Advertisement

Annals of Operations Research

, Volume 275, Issue 2, pp 607–621 | Cite as

Robust newsvendor problems: effect of discrete demands

  • Anh NinhEmail author
  • Honggang Hu
  • David Allen
Original Research
  • 138 Downloads

Abstract

Distribution-free newsvendor models often assume continuous demand distributions to facilitate analysis and computation. However, in practice, discrete demand is a natural phenomenon. So far, there exists no analytical and computational results in the literature under this setting. Thus, the goal of this paper is to investigate the newsvendor problems with partial information when the demand is discrete and solve them using the so-called discrete moment problems. Numerical results are presented to illustrate the value of discrete information.

Keywords

Discrete moment Newsvendor problems Stop-loss Discrete demand Shape information 

Notes

Acknowledgements

The authors would like to thank the referees for the constructive comments and discussions that led to this improved version of the paper.

References

  1. Alizadeh, F., & Goldfarb, D. (2003). Second-order cone programming. Mathematical Programming, 95(1), 3–51.Google Scholar
  2. 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.Google Scholar
  3. Axsäter, S. (2013). When is it feasible to model low discrete demand by a normal distribution? OR spectrum, 35(1), 153–162.Google Scholar
  4. Courtois, C., & Denuit, M. (2009). Moment bounds on discrete expected stop-loss transforms, with applications. Methodology and Computing in Applied Probability, 11(3), 307–338.Google Scholar
  5. Gallego, G., & Moon, I. (1993). The distribution free newsboy problem: Review and extensions. Journal of the Operational Research Society, 44(8), 825–834.Google Scholar
  6. Kumaran, V., & Swarnalatha, R. (2017). Bounds for the probability of union of events following monotonic distribution. Discrete Applied Mathematics, 223, 98–119.Google Scholar
  7. Lemke, C. E. (1954). The dual method of solving linear programming problem. Naval Research Logistics Quarterly, 1(1), 36–47.Google Scholar
  8. Mádi-Nagy, G., & Prékopa, A. (2004). On multivariate discrete moment problems and their applications to bounding expectations and probabilities. Mathematics of Operations Research, 29(2), 229–258.Google Scholar
  9. Mádi-Nagy, G. (2008). On multivariate discrete moment problems: Generalization of the bivariate min algorithm for higher dimensions. SIAM Journal on Optimization, 19(4), 1781–1806.Google Scholar
  10. Mádi-Nagy, G. (2012). Polynomial bases on the numerical solution of the multivariate discrete moment problem. Annals of Operations Research, 200(1), 75–92.Google Scholar
  11. Natarajan, K., Sim, M., & Uichanco, J. (2018). Asymmetry and ambiguity in newsvendor models. Management Science, 64(7), 2973–3468.Google Scholar
  12. Ninh, A., & Prékopa, A. (2013). The discrete moment problem with fractional moments. Operations Research Letters, 41(6), 715–718.Google Scholar
  13. Ninh, A., & Prékopa, A. (2015). Log-concavity of compound distributions with applications in stochastic optimization. Discrete Applied Mathematics, 161(18), 3017–3027.Google Scholar
  14. Ninh, A., & Pham, M. (2018). Logconcavity, twice-logconcavity and Turán-type inequalities. Annals of Operations Research,.  https://doi.org/10.1007/s10479-018-2923-y.Google Scholar
  15. Perakis, G., & Guillaume, R. (2008). Regret in the newsvendor model with partial information. Operations Research, 56(1), 188–203.Google Scholar
  16. Prékopa, A. (1988). Boole-Bonferroni inequalities and linear programming. Operations Research, 36(1), 145–162.Google Scholar
  17. Prékopa, A. (1990a). Sharp bounds on probabilities using linear programming. Operations Research, 38(2), 227–239.Google Scholar
  18. Prékopa, A. (1990b). The discrete moment problem and linear programming. Discrete Applied Mathematics, 27(3), 235–254.Google Scholar
  19. Prékopa, A. (1990c). Totally positive linear programming problems. In L. J. Leifmann (Ed.), Functional analysis, optimization and mathematical economics. A collection of papers dedicated to the memory of L. V. Kantorovich (pp. 197–207). New York: Oxford University Press.Google Scholar
  20. Prékopa, A. (1992). Inequalities on expectations based on the knowledge of multivariate moments. Lecture Notes-Monograph Series, 309–331.Google Scholar
  21. Prékopa, A. (1995). Stochastic Programming. Dordrecht: Kluwer Scientific.Google Scholar
  22. Prékopa, A. (1998). Bounds on probabilities and expectations using multivariate moments of discrete distributions. Studia Scientiarum Mathematicarum Hungarica, 34(1), 349–378.Google Scholar
  23. Prékopa, A. (2001). Discrete higher order convex functions and their applications. In N. Hadjisavvas, J. E. Martínez-Legaz, J. P. Penot (Eds.), Generalized Convexity and Generalized Monotonicity. Lecture Notes in Economics and Mathematical Systems (Vol. 502). Berlin, Heidelberg: Springer.Google Scholar
  24. Prékopa, A. (2009). Inequalities for discrete higher order convex functions. Journal of Mathematical Inequalities, 3(4), 485–498.Google Scholar
  25. 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), 521–75.Google Scholar
  26. Prékopa, A., Subasi, M., & Subasi, E. (2008). Sharp bounds for the probability of the union of events under unimodality condition. European Journal of Pure and Applied Mathematics, 1(1), 60–81.Google Scholar
  27. Scarf, H. (1958). A min-max solution of an inventory problem. In Studies in the Mathematical Theory of Inventory and Production. Stanford University Press.Google Scholar
  28. Subasi, E., Subasi, M., & Prékopa, A. (2009). Discrete moment problems with distributions known to be unimodal. Mathematical Inequalities and Applications, 12(3), 587–610.Google Scholar
  29. Swaminathan, J. M., & Shanthikumar, J. G. (1999). Supplier diversification: Effect of discrete demand. Operations Research Letters, 24(5), 213–221.Google Scholar
  30. Swarnalatha, R., & Kumaran, V. (2017). Bounds for the probability of the union of events with unimodality. Annals of Operations Research,.  https://doi.org/10.1007/s10479-017-2629-6.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCollege of William and MaryWilliamsburgUSA
  2. 2.Warrington College of BusinessUniversity of FloridaGainesvilleUSA

Personalised recommendations