We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Advertisement

An adaptive approach for determining batch sizes using the hidden Markov model

  • 279 Accesses

  • 1 Citations

Abstract

Determining an optimal batch size is one of the most classic problems in manufacturing systems and operations research. A typical approach is to construct and solve mathematical models of a batch size under several assumptions and constraints in terms of time, cost, or quality. In spite of the partly success in somewhat static processes, wherein the system variability does not change as the process runs, recent proliferation of data-driven process analysis techniques offers a new way of determining batch sizes. Taking into account for dynamic changes in variability in the middle of the process, we suggest a model to determine batch size which can adapt to changes in the process variability using the hidden Markov model which exploits sequence of product quality data obtained points of recalibration dynamically by continuously predicting the level of process variability which is inherent in a system but is unknown explicitly. The proposed model enables to determine points of recalibration dynamically by continuously predicting the level of process variability which is inherent in a system but is unknown explicitly. For the illustrative purpose, a system which consists of a material handler and a machining processor is considered and numerical experiments are conducted. It is shown that the proposed model can be useful in determining batch sizes while assuring desired product quality level as well.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Abbreviations

\(N^{\left( c \right) }\) :

The number of pieces of equipment in a process such as material processors and material handlers

\({\varvec{\Theta }}_i \) :

A set of parameters of a process state distribution when processing product i such that \({\varvec{\Theta }}_i =\left( {{\varvec{\theta }}_i^{c_1 } ,{\varvec{\theta }}_i^{c_2 } ,\ldots ,{\varvec{\theta }}_i^{c_{N^{\left( c \right) }} } } \right) \)

\({\varvec{\theta }}_i^{c_k } \) :

A set of parameters of the equipment, \(c_k \), state distribution when processing product i such that \({\varvec{\theta }}_i^{c_k } =\left( {\theta _i^{c_k ,L} ,\theta _i^{c_k ,D} } \right) \), where \(\theta _i^{c_k ,L} \) and \(\theta _i^{c_k ,D} \) represent parameters of distribution of location and dimensional values, respectively

\(d_{i,f}^D \) :

Deviation of an observed actual dimensional value from a nominal dimensional specification of feature f of product i

\(d_{i,f}^L \) :

Deviation of an observed actual location value from a nominal location specification of feature f of product i

\(F_i \) :

The number of features of product i

\(O_{i,f} \) :

Observation for feature f of product i

\(p\left( {G_{i,f}^D } \right) \) :

Probability that feature f of product i is a good feature with regard to dimensional specification

\(p\left( {G_{i,f}^L } \right) \) :

Probability that feature f of product i is a good feature with regard to location specification

\(PD_{i+1} \) :

Predicted probability that \(\left( {i+1} \right) \)th product is nonconforming product given observed sequence of quality data from beginning of the process to ith product

\(S_{i,f}^D \) :

Discretized state variable for \(d_{i,f}^D \)

\(S_{i,f}^L \) :

Discretized state variable for \(d_{i,f}^L \)

\(S_i^{c_k } \) :

Discretized state variable for equipment \(c_k \) when processing product i

\(T_{i,f}^{D+} \) :

Upper dimensional tolerance for feature f of product i

\(T_{i,f}^{D-} \) :

Lower dimensional tolerance for feature f of product i

\(T_{i,f}^L \) :

Location tolerance for feature f of product i

\(V_{i,f}^D \) :

Nominal value of dimensional specification for feature f of product i

\(V_{i,f}^L \) :

Nominal value of location specification for feature of product

References

  1. Acosta-Mejia, C. A., Pignatiello, J. J., & Rao, B. V. (1999). A comparison of control charting procedures for monitoring process dispersion. IIE Transactions, 31(6), 569–579.

  2. Ahmad, S., Riaz, M., Abbasi, S. A., & Lin, Z. (2013). On monitoring process variability under double sampling scheme. International Journal of Production Economics, 142(2), 388–400.

  3. Bishop, C. M. (2006). Pattern recognition and machine learning. Berlin: Springer.

  4. Du, S., Lv, J., & Xi, L. (2012). A robust approach for root causes identification in machining processes using hybrid learning algorithm and engineering knowledge. Journal of Intelligent Manufacturing, 23(5), 1833–1847.

  5. Han, B., Liu, C. L., & Zhang, W. J. (2016). A method to measure the resilience of algorithm for operation management. IFAC-PapersOnLine, 49(12), 1442–1447.

