Unbiased estimation of maximum expected profits in the Newsvendor Model: a case study analysis

Halkos, George and Kevork, Ilias (2012): Unbiased estimation of maximum expected profits in the Newsvendor Model: a case study analysis.

Abstract

In the current paper we study a real life inventory problem whose operating conditions match to the principles of the classical newsvendor model. Applying appropriate tests to the available sample of historical demand data, we get the sufficient statistical evidences to support that daily demand is stationary, uncorrelated, and normally distributed. Given that at the start of each day, the selling price, the purchasing cost per unit, and the salvage value are known, and do not change through the whole period under investigation, we derive exact and asymptotic prediction intervals for the daily maximum expected profit. To evaluate their performance, we derive the analytic form of three accuracy information metrics. The first metric measures the deviation of the estimated probability of no stock-outs during the day from the critical fractile. The other two metrics relate the validity and precision of the two types of prediction interval to the variability of estimates for the ordered quantity. Both theoretical and empirical analysis demonstrates the importance of implications of the loss of goodwill to the adopted inventory policy. Operating the system at the optimal situation, this intangible cost element determines the probability of no stock-outs during the day, and assesses the precision of prediction intervals. The rising of the loss of goodwill leads to smaller estimates for the daily maximum expected profit and to wider prediction intervals. Finally, in the setting of the real life newsvendor problem, we recommend the asymptotic prediction interval since with samples over 25 observations this type of interval has higher precision and probability to include the daily maximum expected profit almost equal to the nominal confidence level.

• Item Type: MPRA Paper
• Language: English
• Keywords: Newsvendor model; Loss of goodwill; Target inventory measures; Prediction interval; Accuracy information metric
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; M - Business Administration and Business Economics ; Marketing ; Accounting ; Personnel Economics > M1 - Business Administration > M11 - Production Management; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C44 - Operations Research ; Statistical Decision Theory; D - Microeconomics > D2 - Production and Organizations > D24 - Production ; Cost ; Capital ; Capital, Total Factor, and Multifactor Productivity ; Capacity

References

Abramowitz, M., & Stegun, I.E. (1972). Handbook of mathematical functions. National Bureau of Standards, Washington, D.C.

Akcay, A., Biller, B., & Tayur, S. (2011). Improved inventory targets in the presence of limited historical data. Manufacturing & Service Operations Management, DOI 10.1287/ msom. 1100.0320.

Ali, M. M., Boylan, J. E., & Syntetos, A. A. (2011). Forecast errors and inventory performance under forecast information sharing. International Journal of Forecasting, doi:10.1016/j.ijforecast.2010.08.003.

Benzion, U., Cohen, Y., Peled, R., & Shavit, T. (2008). Decision making and the newsvendor problem: an experimental study. Journal of the Operational Research Society, 59, 1281–1287.

Benzion, U., Cohen, Y., & Shavit, T. (2010). The newsvendor problem with unknown distribution. Journal of the Operational Research Society, 61, 1022–1031.

Beutel, A. L., & Minner, S. (2011). Safety stock planning under causal demand forecasting. International Journal of Production Economics, doi:10.1016/j.ijpe.2011.04.017.

Choi, T. M, Chiu, C. H., & Chester, K. M. (2011). A fast fashion safety-first inventory model, Textile Research Journal, 81, 819–826.

Feng, T., Keller, L. R., & Zheng, X. (2011). Decision making in the newsvendor problem: A cross-national laboratory study. Omega, 39, 41–50.

Halkos, G. E., & Kevork, I. S. (2005). A comparison of alternative unit root tests. Journal of Applied Statistics, 32, 45–60.

Halkos, G. E., & Kevork, I. S. (2012a). The classical newsvendor model under normal demand with large coefficients of variation. MPRA Paper No. 40414, Online at http://mpra.ub.uni-muenchen.de/40414/.

