Halkos, George and Kevork, Ilias (2011): Nonnegative demand in newsvendor models:The case of singly truncated normal samples.

PDF
MPRA_paper_31842.pdf Download (347Kb)  Preview 
Abstract
This paper considers the classical newsvendor model when demand is normally distributed but with a large coefficient of variation. This leads to observe with a nonnegligible probability negative values that do not make sense. To avoid the occurrence of such negative values, first, we derive generalized forms for the optimal order quantity and the maximum expected profit using properties of singly truncated normal distributions. Since truncating at zero produces nonsymmetric distributions for the positive values, three alternative models are used to develop confidence intervals for the true optimal order quantity and the true maximum expected profit under truncation. The first model assumes traditional normality without truncation, while the other two models assume that demand follows (a) the lognormal distribution and (b) the exponential distribution. The validity of confidence intervals is tested through MonteCarlo simulations, for low and high profit products under different sample sizes and alternative values for coefficient of variation. For each case, three statistical measures are computed: the coverage, namely the estimated actual confidence level, the relative average half length, and the relative standard deviation of half lengths. Only for very few cases the normal and the lognormal model produce confidence intervals with acceptable coverage but these intervals are characterized by low precision and stability.
Item Type:  MPRA Paper 

Original Title:  Nonnegative demand in newsvendor models:The case of singly truncated normal samples 
Language:  English 
Keywords:  Inventory Management; Newsvendor model; Truncated normal; Demand estimation; Confidence intervals; MonteCarlo simulations 
Subjects:  C  Mathematical and Quantitative Methods > C4  Econometric and Statistical Methods: Special Topics > C44  Operations Research; Statistical Decision Theory C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models; Multiple Variables > C34  Truncated and Censored Models; Switching Regression Models C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C15  Statistical Simulation Methods: General 
Item ID:  31842 
Depositing User:  Nickolaos Tzeremes 
Date Deposited:  26. Jun 2011 10:21 
Last Modified:  12. Feb 2013 20:45 
References:  Agrawal, N., and Smith, SA., 1996. Estimating Negative Binomial Demand for Retail Inventory Management with Unobservable Lost Sales. In Naval Research Logistics, Vol. 43, pp. 839–861. Barr, DR., and Sherrill, ET., 1999. Mean and Variance of Truncated Normal Distributions. In The American Statistician, Vol. 53, No. 4, pp. 357361. BarndorffNielsen, O., 1978. Information and Exponential Families, Wiley, Norwich. Bebu, I., and Mathew, T., 2009. Confidence Intervals for Limited Moments and Truncated Moments in Normal and Lognormal models. In Statistics and Probability Letters, Vol. 79, pp. 375380. Bell, PC., 2000. Forecasting Demand Variation when there are Stockouts. In Journal of the Operational Research Society, Vol 51, pp. 358–363. Braden, D., and Freimer, M., 1991. Informational Dynamics of Censored Observations. In Management Science Vol. 37, No. 11, pp. 13901404. Castillo del J. and Puig P., 1999. The Best Test of Exponentiality against Singly Truncated Normal Alternatives. In Journal of the American Statistical Association, Vol. 94, No. 446, pp. 529532. Cohen, AC., 1950. Estimating the Mean and Variance of Normal Populations from Singly Truncated and Doubly Truncated Samples. In The Annals of Mathematical Statistics, Vol. 21, pp. 557569. Cohen, AC., 1961. Tables for Maximum Likelihood Estimates: Singly Truncated and Singly Censored Samples. In Technometrics, Vol. 3, pp. 535541. Cohen, AC., 1991. Truncated and Censored Samples: Theory and applications. Marcel Dekker, New York. Conrad, SA., 1976. Sales Data and the Estimation of Demand. In Operational Research Quarterly, Vol. 27, pp. 123–127. Consortini, A., and Conforti G., 1984. Detector Saturastion Effect on higherorder Moments of Intensity Fluctuations in Atmospheric Laser Propagation Measurement. In Journal of Optical Society of America A, Vol. 1, No. 11, pp. 10751077. Davis, DJ., 1952. An Analysis of Some Failure Data. In Journal of the American Statistical Association, Vol. 47, pp. 113150. Efron, B., 1978. The Geometry of Exponential Families. In Annals of Statistics, Vol. 6, pp. 362 376. Fisher, RA., 1931. Properties and Applications of Hh Functions. In Introduction to Mathematical Tables, British Association for the Advancement of Science, Vol. 1, pp. xxvixxxv. Gupta, A., 1952. Estimation of the Mean and the Standard Deviation of the Normal Population from a Censored Sample. In Biometrica, Vol. 39, 260273. Hald, A., 1949. Maximum Likelihood Estimation of the Parameters of a Normal Distribution Which is Truncated at a Known Point. In Skandinavisk Aktuarietidskrifi, Vol. 32, pp. 119 134. Halperin, M., 1952a. Maximum Likelihood Estimation in Truncated Samples. In Annals of Mathematical Statistics, Vol. 23, pp. 226238. Halperin, M., 1952b. Estimation in the truncated normal distribution. In Journal of the American Statistical Association, Vol. 47, 457465. Harpaz, G., Lee, W., and Winkler, R.L., 1982. Learning, Experimentation, and the Optimal Output Decisions of a Competitive Firm. In Management Science, Vol. 28, pp. 589603. Hill, RM., 1997. Applying Bayesian Methodology with a Uniform Prior to the Single Period Inventory Model. In European Journal of Operational Research, Vol. 98: pp. 555–562. Jawitz, JW., 2004. Moments of Truncated Continuous Univariate Distributions. In Advances in Water Resource Vol. 27, pp. 269_281. Johnson, NL., Kotz, S., and Bakakrishnan, N., 1994. Continuous Univariate Distributions. 2nd Edition. Wiley New York. Kevork, IS., 2010. Estimating the Optimal Order Quantity and the Maximum Expected Profit for SinglePeriod Inventory Decisions. In Omega, The International Journal of Management Science, Vol. 38, pp. 218227. Khouja, M., 1999. The SinglePeriod (Newsvendor) Problem: Literature Review and Suggestions for Future Research. In Omega, The International Journal of Management Science, Vol. 27, pp. 537–553. Knight, K., 1999. Mathematical Statistics. Taylor & Francis Ltd. Lau, H., 1997. Simple Formulas for the Expected Costs in the Newsboy Problem: an educational note. In European Journal of Operational Research, Vol. 100, pp. 557–61. Lee, A., 1915. Tables of Gaussian ‘Tail’ Functions When the ‘Tail’ is Larger Than the Body. In Biometrika,Vol. 10, pp. 208215. Letac, G. and Mora, M., 1990. Natural Real Exponential Families with Cubic Variance Functions. In Annals of Statistics, Vol. 18, pp. 137. Maddala, GS., 1983. LimitedDependent and Qualitative Variables in Econometrics. Cambridge University Press, Cambridge. Nahmias, S., 1994. Demand Estimation in Lost Sales Inventory Systems. In Naval Research Logistics, Vol. 41, pp. 739–757. Nauman, EB., and Buffham, BA., 1983. Mixing in Continuous Flow Systems. Wiley: New York. Schweitzer, ME., and Cachon, GP., 2000. Decision Bias in the Newsvendor Problem with a Known Demand Distribution: Experimental Evidence. In Management Science, Vol. 46, pp. 404–420. Strijbosch, LWG., and Moors, JJA., 2006. Modified Normal Demand Distributions in (R,S)Inventory Control. In European Journal of Operational Research, Vol. 172, 201212. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/31842 