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Quadratic geometric programming with application to machining economics. (English) Zbl 0558.90074
Geometric Programming is extended to include convex quadratic functions. Generalized Geometric Programming is applied to this class of programs to obtain a convex dual program. Machining economics problems fall into this class. Such problems are studied by applying this duality to a nested set of three problems. One problem is zero degree of difficulty and the solution is obtained by solving a simple system of equations. The inclusion of a constraint restricting the force on the tool to be less than or equal to the breaking force provides a more realistic solution. This model is solved as a program with one degree of difficulty. Finally the behavior of the machining cost per part is studied parametrically as a function of axial depth.

MSC:
90C25 Convex programming
90C99 Mathematical programming
90B30 Production models
90C20 Quadratic programming
49N15 Duality theory (optimization)
90C55 Methods of successive quadratic programming type
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[1] C.S. Beightler and D.T. Phillips,Applied geometric programming, Wiley, New York, 1976). · Zbl 0344.90034
[2] B.N. Colding, ”A three-dimensional tool life equation-machining economics”,Transactions of the American Society of Mechanical Engineers Series B 81 (1959) 239–250.
[3] R.J. Duffin, E.L. Peterson and C. Zener,Geometric programming (Wiley, New York, 1967). · Zbl 0171.17601
[4] J.G. Ecker, ”Geometric programming methods, computations and applications”,SIAM Review 22 (1980) 338–362. · Zbl 0438.90088 · doi:10.1137/1022058
[5] W.F. Hastings, P.L.B. Oxley and M.G. Stevenson, ”Predicting a material’s machining characteristics using flow stress properties obtained from high-speed compression tests”,Proceedings of the Institute of Mechanical Engineers 188 (1974) 245–252. · doi:10.1243/PIME_PROC_1974_188_027_02
[6] C.L. Hough, Jr. and R.E. Goforth, ”Quadratic posylognomials: An extension of posynomial geometric programming”,American Institute of Industrial Engineers Transactions 13 (1981) 47–54.
[7] C.L. Hough, Jr. and R.E. Goforth, ”Optimization of the second order machining economics problem by extended geometric programming part I–Unconstrained”,American Institute of Industrial Engineers 13 (1981) 151–159.
[8] C.L. Hough, Jr. and R.E. Goforth, ”Optimization of the second-order logarithmic machining economics problem by extended geometric programming part II–Posynomial constraints”,American Institute of Industrial Engineers 13 (1981) 234–242.
[9] T.R. Jefferson and C.H. Scott, ”Avenues of geometric programming”,New Zealand Operational Research 6 (1978) 109–136.
[10] T.R. Jefferson and C.H. Scott, ”Avenues of geometric programming–Applications”,New Zealand Operational Research 7 (1979) 51–68.
[11] E.L. Peterson, ”Geometric programming”,SIAM Review 18 (1976) 1–52. · Zbl 0331.90057 · doi:10.1137/1018001
[12] E.L. Peterson and J.G. Ecker, ”Geometric programming: Duality in quadratic programming andl p -approximation I”, in: H.W. Kuhn and A.W. Tucker, eds.,Proceedings of the International Symposium on Mathematical Programming (Princeton University Press, Princeton, 1970) pp. 445–480. · Zbl 0228.90039
[13] R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, 1970). · Zbl 0193.18401
[14] S.M. Wu, ”Tool life testing by response surface methodology–parts 1 and 2”,Transactions of the American Society of Mechanical Engineers Series B 86 (1964) 105–115.
[15] N. Zlatin, M. Field, V.A. Tipnis, R.G. Garrison, S. Bueschar and J.B. Kohis, ”Establishment of production machinability data”,Air Force Materials Laboratory Report AFML-TR-75-120 (August 1975).
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