### Summary

- The 6 Sigma and Operational Excellence. Just common sense?
- How many values are needed to have a representative sample?
- Exploit the data to optimize the control of a process
- Statistical modeling: The need for a reliable approach
- Bayesian approach in cosmetical research: Application to a meta-analysis on the anti-pigmenting effect of vitamin C
- Comparability, equivalence, similarity ... How statistics can help us to demonstrate it. And soon the end of a "blind test" for health authorities and industrialists
- Maintaining the validated status, a step in the validation cycle
- Process validation strategy and implementation of GMP Annex 15 and FDA guidance. Continuous Process Verification (CPV)

"*In God we trust, all others must bring data*", is a quote often attributed to Edwards Deming (the creator of the eponymous cycle ^{(1)} ) or Jack Welsh (the emblematic President of General Electric in the 1990s), which by itself embodies the spirit of the Six Sigma methodology: using data to control processes. Yet, Six Sigma suffers from its name - pronouncing it is difficult - and moreover conjures up acronyms - DMAIC - and roles - Green Belt, Black Belt, Champion - which can make this methodology seem obscure or even on the limit of the 'sect'.

In its 2000^{(2)} Annual Report, General Electric (GE) executives stated that "*the Company not only posted its highest revenues ever, but grew them at one of the highest rates in its history. [...] Through the rigorous pursuit of four big Company-wide initiatives - Globalization, Services, Six Sigma Quality and Digitization – we’ve changed not only where we work and what we sell, but how we work, think and touch our customers* ".

**Six Sigma cannot simply be summed up as common sense. Let us take a new look at it and understand how it can contribute to better mastery of manufacturing processes.**

**1. THE HISTORY OF 6 SIGMA**

**1.1 Craftsmanship: an Ancestral Mode of Production**

Operational Excellence does not originate from Toyota and the Toyota Production System (TPS), nor from Motorola and 6 Sigma. It results from a long evolution of production methods enabling craftsmanship and then industry to continuously progress.

Until the industrial revolution of the nineteenth century, Western societies were predominantly agrarian and artisanal. To manufacture an item, you basically need to use a craftsman (a resource). This unique relationship (one item = one craftsman) has the advantage that each item manufactured is unique and corresponds exactly to the customer's request. However, volumes are low and the costs high: one item = one cost, two items = two costs…. Although constrained by the resource until the arrival of mechanization, artisanal production continuously improved its organization.

**1.2 The Renaissance: First Steps towards new Production Methods**

In July 1574, King Henry III of France visited the city of Venice. On 24 July, " *after making some purchases incognito, the king went to the arsenal. This heart of Venice's commercial and military maritime power - overcame its natural shyness and fundamental distrust to show the sovereign the secrets of its strength - its best 'arsenalotti' (workers who achieve the feat of fitting out a galley in the time taken by the king to enjoy the light meal provided) and its incomparable organization combining efficiency and responsiveness*« ^{(3)}King Henry III then discovered an incredible breakthrough in the way to manufacture: **sequential work**... which nowadays seems so natural.

At the end of the eighteenth century, these spectacular advances in manufacturing continued with the implementation of the Gribeauval system in the French artillery: " *I owe my victories to this genius of Gribeauval*« ^{(4)} declared Napoleon I. Jean Baptiste de Gribeauval [1715 - 1789] was appointed first French artillery inspector in 1776. He undertook a complete overhaul of his organization and *imposed unique measures in all the provinces of France for the manufacture of weapons, a real revolution at a time when each region, and even each city, had its own measures! In foundries and arsenals, it also required the interchangeability of all parts and accessories between themselves, the quantified and standardized definition of a tolerance threshold for all machined parts and rigorous manufacturing controls, carried out according to precise specifications and thanks to control boxes containing templates common to all arsenal*« ^{(5)} . The Gribeauval system is based on two principles: **standardization and the mobility of parts.**

**1.3 Ford and Taylor's Industrial Revolution**

Inventor Eli Whitney [1765 - 1825] took up these principles and popularized them in the United States in the late eighteenth and early nineteenth centuries. He invented a cotton gin machine used to separate the cotton seed from its fibre, a tedious and costly task traditionally carried out manually by slaves in the southern United States. Beyond the technological advance, he offered an innovative marketing model: Eli Whitney sold the gin in standard detachable parts that were easy for the buyer to assemble. The foundations of **mass production** were laid: " *to make each part so very alike that any part of one can be used in any other*". Frederick Taylor [1856 - 1915] took inspiration from these technical advances in production, formalizing them in the decomposition of work into elementary tasks with fixed execution times. Meeting Henry Ford [1863 - 1947] was decisive. By combining Frederick Taylor's work with assembly-line production, the latter moves from the world of the craftsman (low volume and high costs for custom products) to **mass production** (high volume and low costs for standard products).

