Skip to main content
SpringerLink
Log in

Discrete-Time Inventory Problems with Lead-Time and Order-Time Constraint

  • Chapter
  • First Online:
Optimization, Control, and Applications of Stochastic Systems

Part of the book series: Systems & Control: Foundations & Applications ((SCFA))

  • 1384 Accesses

Abstract

We consider a discrete-time inventory problem with infinite horizon, with a fixed lead time of several periods. When there is no restriction on the times of order, in particular when it is possible to make an order while some previous order is not yet delivered, then it is possible to reduce the problem to a standard one with no lead time (more precisely a lead time of one period) by considering the inventory position instead of the real inventory. This is a well-known result, which requires however the possibility of backlog. We consider here the same problem, this time imposing a constraint on the order times. It is not possible to make an order while some previous deliveries are pending. With this constraint, the problem cannot be reduced to a standard one for the inventory position.We solve the problem and show cases when the optimal policy is of (s, S) type.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Archibald, B.C.: Continuous review (s, S) policies with lost sales. Management Science. 27, 1171–1177 (1981).

    Article  MATH  Google Scholar 

  2. Benkherouf, L., Bensoussan, A.: On the Optimality of (s, S) inventory policies: A Quasi-Variational Inequalities Approach. SIAM Journal on Control and Optimization. 48, 756–762 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  3. Benkherouf, L.: On the optimality of (s, S) policies: A Quasi-variational Approach. Journal of Applied Mathematics and Stochastic Analysis. doi 158193, 1–9 (2008).

    Article  Google Scholar 

  4. Bensoussan, A., Çakanyidirim, M., Moussaoui, L.: Optimality of (s, S) policies for continuous-time inventory problems with order constraint. Annals of Operations Research. (In Press).

    Google Scholar 

  5. Bensoussan, A., Liu, R.H, Sethi, S.P.: Optimality of an (s, S) policy with compound Poisson and diffusion demand: A QVI approach. SIAM Journal on Control and Optimization. 44, 1650–1676 (2006).

    Google Scholar 

  6. Beyer, D., Sethi, S.P.: The classical average-cost inventory models of Iglehart and Veinott-Wagner revisited. Journal of Optimization Theory and Applications. 101, 523–555 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Federgruen, Schechner, Z.: Cost formulas for continuous review inventory models with fixed deliver lags. Operations Research. 31, 957–965 (1983).

    Google Scholar 

  8. Fox, E.J., Cox,E.L., Metters, R.: Optimal Inventory Policy with Two Suppliers. Operations Research. 54, 389–393 (2006).

    Article  MATH  Google Scholar 

  9. Hadley, G.J., Within, T.M.: Analysis of Inventory Systems. Prentice-Hall, New-Jersey (1963).

    MATH  Google Scholar 

  10. Iglehart, D.L.: Optimality of (s, S) policies in the infinite horizon dynamic inventory problem. Management Science. 9,259–267 (1963).

    Article  Google Scholar 

  11. Kim, D.E., Park, K.S.: (Q, r) inventory model with a mixture of lost sales and time-weighted backorders. Journal of the Operational Research Society. 36, 231–238 (1985).

    MATH  Google Scholar 

  12. Moinzadeh, K., Nahmias, S.: A continuous review for an inventory system with two supply modes. Management Science. 34, 761–773 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  13. Sahin, I.: On the stationary analysis of continuous review (s, S) inventory systems with constant lead times Operations Research. 27, 717–729 (1971).

    Google Scholar 

  14. Sahin, I.: On the objective function behavior in (s, S) inventory models Operations Research. 30, 709–724 (1982).

    Google Scholar 

  15. Sethi, S.P., Cheng, F.: Optimality of (s, S) policies in inventory models with Markovian demand Operations Research. 45, 931–939 (1997).

    Google Scholar 

  16. Scarf, H.: The optimality of (s, S) policies in the dynamic inventory problem. Mathematical Methods in social Sciences, ed. by K. Arrow, S. Karlin, P. Suppes, Stanford University Press, Stanford, Chap.13, (1960).

    Google Scholar 

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

    Article  MATH  Google Scholar 

  18. Veinott, A.F.: On the optimality of (s, S) inventory policies: New conditions and a new proof. J. Siam. Appl. Math. 14, 1067–1083 (1966).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Alain Bensoussan would like to acknowledge the support of WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (R31-20007).

The authors would also like to thank an anonymous referee for comments on an earlier version of the paper.

Author information

Authors and Affiliations

  1. Department of Statistics and Operations Research, Kuwait University, Safat, 13060, Kuwait

    Lankere Benkherouf

  2. International Center for Decision and Risk Analysis, School of Management, University of Texas at Dallas, Richardson, TX, 75080, USA

    Alain Bensoussan

  3. The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Alain Bensoussan

  4. Ajou University, San S. Woncheon-Dong, Yeontong-Gu, Suwon, 443–749, Korea

    Alain Bensoussan

Authors
  1. Lankere Benkherouf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alain Bensoussan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alain Bensoussan .

Editor information

Editors and Affiliations

  1. , Department of Probability and Statistics, Center for Research in Mathematics, Jalisco s/n, Guanajuato, 36000, Mexico

    Daniel Hernández-Hernández

  2. , Department of Mathematics, University of Sonora, Rosales s/n, Hermosillo, 83000, Sonora, Mexico

    J. Adolfo Minjárez-Sosa

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Benkherouf, L., Bensoussan, A. (2012). Discrete-Time Inventory Problems with Lead-Time and Order-Time Constraint. In: Hernández-Hernández, D., Minjárez-Sosa, J. (eds) Optimization, Control, and Applications of Stochastic Systems. Systems & Control: Foundations & Applications. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8337-5_2

Download citation

Publish with us

Policies and ethics

Search

Navigation