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Turnpike Sets and Their Analysis in Stochastic Production Planning Problems


  • S. Sethi

    (Faculty of Management, University of Toronto, Toronto, Ontario, Canada M5S 1V4)

  • H. M. Soner

    (Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213)

  • Q. Zhang

    (Faculty of Management, University of Toronto, Toronto, Ontario, Canada M5S 1V4)

  • H. Jiang

    (Faculty of Management, University of Toronto, Toronto, Ontario, Canada M5S 1V4)


This paper considers optimal infinite horizon stochastic production planning problems with capacity and demand to be finite state Markov chains. The existence of the optimal feedback control is shown with the aid of viscosity solutions to the dynamic programming equations. Turnpike set concepts are introduced to characterize the optimal inventory levels. It is proved that the turnpike set is an attractor set for the optimal trajectories provided that the capacity is assumed to be fixed at a level exceeding the maximum possible demand. Conditions under which the optimal trajectories enter the convex closure of the set in finite time are given. The structure of turnpike sets is analyzed. Last but not least, it is shown that the turnpike sets exhibit a monotone property with respect to capacity and demand. It turns out that the monotonicity property helps in solving the optimal production problem numerically, and in some cases, analytically.

Suggested Citation

  • S. Sethi & H. M. Soner & Q. Zhang & H. Jiang, 1992. "Turnpike Sets and Their Analysis in Stochastic Production Planning Problems," Mathematics of Operations Research, INFORMS, vol. 17(4), pages 932-950, November.
  • Handle: RePEc:inm:ormoor:v:17:y:1992:i:4:p:932-950
    DOI: 10.1287/moor.17.4.932

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    Cited by:

    1. Owen Q. Wu & Hong Chen, 2010. "Optimal Control and Equilibrium Behavior of Production-Inventory Systems," Management Science, INFORMS, vol. 56(8), pages 1362-1379, August.
    2. Dylan Possamai & H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs: the multidimensional case," Papers 1212.6275,, revised Jan 2013.
    3. Dylan Possamai & Nizar Touzi & H. Mete Soner, 2012. "Large liquidity expansion of super-hedging costs," Papers 1208.3785,, revised Apr 2015.
    4. Nicole Bäuerle, 2001. "Discounted Stochastic Fluid Programs," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 401-420, May.
    5. H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs," Papers 1202.6131,, revised Jun 2013.
    6. S. P. Sethi & W. Suo & M. I. Taksar & Q. Zhang, 1997. "Optimal Production Planning in a Stochastic Manufacturing System with Long-Run Average Cost," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 161-188, January.
    7. S. P. Sethi & H. Yan & H. Zhang & Q. Zhang, 2002. "Optimal and Hierarchical Controls in Dynamic Stochastic Manufacturing Systems: A Survey," Manufacturing & Service Operations Management, INFORMS, vol. 4(2), pages 133-170.


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