**1.4 The birth of probabilities and statistics**

In parallel with these technical advances, mathematics was also undergoing a revolution with the birth of *probability calculation or the quantification of 'chance' * (from Arabic al zahr meaning ... the die^{(6)}). Luca PACCIOLI (fifteenth-century Italian Franciscan religious, mathematician and founder of double-entry accounting [1445-1517]) raises the problem of parties in his book « *Summa of arithmetica, geometria, proportioni and proportionalita* "(Venice, 1494):" *A and B play balla [the ball to the prisoner]. They agree to continue until one of them has won six rounds. In fact, the game stops when A has five points and B three points. How should the starting bet be split?*« ^{(7)}. Reformulated otherwise, the question is: what decision to take to allocate the bet knowing that A has five points, B three points and that the winner is the one who has six points? This problem was repeated by many Italian mathematicians: Pacioli in 1494, Forestani in 1603, Calandri, Cardano in 1539, Tartaglia in 1556, Peverone in 1558, Pagani in 1591, and in French Gosselin in 1578^{(8)}. But especially by Blaise PASCAL [1623-1668] and Pierre de FERMAT [≈1610 - 1665] who during their 1654 summer correspondence establish Pascal's famous triangle - or binomial law - to compute **the probability of theoretical occurrence** an event. Jacques BERNOULLI [1654-1705] continues in this way with the law of large numbers drawn from his experience "the urn of Bernoulli". He then apprehends the probability of real occurrence in order to estimate the uncertainty: from observations of frequency Jacques BERNOULLI **esteem the reality**.

Concretely, if I extract several samples of black and white balls from the urn, I can then estimate the actual proportion of black and white balls, i.e. calculate " *the probability that the error between an observed value and the true value lies within a given limit*« ^{(9)} . French mathematician Abraham de Moivre [1667 - 1754] took an interest in the work of Blaise Pascal and Pierre de Fermat and more particularly in « *the convergence of random variables, from the following perspective: to what extent can we be sure that when we roll a die a large number of times, the observed frequency of appearance of the number 'six' tends towards theoretical probability? *« ^{(10)} . Abraham de Moivre notes a dispersion of the results around the mean, drawing a bell-shaped distribution: the normal Laplace-Gauss distribution. He calls this dispersion the ** standard deviation** . " *This measure is of crucial importance in determining whether a set of observations includes a sufficiently representative sample of the universe to which they belong*« ^{(11)}.

**1.5 Motorola and Statistical Process Control**

This approach to prediction using dispersion not only revolutionized mathematics but two centuries later also had a crucial impact on mastering manufacturing processes. Walter A. Shewhart - an American physicist [1891 - 1967] in charge of the technical department at Bell Telephone Laboratories – implemented control charts to ensure the quality of products manufactured at Western Electric's Hawthorne plant. The results of his work were published in 1931 in his book *"Economic Control of Quality of Manufactured Product"*a reference still today for mastering manufacturing processes. It **represents the birth of Quality Management.**.

Mass production revolutionized manufacturing methods and, above all, made it possible to offer as many products as possible that were previously inaccessible or reserved for an elite. Mass production is good, but risky: manufacturing 1 similar vehicles entails the risk of repeating the same mistake 000 times. Very quickly, manufacturers became confronted with quality problems: product volumes increased ... as did the errors. Walter A. Shewhart's works provided tools to master manufacturing processes using statistics, the best-known being PDCA: Plan, Do, Check, Act or the Shewhart Cycle.

Edwards DEMING [1900 - 1993] came to popularize PDCA among Japanese industrialists in the 9001s. Nowadays, the PDCA is better known as the 'Deming Cycle' and is the cornerstone of the ISO 2015:XNUMX standard for Quality Management Systems: " *The 2015 version is more oriented towards 'results' than 'means'. Some documents are no longer required, such as the quality manual or system procedures. It is now up to the teams to determine their own operating means and tools. It is an opportunity to save time and promote your priorities. ISO 9001 and ISO 14001 standards adopt a common structure, organized according to PDCA (Plan-Do-Check-Act)*« ^{(12)}.

