Volume List  / Volume 11 (2)



DOI: 10.7708/ijtte.2021.11(2).11

11 / 2 / 323-340 Pages


Md Abu Helal - Southern Arkansas University – Magnolia, Arkansas, USA -

Alain Bensoussan - The University of Texas at Dallas – Richardson, Texas, USA -

Viswanath Ramakrishna - The University of Texas at Dallas – Richardson, Texas, USA -

Suresh P. Sethi - The University of Texas at Dallas – Richardson, Texas, USA -


This article is concerned with stochastic inventory control problems with backlog sales in stock-out situations. We examine an infinite horizon model for piecewise linear concave ordering costs. Unlike finite horizons, however, infinite horizons lead to a functional equation for the value function. Such functional equations are solved numerically. Here we give a rigorous theory which explicitly solves this functional equation. We consider both the scenario in which an optimal selection can be made among two suppliers, as well as the scenario in which inventory can be purchased with incremental quantity discounts from a single supplier. We provide conditions that guarantee the optimality of standard (s,S) policy. Moreover, when these conditions fail to hold, we show that an extended four parameter policy is optimal.

Download Article

Number of downloads: 447


Benjaafar, S.; Chen, D.; Yu, Y. 2018. Optimal policies for inventory systems with concave ordering costs, Naval Research Logistics 65: 291-302.


Bensoussan, A.; Helal, M.A.; Ramakrishna, V.; Sethi, S.P. 2020. Inventory policies for piecewise concave linear ordering cost. Working Paper, The University of Texas at Dallas, USA.


Bensoussan, A. 2011. Dynamic programming and inventory control. Volume 3 of Studies in Probability, Optimization and Statistics, IOS Press. 369p.


Beyer, D.; Sethi, S.P.; Taksar, M. 1998. Inventory Models with Markovian Demands and Cost Functions of Polynomial Growth, Journal of Optimization Theory and Application 98(2): 281-323.


Chen, X.; Simchi-Levi, D. 2012. Pricing and Inventory Management. The Oxford Handbook of Pricing Management. Oxford University Press, 1: 784-824. https://doi.org/10.1093/oxfordhb/9780199543175.013.0030.


Chen, Y.; Ray, S.; Song, Y. 2005. Optimal pricing and inventory control policy in periodic review systems with fixed ordering cost and lost sales, Naval Research Logist 53(2): 117-136.


Huh. W.; Janakiraman, G. 2008. (s,S) optimality in joint inventory pricing control: an alternative approach, Operations Research 56(3): 783-790.


Karlin, S.; Fabens, A. 1960. The (s,S)-Inventory Model Under Markovian Demand Process. In Mathematical Methods in the Social Sciences, Edited by K. Arrow, S. Karlin and P. Suppes, Stanford University Press, Stanford, California, 159-175.


Ozekici, S.; Parlar, M. 1995. Periodic review inventory models in random environments. Working Paper, McMaster University, Hamilton, Ontario. Porteus, E.L. 1971. On the optimality of generalized (s,S) Polices, Management Science 17: 411-426.


Scarf, H. 1960. The Optimality of (S, s) Policies in the Dynamic Inventory Problem. Mathematical Methods in the Social Sciences, Stanford University Press, Stanford, CA, 196-202.


Sethi, S.P.; Cheng, F. 1997. Optimality of (s, S) Policies in Inventory Models with Markovian Demand Processes, Operations Research 45(6): 931-939.


Singha, K.; Buddhakulsomsiri, J.; Parthanadee, P. 2017. Mathematical Model of (R, Q) Inventory Policy under Limited Storage Space for Continuous and Periodic Review Policies with Backlog and Lost Sales, Mathematical Problems in Engineering 2017: 1-9, Article ID 4391970.


Song, J.S.; Zipkin, P. 1993. Inventory Control in a Fluctuating Demand Environment, Operations Research 41(2): 351-370.


Song, Y.; Ray, S.; Boyaci, T. 2009. Optimal dynamic joint inventory-pricing control for multiplicative demand with fixed order costs and lost sales, Operations Research 57(1): 245-250.


Veinott, Jr. A.F. 1966. On the Optimality of (s,S) Inventory Policies: New Conditions and a New Proof, SIAM Journal on Applied Mathematics 14(5): 1067-1083.

Quoted IJTTE Works

Related Keywords