Interaction Graphs for a Two-Level Combined Array Experiment Design by Dr

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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

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Research Statistical Methods

Reviewed Article

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.

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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, graph “B” depicts interac- searchers used orthogonal arrays based smaller number of runs as by using the tions among factors U1-5, and graph “C” on 16 experimental runs as the basis for cross product array. Since the noise depicts interactions among factors W1-5. the experiment and application of factors are not usually controllable, Each graph displays a certain set of linear graphs. Wu and Chen (1992) main effects and two factor interactions characteristics that describe relation- then developed interaction graphs of noise factors are less important than ships, or interactions, among the factors. using the criterion of minimum aberra- control-to-noise interactions, Chen et al It is common practice during the tion. Robinson (1993) followed by (1993). More recently, Borror et al design of an experiment for experi- suggesting other modifications of (2002) studied statistical designs for menters to assign factors so as to Taguchi’s linear graphs. The remainder experiments involving noise factors exhibit interactions if they are known of work related to interaction graphs In practice, both control-to-control to exist and are of interest. It is also available in the literature to date and control-to-noise interactions are common during the design of an pertains to orthogonal arrays corre- normally important. Control-to-control experiment for experimenters to sponding to a single set of factors. interactions play an important role in rename selected factors so as to exhibit Taguchi, however, divided the factors product design and production pro- different interactions within the same into two categories, that is control cesses while control-to-noise interac- fractional factorial experiment in order factors and noise factors. tions play their role in product perfor- to get more information from fewer mance and are associated with varia- experimental runs. Relating Taguchi’s Work To Non- tion. The structure of these interactions When experimenters rename the Isomorphic Interaction Graphs determines the nature of non-homoge- factors to observe different interactions Taguchi introduced the technique neity of process variance that charac- within a fractional factorial experiment, of robust parameter design to reduce terizes product design problems. and the renaming does not change any performance variation in products and Accordingly, a “good” product design of the graph characteristics, the processes by selecting the setting of is one where all desired control-to- interaction graphs are considered to be control factors or design parameters so control and control-to-noise interac- isomorphic. Graph “A” and “B” of that performance is insensitive to noise tions can be estimated by using a Figure 1 are thus isomorphic as the factors such as environmental condi- minimum number of runs. This can be characteristics depicted before and

3 Journal of Industrial Technology • Volume 18, Number 4 • August 2002 to October 2002 • www.nait.org after renaming of the factors are alias structure. These defining relations interactions along with all conceptually the same. are as follows: clear two-factor interactions. When experimenters rename the STEP 4: Calculate interaction matrix factors to observe different interactions a) I=ABCE=BCdf for finding total number of within a fractional factorial experiment, non-isomorphic interaction and the renaming does change one or AE AB AC Bd Bf Ad Af graphs, see Li, Washio, Iida more of the graph characteristics, the BC CE BE Cf Cd Ef dE and Tanimoto (1991). The interactions are considered to be non- two columns and rows of the isomorphic. Comparing either of graph b) I=ABCe=BCDf interaction matrix are headed “A” or “B” (i.e., the isomorphic alphabetically (representing interaction graphs) to graph “C” Ae AB AC BD Bf AD Af the control and the noise reveals the change of several character- BC Ce Be Cf CD - De factors). Then the ijth entry in istics and thus graphs “A” and “B” are Df the interaction matrix is 1 if non-isomorphic to graph “C.” the interaction between ith An Algorithm For Developing row and jth column belongs to An Algorithm For Developing Interaction Graphs For A the set formed in Step 3 and A Non-Isomorphic Alias Struc- Combined Array 0 otherwise. ture For A Different Number The following steps are required STEP 5: Calculate the column total of Of Control And Noise Factors for developing non-isomorphic interac- the interaction matrix which In order to define a non-isomorphic tion graphs. The method for developing are called patterns. These alias structure for a given relation the the algorithm is based on the technique patterns represent the number following two criterion must be met given by Li, Washio, Iida and Tanimoto of interactions associated with within the alias structure where C = a (1991) and Wu and Chen (1992). each factor, i.e., the number control factor and N = a noise factor: of lines (edges) starting from a) Count the number of clear CxC, STEP 1: Consider a defining relation each factor (node). CxN & NxN interactions. with specified number of STEP 6: Calculate extended patterns Σ b) Count the number of alias CxC control and noise factors. which are defined as Di= with CxC; CxC with CxN & STEP 2: Based on the defining dij, where dij’s the pattern of CxN with CxN interactions relation allocate both control all factors adjacent to ith (aliasing of any two factor and noise factors to the factor. interaction NxN interaction is columns of orthogonal arrays STEP 7: Arrange in descending order assumed to be clear two factor and developed alias structure. the patterns and extended interaction). STEP 3: Form a set of two-factor patterns. Since there are two interactions by selecting one types of factors, divide the For a given defined relation, we first two-factor interaction from pattern in two groups, one construct an alias structure with a pre- each aliased two-factor corresponding to the control defined number of control and noise factors. Next we rename the control and noise factors and observe the change in Figure 1. Isomorphic vs. Non-Isomorphic Interaction Graphs the alias structure on the basis of the U criterion discussed above. It has been 1 W observed that the alias pattern changes 1 with the renaming of factors and for different numbers of control and noise factors. This method gives us all possible V1 V2 non-isomorphic alias structure for W W different numbers of control and noise 2 3 U U U factors for a given defining relation. 3 4 5