Bill Smith [1929 - 1993], MOTOROLA's Director of Quality Assurance, successfully connected statistical approaches to large-scale manufacture by creating **6 Sigma** , strongly inspired by Toyota's Total Quality Management (TQM). He convinced Motorola's management of his statistical approach in order to reduce defects in production lines and thereby increase quality ... and results. The principle is statistically simple. For data following a normal distribution, Six Sigma considers that within a range of six standard deviations applied on either side of the mean, 99.99966% of the data^{(13)} can be found (see adjacent figure). In other words, if a production line produces 1,000,000 parts and its quality level is Six Sigma, it will generate 3.4 defects for 1,000,000 parts produced - the objective that MOTOROLA set itself in the 1980s. **Thus, Six Sigma was born.**.

**1.6 The Dissemination of the Culture of Operational Excellence by General Electric**

During the 6s, General Electric and its emblematic CEO Jack WELSH generalized Six Sigma to all the conglomerate's processes thanks to the Change Acceleration Process (CAP) approach *Change Acceleration Process* (CAP) and **the equation of effectiveness of change** « *E = Q x A* ": The effectiveness (E) of any initiative is equal to the product of the quality (Q) of the technicality, the strategy and the acceptance (A) of this strategy by the stakeholders. General Electric is focusing on increasing acceptance to sustain its initiative by training employees at 6 Sigma. Combining the PDCA approach *Change Acceleration Process*, General Electric formalizes its approach by the DMAIC methodology: Define, Measure, Analyze, Improve, Control (in French, Define, Measure, Analyze, Improve and Control). Levels are created (Green Belt, Black Belt and Master Black Belt) according to the mastery of the methodology. These "belts", once trained, lead problem-solving projects, train others " *belts*", Contribute to process improvement ... a real policy of**Operational effectiveness** which is spreading to all levels of the company. The results are not overdue as indicated in the caption of this article: in its annual report of 2000^{(14)}, the executives of General Electric (GE) say that " *GE not only achieved its best ever result, but it had the highest growth rate in its history [...] thanks to the rigorous pursuit of four major initiatives: globalization, services, 6 Sigma quality and digitalization. We have changed not only our work environment and what we sell, but also the way we work, think and respond to our customers*".

With statistical mastery of processes, the empirical approach (*for each problem the objective is to find a solution*) becomes a scientific approach (*for each problem the objective is to find the root causes*) , which is summarized by the equation Y = f(X1, X2, X3, ... Xn) + ε where :

- 'Y' is the unit of output from the process and delivered to the customer;
- 'f' is the process that generates the 'Y';
- "(X
_{1}, X_{2}, X_{3}, ... X_{n}) are the known root causes that influence the 'f' and therefore have a cause-and-effect relationship with the 'Y', - 'ε' is noise, i.e. what cannot be explained.

Let's take the example of making a chocolate cake:

- "Y" is the chocolate cake,
- 'f' represents all the tasks to be performed according to the recipe;
- "(X
_{1}, X_{2}, X_{3}, ... X_{n}) are the quantity of flour, sugar, eggs, the cooking time, oven temperature...; - 'ε' is what cannot be explained:
*my grandmother makes the best cakes*".

If you check the cake when it comes out of the oven, you can tell if it is good or bad ... but it will be too late if it is bad. You will have to cook a new one, hoping that this time it will be good. On the other hand, if you know that you must put between 50 - 55 grams of flour, 100 - 110 grams of sugar, 100 - 105 grams of butter, 200 - 210 grams of chocolate, 3 eggs and that the oven temperature must be between 180°C and 185°C and the baking time between 25 minutes and 30 minutes, then you will have a very high probability of success when making your cake ... close to ε. **The process output (Y) is determined by the inputs (X) when ε is small. Therefore, if you control the Xs, then you can reliably predict Y.**.

