1. Impact of Statistical Process Control (SPC) on the Performance of Production Systems
M. Colledani, T. Tolio
Dipartimento di Meccanica
Sezione Tecnologie Meccaniche e Produzione

2. Outline of the presentation
1- Literature review
2- Problem definition
3- Isolated machine with local monitoring
4- Two machines one buffer with local monitoring
5- Two machines one buffer with remote monitoring
6- Long lines with local monitoring
7- Numerical results
8- Conclusion and future research

3. 1- Literature Review
- Montgomery,D.C, Introduction to Statistical Process Control, John Wiley and Sons, Inc, 1991.
- Ho C., Case K., Economic Design of Control Charts: A Literature Review for , Journal of Quality Technology, 26,: 39-53,1994.
- Raz T.,” A Survey of Models for Allocating Inspection Effort in Multistage Production Systems”, Journal of Quality Technology, , 1986.
- Dallery, Y.,Gershwin, S.B., “Manufacturing Flow Line Systems: A Review of Models an Analytical Results, Queueing Systems Theory and Applications, Special Issue on Queueing Models of Manufacturing Systems, 12(1-2)
- Gershwin S.B., Matta A. and Tolio T., Analisys of Two Machine Lines with Multiple Failure Modes, IIE Transaction, 34(1) : , 2002.
- Gershwin S.B., Kim J.,Integrated Quality and Quantity Modeling of a Production Line”, OR Spectrum, 2005.
- Colosimo B.M., Semeraro Q. and Tolio T. “Designing X bar Control Charts in Multistage Serial Manufacturing System, CIRP Journal of Manufacturing Systems, 31-6, 2002
- Tempelmeier H., Burger M, Performance Evaluation of Unbalanced Flow Lines with General Distributed Processing Times, Failures and Imperfect Production, IIE Transactions,33, , 2000.
- Helber S. Performance Anaiysis of Flow Lines with Non-Linear Flow of Material, Springer, 1999.
- Inman R., Blumenfeld D., Huang N. and Li J..,” Designing Productivity Systems for Quality: research opportunities from an automotive industry perspective “, IJPR, 41(9), 2003.

4. 2- Process in Statistical control
PROCESS IN CONTROL each quality measure is in a statistical control state.
STATISTICAL CONTROL STATE is a state where all the variations within the observed data can be related to a set of causes not identifiable which do not change over time (i.e. the distribution is stable)
Example t

5. 2- Specifications and bad parts
Even if the process is in control it can produce bad parts.
Upper Specification Limit - USL t
Lower Specification Limit - LSL
However, if the process goes out of control the number of bad parts produced changes (in general, infinite out of control modes are possible).
USL t
LSL

6. 2- Detecting out of control states
In order to understand if the process is in control or out of control, we can sample the produced parts (in the extreme case we can have 100% inspection). Then we measure the parts in the sample and we perform a statistical test with the following hypotheses.
H0: the process is in control
H1: the process is out control
The outcome of the test is subject to two types of errors:
a error: the process is in control but the test detects an out of control (false alarm)
b error: the process is out control but the test does not detect it (out of control not detected)

7. 2- Detecting out of control states
If we repeat the test many times and each test has the same a and b errors than we can evaluate the average number of samples we have to take in order to have an alarm (ARL = Average Run Length).
If the process is in control
If the process is out of control t t
For example:

8. 2- Detecting out of control states
If we consider a single machine in isolation and a control chart attached to it then how many parts does the machine produce before getting an alarm?
Let us define:
m the sample size.
h the number of parts produced between two samples.
If the process is in control
If the process is out of control

9. 2- Inspection stations
Testing allows to draw information on the process but also on the inspected parts.
Inspected parts which are within the specification limits may proceed downstream (if testing does not destroy the parts).
Inspected parts which are outside the specification limits may be either scrapped or reworked
Therefore the logic at an inspection station decides two things:
control charts which send the alarms related to the out of control conditions
scrap/rework policies which decide the final destination of the inspected parts
Out of control
good
(If m=1 and h=0 than 100% inspection is performed).
rework
scrap

10. 2- Some Assumptions of the Model
The flow of material in the system is considered as discrete.
Each machine is characterized by the same processing time, scaled to time unit.
Buffers have finite capacity.
Machines can be of three types: manufacturing machines, inspection machines or integrated machines.
Inspection machines are perfectly accurate.
Failures and shifts to out of control are Operation Dependent.
Once an out of control has been detected, the time to repair it is geometrically distributed.
Machines can fail in different modes. We identify two classes of failures:
f type local failures: are those for which the repairing intervention also set the machine to the in control state;
f type local failures: are those for which the repairing intervention reset the machine to conditions it had before the failure occurred.