V3 V4 V5 Example Consider a 26-2 fractional factorial experiment with defining relation

I=ABCE=BCDF. Suppose there are W 4 W 5 four (4) control factors and two (2) U2 noise factors. This gives two non- isomorphic defining relations of same Graph “A” Graph “B” Graph “C ” word length pattern, but of different

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factors and the other corre- There are eight (8) possible I=BcdeF=AcdeG, two non-isomorphic sponding to the noise factors. combinations of eligible, but not clear interaction graphs corresponding to the STEP 8: Repeat the steps 4 to 7 for all two-factor interactions. Consider one defining relation I=BCDef=ACDeg and combinations. Different of the combinations annexing the clear one non-isomorphic interaction graph combinations will give rise to two-factor interactions as: for I=bCDEf=ACDEg. The list of non-isomorphic graphs if the patterns and extended patterns are patterns or the extended AC Ad Ae BC Bd Be Cd given in Table 4 (page 6). patterns are distinct. Ce CF Cg dF eF Fg Bg A detailed list showing the number STEP 9: Corresponding to each BF of non-isomorphic interaction graphs distinct combination, develop for various two level fractional factorial an interaction graph. The interaction matrix, pattern and designs composed of a different extended pattern for the above combi- number of control factors and noise The clear interaction between nation is shown in Table 2. factors is given in the Appendix. A control-to-control factors and control- The sorted pattern and extended complete catalogue of interaction to-noise factors are represented by pattern are [3 5 5 6 3 4 4 ] and [14 22 graphs is available with the authors. solid lines and the eligible, but not 22 24 16 19 19 ] respectively. clear interaction between factors are Proceeding in this manner, we get Use And Interpretation Of represented by broken lines. three distinct patterns and extended Non-Isomorphic Interaction We explain in greater detail the patterns that are listed in Table 3 (page Graphs algorithm steps for developing non- 6). These patterns and extended patterns Designing an experiment using only isomorphic graphs with the help of an give rise to three non-isomorphic few non-isomorphic interaction graphs example as was discussed in Keats and interaction graphs that are shown in is very simple and efficient. This Montgomery (1991). Figure 2 (page 7) as solid lines indicating technique allows us at a glance to decide clear interactions between factors and as how to allocate different control as well Example dotted lines indicating possible, but not as noise factors to different columns of In casting aircraft and jet turbine clear, interactions between other factors. the treatment matrix such that the engine parts, a preliminary goal is to Similarly, we get two non-isomor- desired interaction can be estimated determine an appropriate alloy chemis- phic interaction graphs corresponding with as few experimental runs as try for the material. The intent is to to the defining relation possible. For instance, in the above achieve certain physical properties, such as high ultimate tensile strength, which is achieved by varying the alloy Table 1. Alias Structure elements of materials. In this experiment, Keats and Montgomery (1991) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 considered seven factors (each at 2 A B AB C AC BC d Ad Bd Cd eF levels) of which four are control factors Fg namely (1) aluminum (2) tritanium (3) chromium and (4) silicon and the rest 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 are noise factors, i.e. (5) heat treatment e AeBe Ce dF CgCF BFg F AF (6) temperature and (7) oxygen level. Ag Bg Now we consider a 27-2 design with 32 runs, having WLP (word length pattern) as (000455), see Chen et al Table 2. Interaction Matrix (1993). This will give a resolution IV design. It can be seen there are four A BCFd e g non-isomorphic defining relations of A 0 0101 1 0 WLP (000455) which are as follows: B 0 0111 1 1 (1) I=BCdeF=ACdeg=ABFg C 1 1011 1 1 (2) I=BCdeF=AcdeG=ABFG F 0 1101 1 1 (3) I=BCDef=ACDeg=ABfg d 1 1110 0 0 (4) I=bCDEf=ACDEg=Abfg e 1 1110 0 0 g 0 1110 0 0 Now consider the first defining relation i.e., I=BCdeF=ACdeg=ABFg. The Pattern 3 5654 4 3 alias structure for the given defining Extended 14 22 24 22 19 19 16 relation is as shown in Table 1. Pattern

