Optimal economic order quantity for buyer–distributor–vendor supply chain with backlogging derived without derivatives

Abstract

In this article, we first complement an inappropriate mathematical error on the total cost in the previously published paper by Chung and Wee [2007, ‘Optimal the Economic Lot Size of a Three-stage Supply Chain With Backlogging Derived Without Derivatives’, European Journal of Operational Research, 183, 933–943] related to buyer–distributor–vendor three-stage supply chain with backlogging derived without derivatives. Then, an arithmetic–geometric inequality method is proposed not only to simplify the algebraic method of completing prefect squares, but also to complement their shortcomings. In addition, we provide a closed-form solution to integral number of deliveries for the distributor and the vendor without using complex derivatives. Furthermore, our method can solve many cases in which their method cannot, because they did not consider that a squared root of a negative number does not exist. Finally, we use some numerical examples to show that our proposed optimal solution is cheaper to operate than theirs.

Keywords:

• supply chain management
• inventory
• arithmetic–geometric mean
• backlogging

Acknowledgements

The authors thank to the four anonymous referees for their constructive comments. This research was supported by the ART for Research from the William Paterson University of New Jersey and by the School of Business and the Tecnológico de Monterrey research fund numbers CAT128 and CAT185.