Materiales y técnicas constructivas

Box, Hunter, and Hunter (2005) assert that statistically designed experiments, which are also known as design of experiments are part and parcel of discovery and problem solving in engineering and other science disciplines (Box, 1993; Montgomery, 2013). In the DoE literature, three different types of experimental designs are mentioned: factorial designs (which can be divided into full factorial and fractional factorial designs), response surface designs, and evolutionary operation (Box et al., 2005; Steinberg, 2014). Montgomery (2013) depicts an experiment-based problem solving cycle (Figure 3.1) in seven essential steps.10 A brief description of each step follows.

10 _{It appears that Montgomery’s model shown in Figure 3.1 does not cover the evolutionary operation} (EVOP) procedure. EVOP concerns adjusting a live process to evolve into a better operating condition by adjusting few (typically two) factors in a cyclical position. As such, some steps in Figure 3.1 do not apply to EVOP, for example, the question of proposing and refining the model (Box 3 in Figure 3.1) does not arise.

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Figure 3.1: The engineering problem solving method (source: Montgomery, 2013)

Like in any problem solving matter, defining the problem and succinctly describing it (Step 1), is strongly emphasised in the literature (Box et al., 2005; Montgomery, 2013). At the early stage of learning about a process, the problem to be solved could be, for example, identifying the important factors that need to be manipulated (out of many possible factors) in future experiments (to achieve the end goal), without wasting too much resources; a fractional factorial experiment, where a large number of factors (variables) are simultaneously manipulated at few selected combinations of factor settings (out of all the possible combinations of factor settings, given the number of levels at each factor is to be manipulated) becomes the experiment of choice in such a situation (Box et al., 2005).11 At the final stages of learning, where a “response

surface” DoE methodology such as the central composite design becomes the natural choice (Box et al., 2005), the problem definition (still in Box 1, Figure 3.1) could be for example, mapping the relationship between the most influential factors (previously identified) and the process output or outcome (e.g. the yield) in the region in which the desired outcome exists (Montgomery, 2013). Three types of the desired outcomes are mentioned in the literature: maximising the response (which is usually the case if the process yield is the response variable), or achieving a target response (which is the case when producing a product that needs to conform to a certain specification), or

11 For example, the engineers may guess that there are as many as seven potential factors that affect their process outcome (for example, in a chemical production process, the yield). If they decide that each factor should be manipulated at two levels (which is typical) there are 27 (= 128) ways in which the seven factors could be manipulated. Just for the purpose of identifying the influential (important) factors, the engineers would not waste resources by way of running 128 experimental runs; instead, they would decide to run, say 1/16th fraction of the full complement of runs, which boils down to just running 8 experimental runs (Montgomery, 2013).

minimising the response (which is the case if the response is not desired, such as the impurity level in a composite material that is produced).

At the very first stage of learning, the experimenters need to identify the potentially important factors (Box 2) from whatever process knowledge they possess (Montgomery, 2013); in the field of quality management, tools such as brainstorming and Pareto Charts are prescribed in the literature to facilitate this process (Dale et al., 2013; Summers, 2010). At the initial stage/s of learning an experimenter would propose a linear model (also no interaction between factors) relating the response with the factors that they identified (Box 2). Thereafter they would collect the data (Box 3) and fit the model to data, in order to refine the model (back to Box 2). The statistical literacy required to perform these tasks include knowledge on the principles of data collection, the awareness of the existence of confounding factors that can affect the results and precautions that need to be taken to minimise their effects (e.g. “blocking”), and analytical skills such as the analysis of variance (ANOVA) and multiple regression (Box et al., 2005; Montgomery, 2013).

Confirming the solution (Box 6 in Figure 3.1), meaning confirming the statistically determined optimal solution with a new set of data, becomes paramount in the final stages of learning because this solution has commercial consequences (Montgomery, 2013; Roy, 2010). The conformation step (Box 6) also epitomises the most fundamental principle in statistical thinking: variation is inevitable in any phenomenon (Hoerl & Snee, 2012).

“Sequential learning” is a principle that is heavily emphasised in conventional DoE literature (Montgomery, 2013; Tanco, Viles, Ilzarbe, & Alvarez, 2009). Sequential learning refers to building knowledge in a sequential fashion by conducting one small experiment (meaning a low commitment of resources) at each cycle of learning, by answering only one of few questions in each cycle (Montgomery, 2013; Tanco et al., 2009). This is also a reason why a clear and concise description of the problem to be resolved (Box 1 in Figure 3.1) is important in each learning cycle. The principle of learning sequentially, (i.e. iteratively) using a small experiment at each time is known as the “keep it simple and sequential” (KISS) principle in the conventional DoE literature (Montgomery, 2013). This principle is based on the proposition that an experimenter would know very little about the process being investigated at the beginning, and

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therefore they should not take the risk committing vast amounts of resources upfront to achieve the final objective (Montgomery, 2013; Tanco et al., 2009). Thus the KISS principle is viewed as a risk minimisation strategy (Box et al., 2005).

Another important principle that is highlighted in the DoE literature is varying all the experimental factors simultaneously, instead of varying one-factor-at-a-time (OFAT), keeping other factors fixed (Czitrom, 1999; Montgomery & Runger, 2010; Tanco, Viles, Ilzarbe, & Álvarez, 2007). It is mentioned in the literature that OFAT experiments are inefficient and unreliable because of possible interactions between factors (Czitrom, 1999; George, Raghunath, Manocha, & Warrier, 2004; Gunter, 1987; Montgomery & Runger, 2010). Characterising the process by establishing a functional relationship (i.e. a mathematical relation of the form y = f(x)) between the response (y)

and the factors (the independent variables x) empirically through data collection and model fitting is a salient feature in conventional DoE (Box et al., 2005; Khuri & Mukhopadhyay, 2010; Montgomery, 2013).

Implicit in the traditional approaches is the notion that it is the statistician (or the statistically literate engineer) who plans the experiments and analyses the results (Mayer & Benjamin, 1992; Roy, 2010; Tay & Butler, 1999). The traditional approach does not require a great deal of prior process knowledge; knowledge emerges through statistical analysis such as ANOVA, regression analysis, model adequacy diagnostics, factorial plots and so on (Gunter, 1987; Montgomery, 2013; Roy, 2010).

It is commonly acknowledged in the literature (e.g. Box, 1993; Montgomery, 2013) that until Genichi Taguchi introduced the RD concept (Taguchi, 1986) and the need to design experiments to reduce the variation of the response, DoE experiments were designed towards achieving a certain response value (on average) only; designing experiments to model response functions to capture variation (dispersion) became common in conventional DoE with the advent of the Taguchi’s RPD concept; concurrently, applications of DoE methods advocated by Taguchi (commonly known as Taguchi methods) also increased (Montgomery, 2013) with one stream getting cross- fertilised from the other to add the richness to each stream. For example, some procedures that are now being considered as standard in Taguchi methods actually come from the recommendations made by statisticians such as Box (e.g. Box, 1988) on working around some of the possible pitfalls of Taguchi’s data analytic framework

(section 3.4.2.3). Today, the academia and the practitioners in the field of quality are blessed with a rich body of knowledge on designing experiments to reduce variation. Six Sigma (see section 2.3.5) is one of the well-established quality systems that use DoE to reduce variation (Allen, 2010; Evans & Lindsay, 2015).

THE ROBUST DESIGN APPROACH