**2. Example of the Application of Six Sigma in Industry ^{(15)}**

**2.1 For each problem the goal is to find root causes**

An industrial manufacturer produces agri-food products for everyday consumption. For one of its ranges of products, its customers regularly report nonconformities in terms of feeling or perception of the product consumed: there is not the expected flavor. This results in destroyed products when the defect is detected by the factory and consumer claims in the opposite case. The goal is to identify the root causes that explain why the process generates defective products (flavor problem) and thus drive the root causes (the Xs) to reliably predict the flavor (the Y) : without methodology, so much to look for a needle in a haystack. In order to avoid this pitfall, the plant begins a process of eliminating the defect found by launching a project to control industrial variability using the 6 Sigma - DMAIC methodology:

**Define:**explain the problem, confirm its relevance to the company's objectives, confirm the potential for benefit, build a project team;**Measure:**understand the associated process, confirm the existence of the problem (reliability of the measurement, performance), consolidate the team;**Analyze:**identify the root causes of the problem, confront them with the reality on the ground, confirm with operational staff their impact on the process;**Improve:**encourage the emergence of solutions to eliminate or significantly reduce the problem;**Control:**ensure the sustainability and mastery of the solution, establish control tools, bring all the operational staff concerned on board.

**The DEFINE and MEASURE** phases explain and calculate the problem quantitatively by identifying the output produced by the process (*Y, here the flavour *) and the customer's requirements, namely their objective and tolerance. If Y does not meet the customer's flavour requirements, then the problem is proven. The calculation of the problem (*or calculation of the process performance*) confirms its existence in figures: we move from feeling " *every time it doesn't work*"to facts" *during the last twenty production runs, fourteen have not respected the customer's requirements*". These two steps avoid starting up problem resolution if there is no problem.

**The ANALYZE phase searches for root causes. ** To do so, process mapping is carried out with the operators and displayed in a room (see adjacent photo) throughout the project. This visual representation invites operators to indicate where they think a parameter can influence the flavour - the green post-it notes in the adjacent photo. Sixty-four potential root causes are identified. A plan to collect seventeen thousand pieces of data over a four-month period is then put in place.

Once the data has been collected, statistical analysis can then begin: a PLS (Partial Least Square) regression method is then used to evaluate the relationship between the response Y and the explanatory variables X (process parameters to be studied): Y = f(X_{1}, X_{2}, X_{3}, ... X_{n}) + ε. PLS regression is particularly suitable when there are many X explanatory variables and some variables correlate with one another. The principle of this method is to construct a model between the response Y and latent variables (each latent variable, also called component, being a linear combination of the initial explanatory variables). The modelling results then highlight the latent variables that best explain/impact the response Y. This method enables confirmation or invalidation of the operators' intuition and five root causes are thereby identified statistically.

**2.2 Control of the Manufacturing Process: Control Charts**

**The IMPROVEMENT phase** moving from statistical analysis to industrial reality. The impact of the five root causes on Y (flavour) is statistically modelled. This modelling calculates the **tolerance limits**, *also called finished product specifications *that must not be exceeded by the root cause in order for the flavour to meet the customer's requirements. From these tolerance limits, the manufacturer can also establish more demanding **industrial specifications** Indeed, the tolerance limits delimit the area beyond which Y will not meet the customer's requirements. Keeping them as they are is risky: if your car is 180 centimetres wide, you will not build a garage 180 centimetres wide but at least 240 centimetres wide.

**The CONTROL phase ** : *from industrial reality to operational application.*Knowing the limits not to be exceeded, the challenge is to ensure that the root cause respects these limits, i.e. that its variation is mastered in order to guarantee customer satisfaction. Mastery of this variation does not only require compliance with tolerance limits or industrial specifications, but mainly the possibility of detecting a trend that raises an alert of possible abnormal behaviour. This is the principle of Walter A. Shewhart's control charts above: statistical mastery of processes. In the illustration above, the root cause evolves in a so-called control zone and shows no abnormal tendency: the data is normally distributed around the mean, with reference to the Laplace-Gauss distribution used by Abraham de Moivre, and are subject only to common causes, i.e. specific to the process itself. The process is under control, the operator is able to visually control it.

In our example, the analysis of the five root causes enables us to calculate tolerance limits, specify industrial specifications and calculate control limits for each root cause. The installation of control charts directly on the production lines enables operators to predict the defect ... and thereby avoid it.

**3. Six Sigma - an Analogy with Management**

Six Sigma is often combined with statistics, which is true because they are very significant. Sometimes it is not easy to understand them: carrying out an experimental design or conducting an analysis of variance (ANOVA) requires good knowledge of the context in which these methods are used, as well as how to interpret the results.

