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.
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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
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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
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DOI: https://doi.org/10.1007/s10845-017-1297-3