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Technological and economical risks quantification by quality loss function for milling on a flexible fabrication system.


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 [], amount of the process's deviation relatively to the target value (Mihail, 2008).


It can be obtained the next correlation equation between the [] precision index and the Taguchi's average risk (Barsan & Popescu, 2003):

M[L.sub.T[(x,T)] = k x (STI/6 x 1[]).sup.2] (3)

where: STI = LSL - USL is the specified tolerance interval.


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], [] 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.



After the computing of the measured data computing, it was observed that the [c.sub.m] and [] indexes values are very low ([c.sub.m] = 0.04 and []=0.35) and [] = 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 []=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 [] 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 [] = 0.03, and for the second case, in which the tolerance field is optimised, [] = 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.




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).


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.


Barsan-Pipu, N. & Popescu, I. (2003). Managementul 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. Group, ISBN: 0-471-41334-8, USA, 2004
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Author:Mihail, Laurentiu-Aurel; Fota, Adriana; Buzatu, Constantin
Publication:Annals of DAAAM & Proceedings
Article Type:Report
Geographic Code:4EUAU
Date:Jan 1, 2009
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