  6. Harris, F. W. (1913). How many parts to make at once. Factory, The Magazine of Management, 10(2), 135–136.

  7. Henzold, G. (2006). Geometrical dimensioning and tolerancing for design, manufacturing and inspection: A handbook for geometrical product specification using ISO and ASME standards. Oxford: Butterworth-Heinemann.

  8. Hwang, K. H., Lee, J. M., & Hwang, Y. (2013). A new machine condition monitoring method based on likelihood change of a stochastic model. Mechanical Systems and Signal Processing, 41(1), 357–365.

  9. Kerzner, H. R. (2013). Project management: A systems approach to planning, scheduling, and controlling. New York: Wiley.

  10. Kuo, T., & Mital, A. (1993). Quality control expert systems: A review of pertinent literature. Journal of Intelligent Manufacturing, 4(4), 245–257.

  11. Liao, W., Li, D., & Cui, S. (2016). A heuristic optimization algorithm for HMM based on SA and EM in machinery diagnosis. Journal of Intelligent Manufacturing,. doi:10.1007/s10845-016-1222-1.

  12. Lu, C. J., Shao, Y. E., & Li, C. C. (2014). Recognition of concurrent control chart patterns by integrating ICA and SVM. Applied Mathematics and Information Sciences, 8(2), 681.

  13. Mahesh, B. P., & Prabhuswamy, M. S. (2010). Process variability reduction through statistical process control for quality improvement. International Journal for Quality Research, 4(3), 193–203.

  14. Miao, Q., & Makis, V. (2007). Condition monitoring and classification of rotating machinery using wavelets and hidden Markov models. Mechanical Systems and Signal Processing, 21(2), 840–855.

  15. Montgomery, D. C. (2007). Introduction to statistical quality control. New York: Wiley.

  16. Murphy, K. P. (2012). Machine learning: A probabilistic perspective. Cambridge: MIT Press.

  17. Natsagdorj, S., Chiang, J. Y., Su, C. H., Lin, C. C., & Chen, C. Y. (2015). Vision-based assembly and inspection system for golf club heads. Robotics and Computer Integrated Manufacturing, 32, 83–92.

  18. Pullan, T. T., Bhasi, M., & Madhu, G. (2010). Application of concurrent engineering in manufacturing industry. International Journal of Computer Integrated Manufacturing, 23(5), 425–440.

  19. Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2), 257–286.

  20. Rosenblatt, M. J., & Lee, H. L. (1986). Economic production cycles with imperfect production processes. IIE Transactions, 18(1), 48–55.

  21. Sarkar, B., & Moon, I. (2011). An EPQ model with inflation in an imperfect production system. Applied Mathematics and Computation, 217(13), 6159–6167.

  22. Shapiro, S. S., & Hahn, G. J. (1967). Statistical methods in engineering. New York: John Wiley and Sons.

  23. Shin, D., Park, J., Kim, N., & Wysk, R. A. (2009). A stochastic model for the optimal batch size in multi-step operations with process and product variability. International Journal of Production Research, 47(14), 3919–3936.

  24. Tai, A. H., Ching, W. K., & Chan, L. Y. (2009). Detection of machine failure: Hidden Markov model approach. Computers and Industrial Engineering, 57(2), 608–619.

  25. Western Electric Corporation. (1958). Statistical quality control handbook. New York: The Company.

  26. Xu, Y., & Ge, M. (2004). Hidden Markov model-based process monitoring system. Journal of Intelligent Manufacturing, 15(3), 337–350.

  27. Yang, Y., Zha, Z. J., Gao, M., & He, Z. (2015). A robust vision inspection system for detecting surface defects of film capacitors. Signal Processing, 124, 54–62.

  28. Yang, W. A., & Zhou, W. (2015). Autoregressive coefficient-invariant control chart pattern recognition in autocorrelated manufacturing processes using neural network ensemble. Journal of Intelligent Manufacturing, 26(6), 1161–1180.

  29. Yu, J. (2012). Multiway discrete hidden Markov model-based approach for dynamic batch process monitoring and fault classification. AIChE Journal, 58(9), 2714–2725.

  30. Zhang, W. J., & Van Luttervelt, C. A. (2011). Toward a resilient manufacturing system. CIRP Annals-Manufacturing Technology, 60(1), 469–472.

Download references

Author information

Correspondence to Dongmin Shin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Joo, T., Seo, M. & Shin, D. An adaptive approach for determining batch sizes using the hidden Markov model. J Intell Manuf 30, 917–932 (2019). https://doi.org/10.1007/s10845-017-1297-3

Download citation

Keywords

  • Adaptive batch size
  • Data-driven process control
  • Hidden Markov model
  • Process variability prediction
  • Product quality data
  • System state prediction