Journal of Industrial Technology • Volume 18, Number 4 • August 2002 to October 2002 • www.nait.org Volume 18, Number 4 - August 2002 to October 2002 Interaction Graphs For A Two-Level Combined Array Experiment Design By Dr. M.L. Aggarwal, Dr. B.C. Gupta, Dr. S. Roy Chaudhury & Dr. H. F. Walker KEYWORD SEARCH Research Statistical Methods Reviewed Article The Official Electronic Publication of the National Association of Industrial Technology • www.nait.org © 2002 1 Journal of Industrial Technology • Volume 18, Number 4 • August 2002 to October 2002 • www.nait.org Interaction Graphs For A Two-Level Combined Array Experiment Design By Dr. M.L. Aggarwal, Dr. B.C. Gupta, Dr. S. Roy Chaudhury & Dr. H. F. Walker Abstract interactions among those variables. To Dr. Bisham Gupta is a professor of Statistics in In planning a 2k-p fractional pinpoint these most significant variables the Department of Mathematics and Statistics at the University of Southern Maine. Bisham devel- factorial experiment, prior knowledge and their interactions, the IT’s, engi- oped the undergraduate and graduate programs in Statistics and has taught a variety of courses in may enable an experimenter to pinpoint neers, and management team members statistics to Industrial Technologists, Engineers, interactions which should be estimated who serve in the role of experimenters Science, and Business majors. Specific courses in- clude Design of Experiments (DOE), Quality Con- free of the main effects and any other rely on the Design of Experiments trol, Regression Analysis, and Biostatistics. desired interactions. Taguchi (1987) (DOE) as the primary tool of their trade. Dr. Gupta’s research interests are in DOE and sam- gave a graph-aided method known as Within the branch of DOE known pling. Bisham received his Ph.D. (Statistics) and Mas- linear graphs associated with orthogo- as classical methods, it is possible for ter of Science in Mathematics from the University of Windsor, Canada in 1972 and 1969 respectively. He nal arrays to facilitate the planning of experimenters to make use of their also received his Master of Science in Mathematics such experiments. Since then the graph- technical skills and in-depth knowledge and Statistics in 1964 and Bachelor of Science in Mathematics in 1962 from Punjab University, India. aided method has been enhanced by of their particular industrial operations various authors such as Li et al (1991), to design more effective, less expensive Wu and Chen (1992), Robinson (1993), experiments using non-isomorphic and Wu and Hamada (2000). In this interaction graphs. Accordingly, in this paper, we propose an algorithm for paper the authors will provide for developing all possible non-isomorphic readers a context for the application of interaction graphs for combined arrays. non-isomorphic interaction graphs, formally define and describe these Introduction graphs as valuable tools, provide an Industrial Technologists (IT’s) are algorithm for developing all possible frequently required to use designed graphs for combined arrays, provide for experiments as problem solving and readers a detailed example developing process improvement tools in their roles and using these graphs, explain how to as “…technical and/or management read and interpret the graphs, and pose oriented professionals…in business, the argument that IT’s and other Dr. H. Fred Walker, CQMgr, CQE, CQA, CSIT is the industry, education, and government.” interested professionals can use non- Department Chair and Graduate Coordinator in the Department of Technology at the University of South- (The National Association of Industrial isomorphic interaction graphs to do their ern Maine. Fred develops and teaches graduate Technology, 1997, [Online]). In fact, it jobs more effectively and efficiently. courses in Applied Research Methods, Engineering Economy, Quality Systems, Manufacturing Strategies, is common practice for IT’s to work and Project Management. He also develops and teaches undergraduate courses Quality, Industrial Sta- with other communities of technical and Background tistics, Cost Analysis and Control, Human Resource managerial professionals as experiment- In planning a 2k-p fractional facto- Management, Project Management, and Technical Writing. Dr. Walker’s industrial experience includes ers in the application of designed rial experiment, prior knowledge may 12 years with airborne weapons systems integration experiments in discrete part manufactur- enable experimenters to pinpoint and automation, supervision and administration, project management, and program management in ing, assembly, and process industries. interactions which should be estimated countries throughout the Pacific Rim, Australia, and Of primary importance to these experi- free of main