However, reducing Six Sigma to statistical tools means believing in the existence of a single technical solution to every problem. A somewhat simplistic relationship of uniqueness: " *We have found the solution, all we have to do is tell the teams to apply it*".

**Why are we trying to solve a problem? To have an effect on customer satisfaction, process performance, efficiency of implementation etc.... A good technical solution will certainly contribute to this ... if applied. In other words, if the teams do not use the solution, there is no point in looking for it.**

Six Sigma is more than just a toolbox for solving problems. It does not provide a " *miracle* solution. It searches for the root cause of the problem, places it under control using operational control and thereby prevents the problem from recurring. In the previous example, the manufacturer does not have a *miracle* solution. On the other hand, operators know the parameters (root causes) that influence the process. They have the means to control them (control charts) and are therefore able to master the process in order to deliver the right product/service to the customer. This combination of the *"how"* (the solution) and the *"Why"* (the cause) increases the impact of the desired effect. It is the combination of the technical quality of the solution and the acceptance of this same solution: **EFFECT = QUALITY X ACCEPTANCE.** By implementing control charts - quantified mastery of the process - the manufacturer in our example has worked on acceptance in order to guarantee the flavour expected by their customers. *"In God we trust, all others must bring data"* ^{(XNUMX)}.

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## Damien BONHOMME – 3CONSEILS

### damien.bonhomme@3conseils.com

## References

**(1)** *In God we trust, all others must bring data*

**(2)** GE Annual Report 2000 - 4 Page - http://www.ge.com/annual00/download/images/GEannual00.pdf

**(3)** Marie Viallon. The honors of Venice to Henry of Valois, King of France and Poland: Study of the Venetian stay of King Henry III in 1574. Annual Congress of RSA, Apr 2010, Venice, Italy.

**(4)** «Gribeauval or the beginnings of industrial standardization» - TRISTAN GASTON-BRETON COMPANY HISTORIAN - THE ECHOES | THE 03 / 08 / 2016

**(5)** «Gribeauval or the beginnings of industrial standardization» - TRISTAN GASTON-BRETON COMPANY HISTORIAN - THE ECHOES | THE 03 / 08 / 2016

**(6)** Peter L. Bernstein's Against the Gods - the remarkable story of risk - 1996 - Wiley Edition p. 13

**(7)** Peter L. Bernstein's Against the Gods - the remarkable story of risk - 1996 - Wiley Edition p. 43

**(8)** THE PROBLEM OF PARTIES MOVES ... MORE AND MORE - Norbert Meusnier - University of Paris VIII - Electronic Journal for History of Probability and Statistics Vol 3 N ° 1 June 2007 - 4 page

**(9)** Peter L. Bernstein's Against the Gods - the remarkable story of risk - 1996 - Wiley Edition p. 125

**(10)** Biography of Abraham de MOIVRE: http://www.bibmath.net/bios/index.php?action=affiche&why=demoivre

**(11)** Peter L. Bernstein's Against the Gods - the remarkable story of risk - 1996 - Wiley Edition p. 127-128

**(12)** AFNOR Certification "ISO9001 Transition Guide: 2015 for Small and Very Small Businesses" - November 2016

**(13)** During the life of a process will occur a decentering (called "shift" in English) which is estimated to 1,5 Sigma. We then speak of a long term Sigma (LT) Vs a short term Sigma (CT) where Sigma LT = Sigma CT - 1,5. For details, see. "Six Sigma: How to apply it" by Maurice PILLET / Eyrolles August 2013 / Page 135

**(14)** GE Annual Report 2000 - 4 Page - http://www.ge.com/annual00/download/images/GEannual00.pdf

**(15)** Names, numbers and results have been changed

**(16)** *In God we trust, all others must bring data*

## Bibliography

**"Against the Gods: The Remarkable Story of Risk"** by Peter L. Bernstein - John Wiley & Sons / 29 September 1998

**"The Machine That Changed the World: The Story of Lean Production- Toyota's Secret Weapon in the Global Car Wars That's Now Revolutionizing World Industry"** by James P. Womack, Dan T. Jones & Daniel Roos – Free Press / 13 mars 2007

**"Economic control of quality of manufactured product"** by Walter A. Shewhart - Martino Fine Books / 25 April 2015

**"Six Sigma: Comment l’appliquer"** by Maurice Pillet - Eyrolles / 29 August 2013