Mathematical Methods of Operations Research

, Volume 83, Issue 3, pp 295–323 | Cite as

Approximations of time-dependent unreliable flow lines with finite buffers

  • S. GöttlichEmail author
  • S. Kühn
  • J. A. Schwarz
  • R. Stolletz


Flow lines process discrete workpieces on consecutive machines, which are coupled by buffers. Their operating environment is often stochastic and time-dependent. For the flow line under consideration, the stochasticity is generated by random breakdowns and successive stochastic repair times, whereas the processing times are deterministic. However, the release rate of workpieces to the line is time-dependent, due to changes in demand. The buffers between the machines may be finite or infinite. We introduce two new sampling approaches for the performance evaluation of such flow lines: one method utilizes an approximation based on a mixed-integer program in discrete time with discrete material, while the other approximation is based on partial and ordinary differential equations in continuous time and with a continuous flow of material. In addition, we sketch a proof that these two approximations are equivalent under some linearity assumptions. A computational study demonstrates the accuracy of both approximations relative to a discrete-event simulation in continuous time. Furthermore, we reveal some effects occurring in unreliable flow lines with time-dependent processing rates.


Unreliable flow line Sampling Mixed-integer program Conservation laws Piecewise deterministic process 

Mathematics Subject Classification

90B15 90B30 



The author S. Kühn was financially supported by Stiftung Rheinland-Pfalz für Innovation, Project EvaC, FKZ 989 and the author S. Göttlich by the BMBF project KinOpt.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • S. Göttlich
    • 1
    Email author
  • S. Kühn
    • 1
  • J. A. Schwarz
    • 2
  • R. Stolletz
    • 2
  1. 1.School of Business Informatics and MathematicsUniversity of MannheimMannheimGermany
  2. 2.Business SchoolUniversity of MannheimMannheimGermany

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