effects and any other Africa. Fred’s consulting experience includes ten years of continual involvement with internationally-based menters is working with design, desired interactions. While working on manufacturers from electronics, pulp and paper, bio- production, and quality personnel to applications of classical DOE methods, medical equipment, printing, farm implements, and machined component industries. This experience has identify and understand the many Taguchi (1987) introduced a graph- enabled Fred to earn Certified Quality Manager, Cer- variables or factors associated with any aided technique known as linear graphs tified Quality Engineer, Certified Quality Auditor, Cer- tified Manufacturing Technologist, and Certified Se- type of industrial operation. It is associated with orthogonal arrays to nior Industrial Technologist designations. Dr. Walker particularly interesting to these experi- facilitate the planning of such experi- received his Ph.D. (Industrial Education and Technol- ogy) and Master of Science in Systems Engineering menters to pinpoint those variables that ments, and as such, Taguchi’s graph from Iowa State University. He also received his Mas- ter of Business Administration (MBA) and Bachelor of most significantly influence or impact aided technique is not to be confused Science in Industrial Technology from California State their industrial operations as well as any with what is commonly known as University – Fresno. 2 Journal of Industrial Technology • Volume 18, Number 4 • August 2002 to October 2002 • www.nait.org Taguchi Methods. Linear graphs are tions, properties of raw materials and achieved by using interaction graphs graphical representations of the any other factors that are hard-to- for two-level combined arrays. allocation of main effects and desired control. Taguchi’s technique, also In this paper, we give an algorithm two-factor interactions to the columns known as Taguchi methods, uses an for developing all possible non- of orthogonal arrays. Linear graphs experimental design consisting of a isomorphic interaction graphs for a facilitate the selection of an aliasing cross product of two arrays, an inner combined array of various two level pattern that in turn enables experiment- array containing the control factors and fractional factorial designs. We do this ers to estimate all main effects and an outer array containing the noise in the context of classical DOE appli- desired two-factor interactions. And it factors. The cross product of the inner cations even though the topic could be should be noted it is important for and outer array often leads to a large applied to Taguchi Methods. These experimenters to be able to quantify the number of observations that are interaction graphs enable one to existence and magnitude of these main generally very expensive to complete. allocate the factors to the columns of effects and interactions to support fact- In an attempt to reduce the number orthogonal arrays, along with the based decision making regarding the of observations needed to support an estimation of main effects and desired experiment design and application. experiment, and thus reduce costs, control-to-control, control-to-noise Since 1987, researchers such as Li, Welch et al (1990), Shoemaker et al interactions assuming all other interac- Washio, Iida and Tanimoto (1991) have (1991), Montgomery (1991), and tions to be negligible. extended Taguchi’s work on classical Borkowski and Lucas (1997) suggested DOE applications of linear graphs by independently an alternative design to “Non-Isomorphic” Defined developing non-isomorphic linear study both the control and noise factors Having introduced the concept of graphs for an experiment involving by using a single array, called a linear interaction graphs in the preced- eight (8) factors. In this scenario, a 2k combined array. A combined array ing paragraphs, it is time to explain the experiment design was crafted wherein structure enables experimenters to nature and meaning of the concepts of no main effect was confounded with estimate both control-to-control and isomorphic and non-isomorphic as they any other main effect or with any two- control-to-noise interactions with fewer relate to interaction graphs. Figure 1 factor interactions, while two-factor observations. Miller et al (1993) used below is provided to help readers interactions were confounded with a combined array in an automobile visualize the explanation. each other. This type of experiment is experiment and demonstrated similar As can be seen in Figure 1 (page 4), commonly referred to as a resolution results could be obtained using the graph “A” depicts interactions among IV design and, in this case, the re- combined array approach and a much factorsV1-5,