Constrained Probabilistic Continuous Review Inventory System with Mixture Shortage and Stochastic Lead Time Demand

Mona F. El-Wakeel, Hala A. Fergany


This paper derives the probabilistic continuous review backorders and lost sales inventory system when the order cost is a function of the order quantity. Our objective is to minimize the expected annual total cost under a restriction on the expected annual holding cost when the lead time demand follows some continuous distributions by using the Lagrangian method. Some published special cases are deduced and an illustrative numerical example with some graphs is added.


[1] Posner, M., & Yansouni, B. (1972). A class of inventory models with customer impatience. Naval Research Logistics Quarterly, 19(3), 483–492.

[2] Montgomery, D. C., Bazaraa, M., & Keswani, A. K. (2006). Inventory models with a mixture of backorders and lost sales. Naval Research Logistics Quarterly, 20(2), 255–263.

[3] Rosenberg, D. (1979). A new analysis of a lotsize model with partial backlogging. Naval Research Logistics Quarterly, 26(2), 349–353.

[4] KYUNG, S. (1982). Inventory model with partial backorders. International Journal of Systems Science, 13(12), 1313–1317.

[5] Rabinowitz, G., Mehrez, A., Chu, C.-W., & Patuwo, B. E. (1995). A partial backorder control for continuous review (r, q) inventory system with poisson demand and constant lead time. Computers &

operations research, 22(7), 689–700.

[6] Zipkin, P. (2000). Foundations of Inventory Management McGraw-Hill.

[7] Fergany, H., & El-Wakeel, M. (2006). Constrained probabilistic lost sales inventory system with continuous distributions and varying order cost. Journal of Association for the Advancement of Modeling &Simulation Techniques in Enterprises, 27, 3–4.

[8] Fergany, A., & El-Wakeel, M. (2006). Constrained probabilistic lost sales inventory system with normal distribution and varying order cost. Journal of Mathematics and Statistics, 2(1), 363–366.

[9] El-Wakeel, M. F. (2012). Constrained backorders inventory system with varying order cost: Lead time demand uniformly distributed. Journal of King Saud University-Science, 24(3), 285–288.

[10] Hadley, G. (1963). Analysis of Inventory Systems: By G. Hadley and TM Whitin. Prentice-Hall.

[11] Fabrycky,W. J., & Banks, J. (1967). Procurement and inventory systems: theory and analysis. Reinhold.

[12] Hillier, F. S., & Lieberman, G. J. (1995). Introduction to operations research, volume 6. McGraw-Hill New York.

[13] Nahmias, S., & Cheng, Y. (1997). Production and operations analysis.


DOI (pdf):

DOI (indexed/included/archived):


  • There are currently no refbacks.

Copyright (c)

Share us to:   


How to do online submission to another Journal?

If you have already registered in Journal A, then how can you submit another article to Journal B? It takes two steps to make it happen:

1. Register yourself in Journal B as an Author

Find the journal you want to submit to in CATEGORIES, click on “VIEW JOURNAL”, “Online Submissions”, “GO TO LOGIN” and “Edit My Profile”. Check “Author” on the “Edit Profile” page, then “Save”.

2. Submission

Go to “User Home”, and click on “Author” under the name of Journal B. You may start a New Submission by clicking on “CLICK HERE”.

We only use the following emails to deal with issues about paper acceptance, payment and submission of electronic versions of our journals to databases:;;;

 Articles published in Advances in Natural Science are licensed under Creative Commons Attribution 4.0 (CC-BY).


Address: 9375 Rue de Roissy Brossard, Québec, J4X 3A1, Canada

Telephone: 1-514-558 6138
Website: Http://; Http://;

Copyright © 2010 Canadian Research & Development Centre of Sciences and Cultures