Skip to main content
SpringerLink
Log in

A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper, we present a robust optimization formulation for dealing with demand uncertainty in a dynamic pricing and inventory control problem for a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting. We introduce a demand-based fluid model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate and all coefficients are time-dependent. A key part of the model is that no backorders are allowed. We show that the robust formulation is of the same order of complexity as the nominal problem and demonstrate how to adapt the nominal (deterministic) solution algorithm to the robust problem.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adida, E., Perakis.: A Nonlinear Fluid Model of Dynamic Pricing and Inventory Control with no Backorders. Submitted for publication. Operations Research Center, MIT (2004)

  2. Anderson, E.: A New Continuous Model for Jobshop Scheduling. Int. J. Syst. Sci. 12, 1469–1475 (1981)

    MATH  Google Scholar 

  3. Anderson, E., Nash, P.: Linear Programming in Infinite Dimensional Spaces. Wiley-interscience, Chichester, England, 1987

  4. Anderson, E., Philpott, A.: A Continuous-Time Network Simplex Algorithm. Networks 19, 395–425 (1989)

    MATH  MathSciNet  Google Scholar 

  5. Arrow, K., Kurz, M.: Public Investment, the Rate of Return, and Optimal Fiscal Policy. The Johns Hopkins University Press, Baltimore, MD, 1970

  6. Bellman, R.: Bottleneck Problems and Dynamic Programming. Proc. Natl. Acad. Sci. 39, 947–951 (1953)

    MATH  Google Scholar 

  7. Bellman, R.: Dynamic Programming. Princeton University press, Princeton NJ, 1957

  8. Ben-Tal, A., El-Ghaoui, L., Nemirovski, A.: Robust Semidefinite Programming. In Semidefinite Programming and Applications. Kluwer Academic Publishers, Waterloo, Canada, 2000

  9. Ben-Tal, A., Nemirovski, A.: Robust Convex Optimization. Math. Oper. Res. 23, 769–805 (1998)

    MATH  MathSciNet  Google Scholar 

  10. Ben-Tal, A., Nemirovski, A.: Robust Solutions to Uncertain Linear Programs. Oper. Res. Lett. 25, 1–13 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ben-Tal, A., Nemirovski, A.: Robust Solutions of Linear Programming Problems Contaminated with Uncertain Data. Math. Program. 88, 411–424 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bertsimas, D., Sim, M.: The Price of Robustness. Oper. Res. 52, 35–53 (2004)

    Article  MathSciNet  Google Scholar 

  13. Bertsimas, D., Thiele, A.: A Robust Optimization Approach to Supply Chain Management. INFORMS Annual Meeting, Atlanta, Georgia (2003)

  14. Bitran, G., Caldentey, R.: An Overview of Pricing Models and Revenue Management. MSOM 5, 203–229 (2003)

    Google Scholar 

  15. Bitran, G., Monstein, S.: Periodic Pricing of Seasonal Product in Retailing. Manag. Sci. 43, 427–443 (1997)

    Google Scholar 

  16. El-Ghaoui, L., Oustry, F., Lebret H.: Robust Solutions to Uncertain Semidefinite Programs. SIAM J. Optim. 9, 33–52 (1999)

    Article  MathSciNet  Google Scholar 

  17. Elmaghraby, W., Keskinocak, P.: Dynamic Pricing: Research Overview, Current Practices and Future Directions. Manag. Sci. 49, 1287–1309 (2003)

    Article  Google Scholar 

  18. Federgruen, A., Heching, A.: Combined Pricing and Inventory Control under Uncertainty. Oper. Res. 47, 454–475 (1999)

    MATH  MathSciNet  Google Scholar 

  19. Gallego, G., van Ryzin, G.: Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons. Manag. Sci. 40, 999–1020 (1994)

    MATH  Google Scholar 

  20. Gallego, G., van Ryzin, G.: A Multiproduct Dynamic Pricing Problem and its Applications to Network Yield Management. Oper. Res. 40, 999–1020 (1997)

    Google Scholar 

  21. Hartl, R., Sethi, S., Vickson, R.: A Survey of the Maximum Principles for Optimal Control Problems with State Constraints. SIAM Rev. 37, 181–218 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kamien, M., Schwartz, N.: Dynamic Optimization: the Calculus of Variations and Optimal Control in Economics and Management. Elsevier Science, New York, NY, 1991

  23. Pullan, M.: An Algorithm for a Class of Continuous Linear Programming Problems. SIAM J. Control Optim. 1, 1558–1577 (1993)

    Article  MathSciNet  Google Scholar 

  24. Pullan, M.: Existence and Duality Theory for Separated Continuous Linear Programs. Math. Model. Syst. 3, 219–245 (1997)

    MATH  MathSciNet  Google Scholar 

  25. Raman, K., Chatterjee, R.: Optimal Monopolist Pricing and Ordering Decisions by a Monopolist. Manag. Sci. 41, 144–162 (1995)

    MATH  Google Scholar 

  26. Sethi, S., Thompson, G.: Optimal Control Theory. Applications to Management Science and Economics. Kluwer Academic Publishers, Boston, MA, 2000

  27. Soyster, A.: Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming. Oper. Res. 21, 1154–1157 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  28. Tyndall, W.: A Duality Theorem for a Class of Continuous Linear Programming Problems. SIAM J. Appl. Math. 13, 644–666 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  29. Yano, C., Gilbert, S.: Coordinated Pricing and Production/Procurement Decisions: A Review. In Managing Business Interfaces. Kluwer Academic Publishers, Boston, MA, 2003

  30. Young, L.: Price, Inventory and the Structure of Uncertain Demand. New Zealand Oper. Res. 6, 157–177 (1978)

    MathSciNet  Google Scholar 

  31. Zabel, E.: Monopoly and uncertainty. Rev. Econ. Stud. 37, 205–219 (1970)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Operations Research Center, MIT, Cambridge, MA, 02139, USA

    Elodie Adida

  2. Sloan School of Management, MIT, Cambridge, MA, 02139, USA

    Georgia Perakis

Authors
  1. Elodie Adida
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georgia Perakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georgia Perakis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Adida, E., Perakis, G. A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders. Math. Program. 107, 97–129 (2006). https://doi.org/10.1007/s10107-005-0681-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-005-0681-5

Keywords

Search

Navigation