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Turnpike Horizons for Production Planning


  • Gerald L. Thompson

    (Carnegie-Mellon University)

  • Suresh P. Sethi

    (University of Toronto)


A quadratic model for production-inventory planning was made famous by Holt, Modigliani, Muth, and Simon in 1960 in (Holt, C. C., F. Modigliani, J. F. Muth, H. A. Simon. 1960. Planning Production, Inventories, and Work Force. Prentice-Hall, Englewood Cliffs, New Jersey.), especially for its application to a paint factory. A discrete control version of a related quadratic production-inventory model was studied by Kleindorfer, Kriebel, Thompson, and Kleindorfer in (Kleindorfer, P. R., C. H. Kriebel, G. L. Thompson, G. B. Kleindorfer. 1975. Discrete optimal control of production plans. Management Sci. 22 261--273.). In the present paper we solve a continuous version of the model in Kleindorfer, Kriebel, Thompson, and Kleindorfer (Kleindorfer, P. R., C. H. Kriebel, G. L. Thompson, G. B. Kleindorfer. 1975. Discrete optimal control of production plans. Management Sci. 22 261--273.) in complete detail. The reason we are able to obtain a complete solution (which can rarely be done in control models) is that the linear decision rule, which is optimal here as in other quadratic models, permits the elimination of the adjoint function from the state variable equation after one differentiation of the latter. Thus the difficult two-point boundary value problem which usually arises in control problems is converted into an ordinary second order differential equation, which is readily solved. One advantage of having a complete solution to the problem is that it is possible to determine turnpike horizon points. These correspond to zeros of the adjoint function, and have the property that if they are known exactly, then a production-inventory plan which is optimal up to the next horizon point also forms part of the overall optimal plan. In the case of cyclic demand these turnpike horizon points usually occur every half cycle. Similar horizons are likely to exist in real production-inventory problems. A planning procedure for a real problem which extends only as far as a suspected horizon has a good chance of producing an optimal or near optimal solution for that period of time. A second advantage of having the complete solution available is that it is possible to develop a practical production-inventory system which intermingles a prediction procedure (such as the use of a finite Fourier series) with the solution procedure so that a comparison between predicted and actual inventories can be made continuously. Whenever the discrepancy between these two becomes sufficiently large, the model suggests proper corrective actions to be taken.

Suggested Citation

  • Gerald L. Thompson & Suresh P. Sethi, 1980. "Turnpike Horizons for Production Planning," Management Science, INFORMS, vol. 26(3), pages 229-241, March.
  • Handle: RePEc:inm:ormnsc:v:26:y:1980:i:3:p:229-241
    DOI: 10.1287/mnsc.26.3.229

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    Cited by:

    1. Archis Ghate & Robert L. Smith, 2009. "Optimal Backlogging Over an Infinite Horizon Under Time-Varying Convex Production and Inventory Costs," Manufacturing & Service Operations Management, INFORMS, vol. 11(2), pages 362-368, June.
    2. Schwartz, Jay D. & Rivera, Daniel E., 2010. "A process control approach to tactical inventory management in production-inventory systems," International Journal of Production Economics, Elsevier, vol. 125(1), pages 111-124, May.
    3. Avi Herbon & Konstantin Kogan, 2014. "Time-dependent and independent control rules for coordinated production and pricing under demand uncertainty and finite planning horizons," Annals of Operations Research, Springer, vol. 223(1), pages 195-216, December.
    4. Kogan, Konstantin, 2021. "Limited time commitment: Does competition for providing scarce products always improve the supplies?," European Journal of Operational Research, Elsevier, vol. 288(2), pages 408-419.
    5. Dobos, Imre, 2003. "Optimal production-inventory strategies for a HMMS-type reverse logistics system," International Journal of Production Economics, Elsevier, vol. 81(1), pages 351-360, January.
    6. Robert L. Smith & Rachel Q. Zhang, 1998. "Infinite Horizon Production Planning in Time-Varying Systems with Convex Production and Inventory Costs," Management Science, INFORMS, vol. 44(9), pages 1313-1320, September.
    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.
    8. Suresh Chand & Vernon Ning Hsu & Suresh Sethi, 2002. "Forecast, Solution, and Rolling Horizons in Operations Management Problems: A Classified Bibliography," Manufacturing & Service Operations Management, INFORMS, vol. 4(1), pages 25-43, September.


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