Abstract
Testing is an indispensable process for ensuring product quality in production systems. Reducing the time and cost spent on testing whilst minimising the risk of not detecting faults is an essential problem of process engineering. The optimisation of complex testing processes consisting of independent test steps is considered. Survival analysis-based models of an elementary test to efficiently combine the time-dependent outcome of the tests and costs related to the operation of the testing system were developed. A mixed integer non-linear programming (MINLP) model to formalize how the total cost of testing depends on the sequence and the parameters of the elementary test steps was proposed. To provide an efficient formalization of the scheduling problem and avoid difficulties due to the relaxation of the integer variables, the MINLP model as a P-graph representation-based process network synthesis problem was considered. The applicability of the methodology is demonstrated by a realistic case study taken from the computer manufacturing industry. With the application of the optimal test times and sequence provided by the SCIP (Solving Constraint Integer Programs) solver, 0.1–5% of the cost of the testing can be saved.
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Abbreviations
- \(C^i(t)\) :
-
Total cost function of test step i
- \(C_f^i(t)\) :
-
Fixed and amortization cost function of test step i
- \(C_p^i(t)\) :
-
Proportional cost function of test step i
- \(C_r^i(t)\) :
-
Repair cost function of test step i
- \(C_w^i(t)\) :
-
Warranty cost function of test step i
- \(S^i(t)\) :
-
Survival function of test step i
- \(W^i(t)\) :
-
Weibull distribution function of test step i
- \(\pi \) :
-
Vector representing the order of the test steps
- t :
-
Variable representing time
- \(t_i\) :
-
Length of the test step i
- \(\bar{t} \) :
-
Vector of the times of activities
- \(x_i\) :
-
Volume of activity i
- \(\bar{x} \) :
-
Vector of the volumes of activities
- \(y_i\) :
-
Existential (binary) variable representing the status of activity i (whether it works or not)
- \(\bar{y} \) :
-
Vector of existential (binary) variables representing the status of activities
- Z :
-
\(n\times n\) matrix representing the orders of tests
- \(c_f^i\) :
-
Constant fixed and amortization cost of test step i for one item
- \(c_p^i\) :
-
Constant proportional cost of test step i for one item
- \(c_r^i\) :
-
Constant repair cost of test step i for one item
- \(c_w^i\) :
-
Constant warranty cost of test step i for one item
- Exl :
-
Set representing the mutually exclusive activities
- \(e_i\) :
-
Unit vector i
- \(k_i\) :
-
Shape parameter of Weibull distribution \(W^i(t)\)
- \(\lambda _i\) :
-
Scale parameter of Weibull distribution \(W^i(t)\)
- \(L_{p_j}\) :
-
Lower bound for final target \(m_j\) (product)
- N :
-
Number of test steps
- \(N^i_{in}\) :
-
Number of items involved in test step i
- \(N_{in}\) :
-
Number of items entering the testing process
- M :
-
Set of entities in PNS problem
- \(\wp (M)\) :
-
The power set of the set M
- \(\bar{M}\) :
-
Sufficiently large number
- P :
-
Set of products in process network synthesis (PNS) problem
- R :
-
Set of raw materials in PNS problem
- \(ratio_{ji}\) :
-
Function representing the difference between the production and consumption rates of entity \(m_j\) during activity \(o_i\)
- O :
-
Set of operating units in the PNS problem
- \(O_1\) :
-
Set of operating units representing test steps in the process
- \(O_2\) :
-
Set of operating units representing logical (not test steps) activities in the process
- \(T^i_{max}\) :
-
Maximum length of test step i
- \(U_{c_j}\) :
-
Upper bound for resource \(m_j\)
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Acknowledgements
This research has been supported by the National Research, Development and Innovation Office (NKFIH), through the project OTKA-116674 (Process mining and deep learning in the natural sciences and process development). The research was supported by the ÚNKP-17-3 New National Excellence Program of the Ministry of Human Capacities.
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Baumgartner, J., Süle, Z., Bertók, B. et al. Test-sequence optimisation by survival analysis. Cent Eur J Oper Res 27, 357–375 (2019). https://doi.org/10.1007/s10100-018-0578-z
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DOI: https://doi.org/10.1007/s10100-018-0578-z