Crossed Array Design Problems in Robust Parameter Experiments

This posting describes problems in using Taguchi Parameter designs with crossed arrays in Robust Parameter Designs.  The two previous postings describe an application of Taguchi Parameter Designs to reduce plasma cutter cycle time.  That is, the Taguchi Design posting and the Taguchi Results posting.  The crossed array designs proposed by Taguchi when used with the maximum allowable factors can’t estimate the interaction effects among the controllable factors.

Robust Parameter Designs

When implementing Robust Parameter Designs, we have two types of factors, i.e., controllable and noise factors.   Controllable factors are those whose values can be set when doing the experiments as well as in the field when operating the system.  The system can be a process or a product.  For the Taguchi example, the controllable factor were Tip Size, Feed Rate, Voltage, and Amperage in a process.  The noise factors can have their values set during the experiments, but not during actual operation of the systems in the field.   The noise factors in the Taguchi example were Air Pressure and Pierce Time.

For a product, the noise factors could be environmental factors such as temperature or humidity that affect performance.   Other components of the system using the product might have variable attributes that affect the product performance.   For example, the electrical power source to the product might have variable properties.  Consider a manufacturing process.   The noise variables might be raw material properties.

Robust Parameter Design is used when systems have noise variables.  The purpose of Robust Parameter design is to choose levels of controllable factors so that the mean and variance of the output response meets system objectives (Montgomery, 2012).   However, Montgomery (2012) and other references present approaches to Robust Parameter Design that are often more effective than Taguchi designs.   That is, in many cases, other Robust Parameter Designs could require fewer experiments and reveal more effective system solutions.

Crossed Array Designs

The Taguchi designs use crossed arrays between the controllable and uncontrollable factors.   That is, the designs consists of an inner array containing the controllable factors and outer array containing the uncontrollable factors (Montgomery 2012 and Chen, Li and Cox 2009).  The arrays are crossed because every treatment combination for the controllable factors are run for every treatment combination for the noise factors.

In the Taguchi designs, interactions among controllable factors may give misleading results.   Hicks and Turner (1999) on page 398 point out that the inner arrays used by Taguchi are all resolution III designs when used with the maximum allowable factors.   That is, no main effect are aliased with other main effects; however, main effects are aliased with two-factor interactions.    Consider the Taguchi Design posting where the controllable factors include tip size and voltage.   Assume that tip size and voltage interact where large values of tip size combined with large values of voltage give an additional bevel magnitude not predictable based on considering tip size and voltage alone.   Then the estimate of the tip size and voltage main effects include this interaction effect in a resolution III design.

The crossed array designs could give large experiments.  What if we wanted to expand the inner array in the Taguchi Design posting to estimate two-factor interaction effects.  To do that we need a resolution V design (Montgomery, 2012).   Then no main effects or two factor interactions are aliased with other main effects or two factor interactions.   We could do that by using a complete factorial design.    With four factors and three levels for each factor that would require a 34 or 81 runs.   For the complete crossed array, we would have 324 runs.    If we reduced the inner array to only include two levels for each factor, the inner array would have 24 or 16 runs.    That would give 64 runs for the complete crossed array design.

The next posting will describe combined array designs which will usually achieve the desired resolution levels with fewer experimental runs.


  1. Chen, J.C., Y. Li, and R. A. Cox (2009). “Taguchi-based Six Sigma Approach to Optimize Plasma Cutting Process: An Industrial Case Study”. International Journal of Advanced Manufacturing Technology 41: 760-769.
  2. Charles R. Hicks and Kenneth V. Turner, Jr. (1999). Fundamental Concepts in the Design of Experiments, Fifth Edition, Oxford University Press, New York.
  3. Douglas Montgomery (2012). Design and Analysis of Experiments, Eighth Edition, John Wiley & Sons, New York, Chapter 12.

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