11. 3- Modelling a single machine in isolation
Wi: operative in control
Di,fi: f type local down state
Di,fi: f type local down state
Ai1: out of control detected but not real
Ai2: out of control detected and real
Oi: out of control non detected
Quality link equations:

12. 3- The Isolated Machine Case
Once the Markov chain has been solved and all the state probabilities have been calculated, the performance measures for the single machine case can be derive as follows:
Total average production rate:
Effective average production rate:
System yield (fraction of conforming parts produced):
Average production rate of parts to be scrapped:
Average production rate of parts to be reworked:

13. 4- The General Case
scrap
scrap
rework
rework
Quality has an impact on production system performance:
- Control charts allow to identify out of control states. The search for a cause for the out of control reduces the up time of the machine.
- Scrap/rework policies allow to identify defective parts and to decide whether to scrap or to rework them.
Total throughput
Yield
The system architecture impacts on the quality system performance:
- The presence of buffers causes a delay in the transmission of the quality signal.
Total throughput
Yield

14. 4- Two machine one buffer with local monitoring
This system is formed by two machines M1 and M2 locally controlled by C1,1 and C2,2.
Building Block evaluation
“Gershwin, Matta, Tolio 2002”
MU(1)
MD(1) B
New blocking and starvation probabilities
Downstream machine M2
Upstream machine M1
Upstream pseudo-machine MU(1)
CONVERGENCE
Stationary state probability distribution
New failure probabilities calculation
Markov chain solution

15. 4- Two machine one buffer with local monitoring
Blocking (starvation) probabilities equations
Monitored machine model
Stationary state probability distribution
Transition probabilities
False alarm state:
Detected out of control state:
Pseudo-machine model
Total average throughput:
System yield:
Average buffer level:

16. 5- Two machine one buffer with remote monitoring
1,2 M 2 B 1 , p r F
Monitored machine Mi
Control chart Ci,q (i<q)
A12: out of control correctly detected by the control chart (i<q)
O11: out of control, not detected state
O21: out of control, detected if the machine was locally monitored (i=q)
Additional states:
The approach remains the same as in the previous case, only the Markov chain is more complicated:
p1,2delay represents the delay of the quality information due to the presence of the finite capacity buffer B. It can be calculated by using the following equation:

17. 6- Long lines with local monitoring – the approach
By solving the two locally monitored machines systems with the presented method and by using decomposition equations the performance of the original line can be estimated.
As for the two locally monitored machines system, the approach follows a two level decomposition, since alternately the Markov chain representing each machine (machine level) and each building block (buffer level) are studied .

18. 6- Long lines with local monitoring - failures
Local failure probabilities: are simply equal to those of the correspondent machine in the original line.
Quality linked failure probabilities: can be evaluated by using quality link equations provided.
Remote failure probabilities: can be evaluated by using the following decomposition equations. They acts exactly in the same way as f type local failure modes for the pseudo-machines.
Upstream pseudo-machines
Downstream pseudo-machines

19. 6- Long lines with local monitoring – the algorithm
An iterative algorithm, inspired to the DDX, has been used to efficiently solve the decomposition equations. It behaves as follows, after the initialization phase:
- Visiting all the upstream pseudo-machines for i=2,..,K-1
- Unknown failures probabilities are calculated by using decomposition equations;
- The performance of the building block l(i) are evaluated by using the two level approach used for the two monitored machines system;
- The same steps are performed visiting all the downstream pseudo-machines for i=K-2,..,1
At the convergence, the system yield can be evaluated as:
The total and the effective average production rates:

20. 7- Numerical Results – two locally monitored machines
More than one hundred test cases with random parameters have been carried out and compared with simulation. Some of those cases randomly selected are reported:

21. 7- Numerical Results – one remotely monitored machine

22. 7- Numerical Results – K locally monitored machines
3 machine cases
10 machine cases
Summary of results

23. 7- System Behavior
h= % insp
h=3
h=5

24. 8- Conclusion and Future Research
- Quality issues and productivity aspects must be jointly considered in the design phase of production systems, since their correlation is evident.
- The proposed method, dealing with the interaction between SPC theory principles and production system design issues, provides accurate results in the performance analysis of such systems.
- New improvement of the method will be the integration of various scrap and rework policies in order to identify the optimal scrap/rework parameters.
The proposed method paves the way to the integrated analysis and solution of other system design problem such as:
Optimal design of control chart parameters;
Optimal allocation of inspection devices;
Optimal allocation of buffer space.

25. Thank you for your attention.

26. 10- System Behavior
The behavior of systems in which the first machine is monitored by the second one have been observed.