Experimental Design - General Ideas

Example 1: Dyestuff manufacture

Box and Draper (1987, page 115) report an experiment concerned with the manufacture of certain dyestuff. Quality in this context can be described in terms of a desired (specified) hue and brightness and maximum fabric strength. Moreover, it is important to know what to change in order to produce a different hue and brightness should the consumers' taste change.

Put another way, the experimenter would like to identify the factors that affect the brightness, hue, and strength of the final product. In the example described by Box and Draper, there are 6 different factors that are evaluated in a 2(6-0) design (the 2(k-p) notation is explained below). The results of the experiment show that the three most important factors determining fabric strength are the Polysulfide index, Time, and Temperature (see Box and Draper, 1987, page 116). One can summarize the expected effect (predicted means) for the variable of interest (i.e., fabric strength in this case) in a so-called cube-plot. This plot shows the expected (predicted) mean fabric strength for the respective low and high settings for each of the three variables (factors).

Example 1.1: Screening designs

In the previous example, 6 different factors were simultaneously evaluated. It is not uncommon, that there are very many (e.g., 100) different factors that may potentially be important. Special designs (e.g., Plackett-Burman designs, see Plackett and Burman, 1946) have been developed and are implemented in the Experimental Design module to screen such large numbers of factors in an efficient manner, that is, with the least number of observations necessary. For example, you can design and analyze an experiment with 127 factors and only 128 runs (observations) and still be able to estimate the main effects for each factor. Thus, you can quickly identify which ones are important and most likely to yield improvements in the process under study.

Example 2: 33 design

Montgomery (1976, page 204) describes an experiment conducted in order to identify the factors that contribute to the loss of soft drink syrup due to frothing during the filling of five-gallon metal containers. Three factors were considered: (a) the nozzle configuration, (b) the operator of the machine, and (c) the operating pressure. Each factor was set at three different levels, resulting in a complete 3⁽³⁾ experimental design (the 3^(k-p) notation is explained below). Moreover, two measurements were taken for each combination of factor settings, that is, the 3^(3-0)design was completely replicated once.

Example 3: Maximizing yield of a chemical reaction

The yield of many chemical reactions is a function of time and temperature. Unfortunately, these two variables often do not affect the resultant yield in a linear fashion. In other words, it is not so that "the longer the time, the greater the yield" and "the higher the temperature, the greater the yield." Rather, both of these variables are usually related in a curvilinear fashion to the resultant yield. Thus, in this example your goal as experimenter would be to optimize the yield surface that is created by the two variables: Time and Temperature.

Example 4: Testing the effectiveness of four fuel additives

Latin square designs are useful when the factors of interest are measured at more than two levels, and the nature of the problem suggests some blocking. For example, imagine a study of 4 fuel additives on the reduction in oxides of nitrogen (see Box, Hunter, and Hunter, 1978, page 263). You may have 4 drivers and 4 cars at your disposal. You are not particularly interested in any effects of particular cars or drivers on the resultant oxide reduction; however, you do not want the results for the fuel additives to be biased by the particular driver or car. Latin square designs allow you to estimate the main effects of all factors in the design in an unbiased manner. With regard to the example, the arrangement of treatment levels in a Latin square design assures that the variability among drivers or cars does not affect the estimation of the effect due to different fuel additives.

Example 5: Improving surface uniformity in the manufacture of polysilicon wafers

The manufacture of reliable microprocessors requires very high consistency in the manufacturing process. Note that in this instance, it is equally, if not more important to control the variability of certain product characteristics than it is to control the average for a characteristic. For example, with regard to the average surface thickness of the polysilicon layer, the manufacturing process may be perfectly under control; yet, if the variability of the surface thickness on a wafer fluctuates widely, the resultant microchips will not be reliable. Phadke (1989) describes how different characteristics of the manufacturing process (such as deposition temperature, deposition pressure, nitrogen flow, etc.) affect the variability of the polysilicon surface thickness on wafers. However, no theoretical model exists that would allow the engineer to predict how these factors affect the uniformity of wafers. Therefore, systematic experimentation with the factors is required to optimize the process. This is a typical example where Taguchi robust design methods would be applied.

Example 6: Mixture designs

Cornell (1990a, page 9) reports an example of a typical (simple) mixture problem. Specifically, a study was conducted to determine the optimum texture of fish patties as a result of the relative proportions of different types of fish (Mullet, Sheepshead, and Croaker) that made up the patties. Unlike non-mixture experiments, the total sum of the proportions must be equal to a constant, for example, to 100%. The results of such experiments are usually graphically represented in so-called triangular (or ternary) graphs. In general, the overall constraint --that the three components must sum to a constant is reflected in the triangular shape of the graph.

Example 6.1: Constrained mixture designs

It is particularly common in mixture designs that the relative amounts of components are further constrained (in addition to the constraint that they must sum to, for example, 100%). For example, suppose we wanted to design the best-tasting fruit punch consisting of a mixture of juices from five fruits. Since the resulting mixture is supposed to be a fruit punch, pure blends consisting of the pure juice of only one fruit are necessarily excluded. Additional constraints may be placed on the "universe" of mixtures due to cost constraints or other considerations, so that one particular fruit cannot, for example, account for more than 30% of the mixtures (otherwise the fruit punch would be too expensive, the shelf-life would be compromised, the punch could not be produced in large enough quantities, etc.). Such so-called constrained experimental regions present numerous problems, which, however, can be addressed with the Experimental Design module. In general, under those conditions, one seeks to design an experiment that can potentially extract the maximum amount of information about the respective response function (e.g., taste of the fruit punch) in the experimental region of interest.

Example 7: Response Optimization

A typical problem in product development is to find a set of conditions, or levels of the input variables, that produces the most desirable product in terms of its characteristics, or responses on the output variables. As an example, consider, the problem of finding the most desirable tire tread compound (reported in Derringer & Suich, 1980). There are a number of Y variables, such as PICO Abrasion Index, 200 percent modulus, elongation at break, and hardness. The characteristics of the product in terms of the response variables depend on the ingredients, the X variables, such as hydrated silica level, silane coupling agent level, and sulfur. The problem is to select the levels for the X's which will maximize the desirability of the responses on the Y's. The solution must take into account the fact that the levels for the X's which maximize one response may not come close to maximizing a different response. Also, for some response variables, the most desirable value is not necessarily the highest value, for example, the most desirable value of elongation falls within a narrow range of the possible values. Such problems can be addressed with the Experimental Design module using Response/desirability profiling and other options for analyzing prediction profiles and response surfaces.