Design for Six Sigma and MonteCarlo Simulation

The objective of this brief article is the presentation of a proposal of implementation of the MonteCarlo simulation technique to the methodology of analysis known with the name of Design for Six Sigma (DFSS).  With the purpose to make the treatment less technical, we will illustrate the phases of our proposal with reference to a specific example, regarding a molding process of a plastic component.

The DFSS is a methodology either for the design of new processes / products, or for the redesign of existing processes / products, whose objective is the attainment of requisite of performance, reliability and costs, typical of the Six Sigma methodology.

One of the DFSS phases called Tolerance Analysis (TA), includes the study of the output variability, of a process, as function of the inputs of the same process. The final purpose of the TA is to identify what inputs have the greatest impact on the output. The main tool to complete a TA is the transfer function, that it the relationship that ties, among them, inputs and output of the process.

To conduct a TA, some different kinds of quantitative techniques are available and among these, the most common are:

·   Worst case

·   Root sum of squares (RSS)

·   Propagation of the error (POE

·   MonteCarlo simulation

Among these techniques, the MonteCarlo simulation is certainly the most efficient, even if not always the most used. This is due to the relative simplicity of implementation of the other techniques and for the fact that to conduct a MonteCarlo simulation, specific computer software are necessary.

The traditional statistic computer software's for the Six Sigma, in fact, don't include inside any explicit capability, that is, immediately usable by the user, of models simulation.

Some Microsoft Excel © add-in are available to realize a MonteCarlo simulation, which have the advantage to add to the complete modeling, [2] typical of Excel, those MonteCarlo simulation functionalities that Excel doesn't possess, or it possesses in very limited way.

Such software's were born in origin to develop some applications within the Risk Analysis and only recently they have been used for technical applications within the DFSS. However, these add-in, suffer unfortunately of some limits, referable to their genesis.

Such limitations become critical in the moment in which there is the interpretation of the results: the risk, in the pure application, of the simulation techniques to the DFSS, with the today's available software, is to reach partial conclusions and sometimes misleading the optimization problem of the process in examination.

To illustrate this point, we will use an alternative simulation of the example "Simulation with Design of Experiments" available on the site http://www.decisioneering.com. 

The case concerns a injection molding process in which three inputs or factors are involved (Mold Temperature, Cycle Time and Hold Pressure), that jointly contribute to determine the output variability (the variable Length, that is the length of the injected manufactured part).  The transfer function used in the original example has been produced using an Excel multiple regression model,  whose coefficients are the solution of a factorial design, [3] with two levels for each of the three factors and five replicates, but after that, a consequential analogous function will be used from a DOE solution, obtained with one of the common statistic software's for DOE (Design Expert © or Minitab ©), to be also able to use correctly the contribution of the experimental error.

In the example, it is assumed that the three input factors are distributed according to a normal distribution (Figure 1).

In the rest of the article we will show how it is possible to optimize the response variability, depending on the controllable factors and accordingly to reduce the number of obtainable defects during the molding process. For this example, we decide to simulate a million values for the inputs, which, through the transfer function, they become a million of simulated values of the output. [4]

In Figure 2 is shown the distribution of the calculated values for the response variable (Length). It can be noticed that, differently from the input factors, for which the Anderson-Darling [5] normality test must be overcome, as an assumption, because it hypothesizes that each of such variable follows a normal distribution, the same normality test, instead, is not overcome by the response variable.

This is obviously due to the fact that, because of the interactions among the factors in the model, the transfer function is not a pure linear function of the factors themselves.

Particularly, it is possible to show that, in consequence of the irrelevancy of the coefficients, related to the interactions among the factors, simulations with a small number of runs (< 2000) can produce a response, whose distribution is interpretable as a normal, but a large number of runs, will always produce a distribution, in which the normality test is not overcome. Such conclusion becomes as more evident as more elevated is the number of simulated values.

As result, the usual practice to extrapolate the results obtained by a simulation made with a lower number of runs to the whole population, assuming that the response is normally distributed, results to be very risky. Particularly, this is important if there is the necessity to have a very precise estimate for a certain quantity, as the number of defective parts, obtained by the simulation (parts with values of the response outside the specification limits). In our example, in fact, the number of parts with response outside the specification limits, results to be equal to 5480 (respectively 3353 smaller than the lower specification limit and 2127 bigger than the upper specification limit, respectively), while if we assume that the response is normally distributed, we would get a number of defective parts equal to 5317 (2664 and 2653 respectively).  This difference of defectiveness, in the two tails of the response distribution, is not marginal, in practical terms too.

In the illustrated case, in fact, a defectiveness of the length of the manufactured part, over the upper specification limit, is still technically surmountable, submitting the same manufactured part to a mechanical reduction of the surplus length. From the other side, however, a defectiveness due to the length to the manufactured part under the lower specification limit, probably involves the only possibility of recycling the material of some extruded parts (if the used polymer allows it).

Here, the key idea is that, since it is needed to estimate the number of defectiveness of the process, which is always express in parts per million (PPM or DPMO), it results risky to simulate only some thousand of values and to extrapolate the achieved results to the million, required by the Six Sigma methodology. Obviously, to simulate million of data involves to have some available powerful and highly efficient Excel © add-in, as that one used in our example but, the advantage, in terms of precision, about the conclusions, that can be drawn, is very high.

A second aspect of remarkable importance, on which we desire to attract the attention of the reader, concerns another of the disadvantages of the traditional software's for the MonteCarlo simulation today available, mentioned at the beginning, that is the impossibility to hold a correct trace of the simulated values of the inputs. This doesn't represent at all a negligible information, since from these values, using the analysis of all the data, in the hyperspace with five dimensions (the response and the four independent variables), it is possible to reconstruct what configurations (simulations) of the inputs have originated the different parts, resulted then defective and to draw interesting considerations, both in terms of analysis and optimization. (Figure 3)

In our example it is possible, in fact, to calculate that the main portion of the defective parts achieved in the simulation, are produced by values of the input factors that satisfy their specification limits (in the Figure 4, a simulated value that is found within the specification limits is pointed out by a sign of equal).

Therefore, the objective of low defectiveness results hardly attainable if we continue to accept, as valid, the actual specification limits of the inputs of the molding process.

The problem is expressed, therefore, in how to choose the new specification limits, that is, the inputs variability, with the purpose to get a fixed number of defective parts.

Our proposal allows to give also a definitive answer to this problem.  In fact, repeating the whole simulation a certain number of times (in our case, six repetitions are illustrated - Figure 5), it is possible to estimate the relationship that joins the input standard deviations with the response standard deviation.

In this way, once fixed the variability assumed acceptable for the response (that is expressed in a certain number of defective parts), it is possible to resolve such relationship in the opposite way, getting, so, the values of the standard deviations of the input factors that allow to reach the desired precision of the response .

To conclude, in this article [6] we have shown how the usual practice implemented with the traditional software's for the MonteCarlo simulation, to simulate few thousand of values and from these to extrapolate conclusions about the best performance of the process (in terms of defectiveness) allows to get only an approximate and partial response.  To reach the correct conclusions, it is necessary to have software that can easily manage simulations of million of data. We have also shown how it is fundamental to keep trace of all the simulated values for all the variable involved in the model, otherwise we risk to lose remarkable information to optimize the process in examination.

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5