Technological and economical risks quantification by quality loss function for milling on a flexible fabrication system.
Mihail, Laurentiu-Aurel ; Fota, Adriana ; Buzatu, Constantin 等
1. INTRODUCTION
The economic loss is often perceived as being directly linked with
the supplementary manufacturing costs, till the product is supplied to
the client. According with this idea, after the product arrival at the
client, the producer will be the most affected by the quality loss.
First of all the producer will must to pay the warranty costs and, after
the warranty period of time the client will be the one who will pay the
losses produced by the product's quality loss. In reality, the main
victim will be the producer as much time it will lose because of the
overall negative reaction of the market. This fact will bring the
diminishing/loose of his reputation on the market. By the process
quality it must be understood the amount by which his internal or
external client is happy with the obtained result. If his required or
not required attendances will not be according with the process's
result, it's obvious that this will not be perceived as a quality
one (Kamen, 1999). Quality Loss Function is useful for the last
enumerated scope. The objective of the Quality Loss Function is the
quantitative appraisal of the losses generated by the process's
technological variability. Taguchi defines the quality as being the loss
generated by a product to the organization after his shipment to the
customer (Taguchi et al., 2004).
Taguchi has proposed as Quality Loss Function a quadratic equation described by the relation 1 (Fowlkes & Creveling, 1995):
[L.sub.T] (x) = [K.sub.[delta]]/[[delta].sup.2] x [(x - T).sup.2]
(1)
where [[k.sub.[delta]] is the loss corresponding to the deviation
[delta] = [absolute value of x-T] relative to the process's target
T. It was determinate in this mode the Taguchi's economical risk
equation, respectively the average loss associated to a process as
consequence of the deviation of his quality characteristic relatively at
the specified target value, and the capability index [c.sub.pm], amount
of the process's deviation relatively to the target value (Mihail,
2008).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
It can be obtained the next correlation equation between the
[c.sub.pm] precision index and the Taguchi's average risk (Barsan
& Popescu, 2003):
M[L.sub.T[(x,T)] = k x (STI/6 x 1[C.sub.pm]).sup.2] (3)
where: STI = LSL - USL is the specified tolerance interval.
2. CASE STUDY FOR THE ECONOMICAL TECHNOLOGICAL RISK QUANTIFICATION
Practically, within this case study it is applied the already
presented theory. More precisely, on the next paragraphs it's
proposed a genuine utilization by witch it is needed the achieving of an
approach for the tolerance design by the [c.sub.m], [c.sub.mk] and
[c.sub.p]m indexes, so, relatively to the process's variability and
targeting relatively to the average value and to the target value of the
studied quality characteristic.
For this reason it was taken in consideration the dimensional
accuracy for the milling of a circular pocket by computer controlled
interpolation for a part manufactured by a composite material based on
synthetic resins, named Necuron 1001 (see the figure 1). The case study
is realized on a prismatic part which is machined on all his faces (so
it is needed 6 operations), on the same machine toll (a DIGMA 700GC
flexible manufacturing system with 3 axes, with a 16 cutting tools
storage system, with an Andronic 400 operation system and with a spindle
speed up to 23000 rpm).
The designed dimension of [PHI][33.sup.-0.02.sub.-0.01] mm assures
a very narrow tolerance field, which, practically, may be realized for
the best case only for non economical conditions.
As input data we have the measurement results of a 50 parts batch,
the nominal dimension and the lower and upper deviations limits of the
tolerance field.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
After the computing of the measured data computing, it was observed
that the [c.sub.m] and [c.sub.mk] indexes values are very low ([c.sub.m]
= 0.04 and [c.sub.mk]=0.35) and [c.sub.pm] = 0.03, situated below the
values required by the 5 Sigma and 6 Sigma performance levels,
respectively 1.67 and 2.00 (the values imposed by the customers that
activate on the modern manufacturing supply chain). In consequence, for
such a tolerance field, designed to be 0.01 mm wide, it will be achieved
adequate capability indexes only by the tolerance field widening. It is
proposed, for example, for achieving the indexes [c.sub.m] = 2.03 and
[c.sub.mk]=2.00, a tolerance field range of 0.47 mm, much bigger that
the initially designed one. So, in this train of ideas, the correct
designed dimension will be [PHI][33.sup.-0.17.sub.-0.30]. It is obvious
that, according with the figure 4, that in this last optimised case the
average quality loss (Taguchi's risk) is diminishing a lot, too. In
the figure 4 the clear grey symbolise the risk for each operation's
phase and the dark grey columns symbolise the cumulated risk. It is
obvious the fact that for the non optimized case the technological and
economical risk values are much bigger as the optimized ones.
For exemplification, for the sixth operation's phase it is
obtained the average risk M[[L.sub.T6](x,T)]=2612681 for a 6 Sigma
specified performance level in the case in which the dimension is not
economical designed. After the optimisation it is obtained the average
loss M[[L.sub.T6](x,T)]= 0.22 for the 6 Sigma specified performance
level. The difference is obvious, the organisation being the one who
wins on a long time interval because of this optimisation. It was
computed the [c.sub.pm] index, for both cases for obtaining an image on
the deviation of the process's quality characteristic relatively to
his initial target value (T = 32.985 mm). So, for the first case, in
which the tolerance field was not an economic one, the index [c.sub.pm]
= 0.03, and for the second case, in which the tolerance field is
optimised, [c.sub.pm] = 2.02, relatively at the target value T = 32.935
mm. Each of the 6 operation's phases was timed for the estimation
of the economical loss [k.sub.[delta]] for the process's target
deviation [delta] = [absolute value of x-T]. Relatively at the
[k.sub.[delta]] value it was computed the k coefficients for each
operation's phase apart. Finally, with the equation 1 it was
calculated the average quality loss. The graphical representation of the
experimental results it can be observed on the figures 2 and 3 for the
initial case and in the figures 4 and 5 for the optimised tolerance
interval.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
The average loss for each operation's phase is represented
with clear gray and with dark grey is represented the cumulated one. It
can be observed, comparatively, between the two figures (2 and 4), which
one is the loss level for the case in which the tolerance field is not
an economical one and for and optimised one. So, for the first case, if
it is taken in consideration only the sixth phase, the average loss is
M[[L.sub.T6](x,T)] = 1222.7 and for the second situation the average
loss is M[[L.sub.T6](x,T)] = 0.23 (a much smaller one).
3. CONCLUSION
As future possible development opportunities of the researches it
can be done a study on the design of an hole basis (H) dimensioning
system, as a consequence of a relatively easy way to machine such an
internal circular dimension.
By utilisation of the Taguchi's Quality Loss Function it can
be quantified simultaneous the process's technological performances
and his costs, too. In such a manner it can be assured a continuous
improvement approach.
4. REFERENCES
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riscului--concepte, metode, aplicatii, (Risk's
Management--concepts, methods, applications) "Transilvania"
University from Brasov Publisher, ISBN: 973-635-180-7, Romania
Fowlkes, W.Y. & Creveling, C.M. (1995). Engineering Methods for
Robust Product Design--using Taguchi Methods in Technology and Product
Development, Addison-Wesley, ISBN: 0201633671, USA, 1999
Kamen, E.W. (1999). Industrial Controls and Manufacturing, Elsevier
Ltd, ISBN: 978-0-12-394850-2, USA, 1999
Mihail, L.A. (2008). Researches on the efficientisation of the
cutting technological system--PhD Thesis, Transilvania University of
Brasov, Romania, 2008
Taguchi, G., Chowdhury, S. & Wu, Y. (2004). Tacughi's
Quality Engineering Handbook, John Willey & Sons Inc. and ASI Cons.
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