Abstract
We consider the robust newsvendor problem where the demand follows a compound Poisson distribution, but its distribution is only partly known. This assumption means that customers arrive according to a Poisson process with a given intensity, while the size of customer demand is another random variable. Yet, the newsvendor only knows the expectation and variance of the demand sizes. Given limited information, a plausible approach, put forth in prior work, is to evaluate the moments of the aggregate demand (from all customers) and then determine the respective order quantity. Instead, this paper suggests to make the best use of all the information contained in the first few moments of demand sizes as well as the structural properties of a compound demand distribution to compute a better ordering quantity. To achieve this goal, we propose a new decision model employing Panjer’s recursion as constraints. The attendant optimization problem is then solved via convex relaxation using McCormick envelopes. Numerical results confirm that the newsvendor can gain a significant increase in expected profit using the new modeling approach to make his/her ordering decision.
References
Andersson, J., Jörnsten, K., Nonås, S. L., Sandal, L., & Ubøe, J. (2013). A maximum entropy approach to the newsvendor problem with partial information. European Journal of Operational Research, 228(1), 190–200.
Archibald, B. C. (1981). Continuous review (s, S) policies with lost sales. Management Science, 27(10), 1171–1177.
Archibald, B. C., & Silver, E. A. (1978). (s, S) policies under continuous review and discrete compound poisson demand. Management Science, 24(9), 899–909.
Axsäter, S. (2013). When is it feasible to model low discrete demand by a normal distribution? OR Spectrum, 35(1), 153–162.
Axsäter, S. (2015). Inventory control, volume 225 of International Series in Operations Research Management Science. Springer.
Chen, X., He, S., Jiang, B., Ryan, C. T., & Zhang, T. (2020). The discrete moment problem with nonconvex shape constraints. Operations Research. https://doi.org/10.1287/opre.2020.1990.
Choi, S., & RuszczyńSki, A. (2008). A risk-averse newsvendor with law invariant coherent measures of risk. Operations Research Letters, 36(1), 77–82.
Dominey, M. J. G., & Hill, R. M. (2004). Performance of approximations for compound poisson distributed demand in the newsboy problem. International Journal of Production Economics, 92(2), 145–155.
Gallego, G., & Moon, I. (1993). The distribution free newsboy problem: Review and extensions. Journal of the Operational Research Society, 44(8), 825–834.
Gallego, G., Ryan, J. K., & Simchi-Levi, D. (2001). Minimax analysis for finite-horizon inventory models. Iie Transactions, 33(10), 861–874.
Gerber, H. U. (1982). On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums. Insurance: Mathematics and Economics, 1(1), 13–18.
Hess, K. T., Liewald, A., & Schmidt, K. D. (2002). An extension of panjer’s recursion. ASTIN Bulletin: The Journal of the IAA, 32(2), 283–297.
Hesselager, O. (1994). A recursive procedure for calculation of some compound distributions. ASTIN Bulletin: The Journal of the IAA, 24(1), 19–32.
Kitaeva, A. V., Subbotina, V. I., & Zmeev, O. A. (2015). The newsvendor problem with fast moving items and a compound poisson price dependent demand. IFAC-PapersOnLine, 48(3), 1375–1379.
Larsen, C., & Thorstenson, A. (2008). A comparison between the order and the volume fill rate for a base-stock inventory control system under a compound renewal demand process. Journal of the Operational Research Society, 59(6), 798–804.
Lemke, C. E. (1954). The dual method of solving the linear programming problem. Naval Research Logistics Quarterly, 1(1), 36–47.
McCormick, G. P. (1976). Computability of global solutions to factorable nonconvex programs: Part i-convex underestimating problems. Mathematical Programming, 10(1), 147–175.
Natarajan, K., Sim M., & Uichanco, J. (2008). Asymmetry and ambiguity in newsvendor models. Working paper.
Ninh, A., Hu, H., & Allen, D. (2019). Robust newsvendor problems: Effect of discrete demands. Annals of Operations Research, 275(2), 607–621.
Panjer, H. H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 22–26.
Perakis, G., & Roels, G. (2008). Regret in the newsvendor model with partial information. Operations Research, 56(1), 188–203.
Petruzzi, N. C., & Dada, M. (1999). Pricing and the newsvendor problem: A review with extensions. Operations Research, 47(2), 183–194.
Prak, D., Teunter, R., Babai, M. Z., Syntetos, A. A., & Boylan, J. E. (2018). Forecasting and inventory control with compound poisson demand using periodic demand data. SOM Research Reports; Vol. 2018, No. 010). Groningen: University of Groningen, SOM research school.
Prékopa, A. (1988). Boole–Bonferroni inequalities and linear programming. Operations Research, 36(1), 145–162.
Prékopa, A. (1990a). Sharp bounds on probabilities using linear programming. Operations Research, 38(2), 227–239.
Prékopa, A. (1990b). The discrete moment problem and linear programming. Discrete Applied Mathematics, 27(3), 235–254.
Prékopa, A. (1995). Stochastic programming. Dordrecht: Kluwer Scientific.
Prékopa, A., Ninh, A., & Alexe, G. (2016). On the relationship between the discrete and continuous bounding moment problems and their numerical solutions. Annals of Operations Research, 238(1–2), 521–575.
Qin, Y., Wang, R., Vakharia, A. J., Chen, Y., & Seref, M. M. H. (2011). The newsvendor problem: Review and directions for future research. European Journal of Operational Research, 213(2), 361–374.
Saghafian, S., & Tomlin, B. (2016). The newsvendor under demand ambiguity: Combining data with moment and tail information. Operations Research, 64(1), 167–185.
Scarf, H. (1958). A min-max solution of an inventory problem. In Studies in the mathematical theory of inventory and production.
Sherbrooke, C. C. (1968). Discrete compound poisson processes and tables of the geometric poisson distribution. Naval Research Logistics Quarterly, 15(2), 189–203.
Sundt, B., & Jewell, W. S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 27–39.
Teunter, R. H., Syntetos, A. A., & Babai, M. Z. (2010). Determining order-up-to levels under periodic review for compound binomial (intermittent) demand. European Journal of Operational Research, 203(3), 619–624.
Turrini, L., & Meissner, J. (2019). Spare parts inventory management: New evidence from distribution fitting. European Journal of Operational Research, 273(1), 118–130.
Yue, J., Chen, B., & Wang, M. C. (2006). Expected value of distribution information for the newsvendor problem. Operations Research, 54(6), 1128–1136.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ninh, A. Robust newsvendor problems with compound Poisson demands. Ann Oper Res 302, 327–338 (2021). https://doi.org/10.1007/s10479-021-03996-3
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-021-03996-3