometric Programming—Part 1: Unconstrained
,”
AIIE Transactions
, Vol.
13
, No.
2
, pp.
151
159
.
8.
Hough
C. L.
, and
Goforth
R. E.
,
1981
, “
Optimization of the Second Order Logarithmic Machining Economics Problem by Extended Geometric Programming—PART 2: Posynomial Constraints
,”
AIIE Transactions
, Vol.
13
, No.
3
, pp.
234
242
.
9.
Phillips, D. T., and Beightler, C. H., 1970, “Optimization in Tool Engineering Using Geometric Programming,” AIIE Transactions, pp. 355–360.
10.
Tipnis
V. A.
, and
Friedman
M. Y.
,
1976
, “
Cutting Rate-Tool Life Characteristic Functions for Metal Removal Processes-Part 2: Verifications and Applications
,”
ASME Journal of Engineering for Industry
, Vol.
98
, No.
2
, pp.
487
496
.
11.
Wu
S. M.
,
1964
, “
Tool Life Testing by Response Surface Methodology—Parts 1 and 2
,”
Transactions of the ASME
, Series B, Vol.
86
, pp.
105
116
.
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