Double precision rational approximation algorithms for the standard normal first and second order loss functions

Abstract We present double precision algorithms based upon piecewise rational approximations for the standard normal first and second order loss functions. These functions are used frequently in inventory management. No direct approximation or closed formulation exists for the standard first and second order loss functions. Current state-of-the-art algorithms require intermediate computations of the cumulative normal distribution or tabulations and they do not compute to full double precision. We deal with both these issues and present direct double precision accurate algorithms which are valid in the full range of double precision floating point numbers.

[1]  Gary R. Waissi,et al.  A sigmoid approximation of the standard normal integral , 1996 .

[2]  K. V. Mardia,et al.  Fisher's repeated normal integral function and shape distributions , 1998 .

[3]  Rasoul Haji,et al.  A multi-product continuous review inventory system with stochastic demand, backorders, and a budget constraint , 2004, Eur. J. Oper. Res..

[4]  Arthur F. Veinott,et al.  Analysis of Inventory Systems , 1963 .

[5]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[6]  William J. Cody,et al.  Algorithm 715: SPECFUN–a portable FORTRAN package of special function routines and test drivers , 1993, TOMS.

[7]  W. Fraser,et al.  On the computation of rational approximations to continuous functions , 1962, CACM.

[8]  Wlodzimierz Bryc,et al.  A uniform approximation to the right normal tail integral , 2002, Appl. Math. Comput..

[9]  Haim Shore.  Simple Approximations for the Inverse Cumulative Function, the Density Function and the Loss Integral of the Normal Distribution , 1982 .

[10]  Ahmed M. M. Khodier,et al.  Restrictive Chebyshev rational approximation and applications to heat-conduction problems , 2003, Appl. Math. Comput..

[11]  S. Psarakis,et al.  APPROXIMATIONS TO THE NORMAL DISTRIBUTION FUNCTION AND AN EXTENDED TABLE FOR THE MEAN RANGE OF THE NORMAL VARIABLES , 2008 .

[12]  Paul H. Zipkin,et al.  Foundations of Inventory Management , 2000 .

[13]  G. Litvinov.  Approximate construction of rational approximations and the effect of error autocorrection. Applications , 2001, math/0101042.

[14]  Jean Marie Linhart,et al.  Algorithm 885: Computing the Logarithm of the Normal Distribution , 2008, TOMS.

[15]  C. Withers,et al.  Repeated integrals of the univariate normal as a finite series with the remainder in terms of Moran's functions , 2012 .

[16]  W. J. Cody,et al.  A Survey of Practical Rational and Polynomial Approximation of Functions , 1970 .

[17]  John P. Mills.  TABLE OF THE RATIO: AREA TO BOUNDING ORDINATE, FOR ANY PORTION OF NORMAL CURVE , 1926 .

[18]  Sven Axsäter,et al.  A simple procedure for determining order quantities under a fill rate constraint and normally distributed lead-time demand , 2006, Eur. J. Oper. Res..

[19]  David F. Pyke,et al.  Inventory management and production planning and scheduling , 1998 .

[20]  H. L. Loeb,et al.  On the Remez algorithm for non-linear families , 1970 .

[21]  C. Dunham.  Convergence of the Fraser-Hart algorithm for rational Chebyshev approximation , 1975 .

[22]  Expressions for the normal distribution and repeated normal integrals , 2006 .

[23]  Özalp Özer,et al.  Bounds, Heuristics, and Approximations for Distribution Systems , 2007, Oper. Res..