A New Proof for the Tight Range of Optimal Order Quantities for the Newsboy Problem with Mean and Standard Deviation

Author(s)
Jinfeng Yue

Affiliation(s)

Department of Management and Marketing, Jennings A. Jones College of Business, Middle Tennessee State University, Murfreesboro, USA.

Department of Management and Marketing, Jennings A. Jones College of Business, Middle Tennessee State University, Murfreesboro, USA.

ABSTRACT

In the classical Newsboy problem, we provide a new proof for the tight range of optimal order quantities for the newsboy problem when only the mean and standard deviation of demand are available. The new proof is only based on the definition of the optimal solution therefore it is the most straightforward method. It is also shown that the classical Scarf’s rule is the mid-point of the range of optimal order quantities. This provides an additional understanding of Scarf’s order rule as a distribution free decision.

In the classical Newsboy problem, we provide a new proof for the tight range of optimal order quantities for the newsboy problem when only the mean and standard deviation of demand are available. The new proof is only based on the definition of the optimal solution therefore it is the most straightforward method. It is also shown that the classical Scarf’s rule is the mid-point of the range of optimal order quantities. This provides an additional understanding of Scarf’s order rule as a distribution free decision.

Cite this paper

J. Yue, "A New Proof for the Tight Range of Optimal Order Quantities for the Newsboy Problem with Mean and Standard Deviation,"*American Journal of Operations Research*, Vol. 2 No. 2, 2012, pp. 203-206. doi: 10.4236/ajor.2012.22023.

J. Yue, "A New Proof for the Tight Range of Optimal Order Quantities for the Newsboy Problem with Mean and Standard Deviation,"

References

[1] G. Hadley and T. Whitin, “Analysis of Inventory Systems,” Prentice-Hall, New Jersey, 1963.

[2] F. Hillier and G. Lieberman, “Introduction to Operations Research,” 5th Edition, Holden-Day, California, 1990.

[3] S. Nahmias, “Production and Operations Analysis,” Irwin, Illinois, 1989.

[4] H. A. Scarf, “A Min-Max Solution of an Inventory Problem,” In: K. J. Arrow, S. Karlin and H. E. Scarf, Eds., Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, California, 1958, pp. 201-209.

[5] G. Gallego and I. Moon, “The Distribution Free Newsboy Problem: Review and Extensions,” Operations Research Society, Vol. 44, 1993, pp. 825-834.

[6] I. Moon and I. Choi, “The Distribution Free Newsboy Problem with Balking,” Operations Research Society, Vol. 46, 1995, pp. 537-542

[7] G. Gallego, “A Minimax Distribution Free Procedure for the (Q, R) Inventory Mode,” Operations Research Letters, Vol. 11, No. 1, 1992, pp. 55-60. doi:10.1016/0167-6377(92)90063-9

[8] I. Moon and G. Gallego, “Distribution Free Procedures for Some Inventory Models,” Operations Research Society, Vol. 45, 1994, pp. 651-658.

[9] G. Gallego, “New Bounds and Heuristics for (Q, R) Policies,” Management Science, Vol. 44, No. 2, 1998, pp. 219-223. doi:10.1287/mnsc.44.2.219

[10] M. Hariga and M. Ben-Daya, “Some Stochastic Inventory Models with Deterministic Variable Lead Time,” European Journal of Operations Research, Vol. 113, No. 1, 1999, pp. 42-51. doi:10.1016/S0377-2217(97)00441-4

[11] J. Yue, B. Chen and M. C. Wang, “Expected Value of Distribution Information for the Newsvendor Problem,” Operations Research, Vol. 54, No. 6, 2006, pp. 1128- 1136. doi:10.1287/opre.1060.0318

[1] G. Hadley and T. Whitin, “Analysis of Inventory Systems,” Prentice-Hall, New Jersey, 1963.

[2] F. Hillier and G. Lieberman, “Introduction to Operations Research,” 5th Edition, Holden-Day, California, 1990.

[3] S. Nahmias, “Production and Operations Analysis,” Irwin, Illinois, 1989.

[4] H. A. Scarf, “A Min-Max Solution of an Inventory Problem,” In: K. J. Arrow, S. Karlin and H. E. Scarf, Eds., Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, California, 1958, pp. 201-209.

[5] G. Gallego and I. Moon, “The Distribution Free Newsboy Problem: Review and Extensions,” Operations Research Society, Vol. 44, 1993, pp. 825-834.

[6] I. Moon and I. Choi, “The Distribution Free Newsboy Problem with Balking,” Operations Research Society, Vol. 46, 1995, pp. 537-542

[7] G. Gallego, “A Minimax Distribution Free Procedure for the (Q, R) Inventory Mode,” Operations Research Letters, Vol. 11, No. 1, 1992, pp. 55-60. doi:10.1016/0167-6377(92)90063-9

[8] I. Moon and G. Gallego, “Distribution Free Procedures for Some Inventory Models,” Operations Research Society, Vol. 45, 1994, pp. 651-658.

[9] G. Gallego, “New Bounds and Heuristics for (Q, R) Policies,” Management Science, Vol. 44, No. 2, 1998, pp. 219-223. doi:10.1287/mnsc.44.2.219

[10] M. Hariga and M. Ben-Daya, “Some Stochastic Inventory Models with Deterministic Variable Lead Time,” European Journal of Operations Research, Vol. 113, No. 1, 1999, pp. 42-51. doi:10.1016/S0377-2217(97)00441-4

[11] J. Yue, B. Chen and M. C. Wang, “Expected Value of Distribution Information for the Newsvendor Problem,” Operations Research, Vol. 54, No. 6, 2006, pp. 1128- 1136. doi:10.1287/opre.1060.0318