Halkos, G. E., & Kevork, I. S. (2012b). Evaluating alternative frequentist inferential approaches for optimal order quantities in the newsvendor model under exponential demand. MPRA Paper No. 39650, Online at http://mpra.ub.uni-muenchen.de/39650/.

Halkos, G. E., & Kevork, I. S. (2012c). Validity and precision of estimates in the classical newsvendor model with exponential and Rayleigh demand. MPRA Paper No. 36460, Online at http://mpra.ub.uni-muenchen.de/36460/.

Harvey, A.C. (1993). Time series models. 2nd Edition, Prentice Hall, An imprint of Pearson Education Limited.

Hayes, R. H. (1969). Statistical estimation problems in inventory control. Management Science, 15, 686–701.

Janssen E, Strijbosch L, & Brekelmans R. (2009). Assessing the effects of using demand parameters estimates in inventory control and improving the performance using a correction function. International Journal of Production Economics, 118, 34–42.

Johnson, N. L., Kotz, S., & Bakakrishnan, N. (1994). Continuous Univariate Distributions. 2nd Edition. Wiley, New York.

Katircioglou, K. (1996). Essays in inventory control. Ph.D thesis, University of British Columbia, Vancouver, BC, Canada.

Keiding, N., Jensen, S. T., & Ranek, L. (1972). Maximum likelihood estimation of the size distribution of liver cell nuclei from the observed distribution in a plane section. Biometrics, 28, 813–829.

Kevork, I. S. (2010). Estimating the optimal order quantity and the maximum expected profit for single-period inventory decisions. Omega, 38, 218–227.

Knight, K. (1999). Mathematical Statistics. Taylor & Francis Ltd.

Lapin, L. L. (1994). Quantitative methods for business decisions with cases, 6th Edition, Duxbury Press, An International Thomson Publishing Company.

Lindgren, B. W. (1976). Statistical theory, 3rd Edition, London: Collier MacMillan International Editions.

Law, H. (1997). Simple formulas for the expected costs in the newsboy problem: an educational note. European Journal of Operational Research, 100, 557–561.

Liyanage, L. H., & Shanthikumar, J.G. (2005). A practical inventory control policy using operational statistics. Operations Research Letters, 33, 341–348.

Mostard, J., & Teunter, R. (2006). The newsboy problem with resalable returns: A single period model and case study. European Journal of Operational Research, 169, 81–96.

Mostard, J., Teunter, R., & de Coster, R. (2011). Forecasting demand for single-period products: A case study in the apparel industry. European Journal of Operational Research, 211, 139–147.

Olivares, M., Terwiesch, C., & Cassorla, L. (2008). Structural estimation of the newsvendor model: An application to reserving operating room time. Management Science, 54, 41–55

Ritchken, P. H., & Sankar, R. (1984). The effect of estimation risk in establishing safety stock levels in an inventory model. The Journal of the Operational Research Society, 35, 1091–1099.

Schweitzer M. E., & Cachon, G. P. (2000). Decision bias in the newsvendor problem with a known demand distribution: experimental evidence. Management Science, 46, 404–420.

Severini, T.A. (2005). Elements of distribution theory. Cambridge University Press.

Silver, E. A., & Rahnama, M. R. (1987). Biased selection of the inventory reorder point when demand parameters are statistically estimated. Engineering Costs and Production Economics, 12, 283–292.

Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory management and production planning and scheduling, 3rd Edition, New York, John Wiley and Sons.

Steinbrecher, G., & Shaw, W. T. (2008). Quantile mechanics. European Journal of Applied Mathematics, 19, 87–112

Su, R. H., & Pearn, W. L. (2011). Product selection for newsvendor-type products with normal demands and unequal costs. International Journal of Production Economics, 132, 214–222.

Syntetos, A. A., Nikolopoulos, K., & Boylan, J. E. (2010). Judging the judges through accuracy-implication metrics: The case of inventory forecasting. International Journal of Forecasting, 26, 134–143.