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 describes how different characteristics of the manufacturing process such as deposition temperature, deposition pressure, nitrogen flow, etc.
However, no theoretical model exists that would allow the engineer to predict how these factors affect the uniformness 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 , 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.
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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 see above. Example 6. 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. Such so-called constrained experimental regions present numerous problems, which, however, can be addressed. With regard to the first question, there are different considerations that enter into the different types of designs, and they will be discussed shortly. In the most general terms, the goal is always to allow the experimenter to evaluate in an unbiased or least biased way, the consequences of changing the settings of a particular factor, that is, regardless of how other factors were set.
In more technical terms, you attempt to generate designs where main effects are unconfounded among themselves, and in some cases, even unconfounded with the interaction of factors. There are several statistical methods for analyzing designs with random effects see Methods for Analysis of Variance. Experimental methods are finding increasing use in manufacturing to optimize the production process. Specifically, the goal of these methods is to identify the optimum settings for the different factors that affect the production process.
Interestingly, many of these experimental techniques have "made their way" from the production plant into management, and successful implementations have been reported in profit planning in business, cash-flow optimization in banking, etc. In many cases, it is sufficient to consider the factors affecting the production process at two levels. For example, the temperature for a chemical process may either be set a little higher or a little lower, the amount of solvent in a dyestuff manufacturing process can either be slightly increased or decreased, etc.
The experimenter would like to determine whether any of these changes affect the results of the production process. The most intuitive approach to study those factors would be to vary the factors of interest in a full factorial design, that is, to try all possible combinations of settings. This would work fine, except that the number of necessary runs in the experiment observations will increase geometrically. Because each run may require time-consuming and costly setting and resetting of machinery, it is often not feasible to require that many different production runs for the experiment.
In these conditions, fractional factorials are used that "sacrifice" interaction effects so that main effects may still be computed correctly. A technical description of how fractional factorial designs are constructed is beyond the scope of this introduction. In general, it will successively "use" the highest-order interactions to generate new factors.
For example, consider the following design that includes 11 factors but requires only 16 runs observations. Reading the design. The design displayed above should be interpreted as follows.
So for example, in the first run of the experiment, set all factors A through K to the plus setting e. For example, you may use actual values of factors e. Randomizing the runs. Because many other things may change from production run to production run, it is always a good practice to randomize the order in which the systematic runs of the designs are performed. As a result, the design does not give full resolution ; that is, there are certain interaction effects that are confounded with identical to other effects.
In general, a design of resolution R is one where no l -way interactions are confounded with any other interaction of order less than R-l. In the current example, R is equal to 3. Thus, main effects in this design are confounded with two- way interactions; and consequently, all higher-order interactions are equally confounded. In this design then, main effects are not confounded with two-way interactions, but only with three-way interactions. What about the two-way interactions? Thus, the two-way interactions in that design are confounded with each other. One way to design such experiments is to confound all interactions with "new" main effects.
Such designs are also sometimes called saturated designs, because all information in those designs is used to estimate the parameters, leaving no degrees of freedom to estimate the error term for the ANOVA.
Plackett and Burman showed how full factorial design can be fractionalized in a different manner, to yield saturated designs where the number of runs is a multiple of 4, rather than a power of 2. These designs are also sometimes called Hadamard matrix designs. Of course, you do not have to use all available factors in those designs, and, in fact, sometimes you want to generate a saturated design for one more factor than you are expecting to test. This will allow you to estimate the random error variability, and test for the statistical significance of the parameter estimates.
One way in which a resolution III design can be enhanced and turned into a resolution IV design is via foldover e. This is a resolution III design, that is, the two-way interactions will be confounded with the main effects. You can turn this design into a resolution IV design via the Foldover enhance resolution option.
The foldover method copies the entire design and appends it to the end, reversing all signs:. Thus, the standard run number 1 was -1, -1, -1, 1, 1, 1, -1 ; the new run number 9 the first run of the "folded-over" portion has all signs reversed: 1, 1, 1, -1, -1, -1, 1. Design generators. Specifically, factor 5 is identical to the factor 1 by factor 2 by factor 3 interaction. Factor 6 is identical to the interaction, and so on.
Remember that the design is of resolution III three , and you expect some main effects to be confounded with some two-way interactions; indeed, factor 10 ten is identical to the 12 factor 1 by factor 2 interaction, and factor 11 eleven is identical to the 13 factor 1 by factor 3 interaction. Another way in which these equivalencies are often expressed is by saying that the main effect for factor 10 ten is an alias for the interaction of 1 by 2. The term alias was first used by Finney, The resulting design is no longer a full factorial but a fractional factorial.
The fundamental identity. Another way to summarize the design generators is in a simple equation. Thus, we also know that factor 1 is confounded with the interaction, factor 2 with the , interaction, and factor 3 with the interaction, because, in each instance their product must be equal to 1. The confounding of two-way interactions is also defined by this equation, because the 12 interaction multiplied by the 35 interaction must yield 1 , and hence, they are identical or confounded.
In some production processes, units are produced in natural "chunks" or blocks.
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You want to make sure that these blocks do not bias your estimates of main effects. For example, you may have a kiln to produce special ceramics, but the size of the kiln is limited so that you cannot produce all runs of your experiment at once.
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In that case you need to break up the experiment into blocks. However, you do not want to run positive settings of all factors in one block, and all negative settings in the other. Otherwise, any incidental differences between blocks would systematically affect all estimates of the main effects of the factors of interest.