# Technical note: A use of the complete squares method to solve and analyze a quadratic objective function with two decision variables exemplified via a deterministic inventory model with a mixture of backorders and lost sales

## Author

Listed:
• Francis Leung, Kit-Nam

## Abstract

Several researchers have recently derived formulae for economic-order quantities (EOQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions. In addition, this algebraic derivation immediately produces an individual formula for evaluating the minimum average annual cost. The purpose of this paper is twofold. Exemplifying a use of the complete squares method through solving and analyzing Montgomery et al.'s [Montgomery, D.C., Bazaraa, M.S., Keswani, A.C., 1973. Inventory models with a mixture of backorders and lost sales. Naval Research Logistics Quarterly 20, 255-263] model, i.e. the EOQ model taking into account the case of partial backordering first we can readily derive global optimal expressions from a non-convex quadratic cost function with two decision variables in an algebraic manner, second we can straightforwardly identify some analytic cases in a way that is not as easy to do this using calculus. A numerical example has been solved to illustrate the solution procedure. Finally, some special cases can be deduced from the EOQ model under study, and concluding remarks are drawn.

## Suggested Citation

• Francis Leung, Kit-Nam, 2008. "Technical note: A use of the complete squares method to solve and analyze a quadratic objective function with two decision variables exemplified via a deterministic inventory model with a mixture of b," International Journal of Production Economics, Elsevier, vol. 113(1), pages 275-281, May.
• Handle: RePEc:eee:proeco:v:113:y:2008:i:1:p:275-281
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## References listed on IDEAS

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1. Jason Chang, S.K. & Chuang, Jones P.C. & Chen, Hsiao-Jung, 2005. "Short comments on technical note--The EOQ and EPQ models with shortages derived without derivatives," International Journal of Production Economics, Elsevier, vol. 97(2), pages 241-243, August.
2. Wee, Hui Ming & Chung, Chun Jen, 2007. "A note on the economic lot size of the integrated vendor-buyer inventory system derived without derivatives," European Journal of Operational Research, Elsevier, vol. 177(2), pages 1289-1293, March.
3. Cardenas-Barron, Leopoldo Eduardo, 2001. "The economic production quantity (EPQ) with shortage derived algebraically," International Journal of Production Economics, Elsevier, vol. 70(3), pages 289-292, April.
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5. Grubbstrom, Robert W. & Erdem, Asli, 1999. "The EOQ with backlogging derived without derivatives," International Journal of Production Economics, Elsevier, vol. 59(1-3), pages 529-530, March.
6. Sphicas, Georghios P., 2006. "EOQ and EPQ with linear and fixed backorder costs: Two cases identified and models analyzed without calculus," International Journal of Production Economics, Elsevier, vol. 100(1), pages 59-64, March.
7. Chu, Peter & Chung, Kun-Jen, 2004. "The sensitivity of the inventory model with partial backorders," European Journal of Operational Research, Elsevier, vol. 152(1), pages 289-295, January.
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## Citations

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Cited by:

1. Leung, Kit-Nam Francis, 2010. "Some comments on "A simple method to compute economic order quantities"," European Journal of Operational Research, Elsevier, vol. 201(3), pages 960-961, March.
2. Leung, Kit-Nam Francis, 2009. "A generalization of sensitivity of the inventory model with partial backorders," European Journal of Operational Research, Elsevier, vol. 196(2), pages 554-562, July.
3. Joaquín Sicilia & Luis San-José & Juan García-Laguna, 2012. "An inventory model where backordered demand ratio is exponentially decreasing with the waiting time," Annals of Operations Research, Springer, vol. 199(1), pages 137-155, October.
4. Zhang, Ren-qian & Kaku, Ikou & Xiao, Yi-yong, 2011. "Deterministic EOQ with partial backordering and correlated demand caused by cross-selling," European Journal of Operational Research, Elsevier, vol. 210(3), pages 537-551, May.
5. Sphicas, Georghios P., 2014. "Generalized EOQ formula using a new parameter: Coefficient of backorder attractiveness," International Journal of Production Economics, Elsevier, vol. 155(C), pages 143-147.
6. Bijvank, Marco & Vis, Iris F.A., 2011. "Lost-sales inventory theory: A review," European Journal of Operational Research, Elsevier, vol. 215(1), pages 1-13, November.
7. Pentico, David W. & Drake, Matthew J., 2011. "A survey of deterministic models for the EOQ and EPQ with partial backordering," European Journal of Operational Research, Elsevier, vol. 214(2), pages 179-198, October.

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