5 Journal of Industrial Technology • Volume 18, Number 4 • August 2002 to October 2002 • www.nait.org example, if we are interested in estimat- ing the interaction between aluminum Table 3. List Of Distinct Patterns And Extended Patterns and tritanium, then we can easily see For 27-2 Designs With I=BCdeF=ACdeg from the graphs in Figure 2 that we must assign aluminum and tritanium to columns (A,C), (B, C), or (C, F). By extension, as experimenters design fractional factorial experiments, as opposed to more costly and time consuming experiments such as full factorial designs, non-isomorphic Table 4. List Of Distinct Patterns And Extended Patterns interaction graphs become the primary means for identifying the interactions of interest, and we visualize those interactions of interest as the solid lines in the interaction graphs. And since we can readily identify the existence or non-existence of a desired interaction, as experimenters, IT’s can readily focus their attention on renaming factors as needed to obtain the interactions of interest to study these interactions with as few experimental runs as possible – saving time and money, getting better information for use in the problem solving or process improvement process, and reducing the chance of experimental error or information overload. Since the financial implications of References Conclusions completing such quantitative analyses Borkowski, J.J., and Lucas, J. M. It is important to note that IT’s are with fewer rather than more experi- (1997). “Designs of mixed resolu- members of technical as well as mental runs can be substantial, IT’s tion for process robustness studies.” managerial communities. As members should, at the very least, be aware of Technometrics. Vol 39, 63–70. of these communities, IT’s have as the tools and their application to Borror, C.M., Montgomery, D.C., and their primary responsibilities, in many industrial operations. Further, knowing Myers, R.H. (2002). “Evaluation of cases, completing the complex analyses the mechanics of how to actually apply statistical designs for experiments needed to better understand industrial these tools can only help IT’s contrib- involving noise variables.” Journal operations in the context of problem ute to the long-term competitiveness of Quality Technology. 54-70. solving and process improvement or and survivability of their academic or Chen, Y., Dun, D. and Wu, C.F.J. making informed decisions based on industrial employers. (1993) Catalogue for 2 level and 3 the results of experimental-based level fractional factorial design with analysis. In order to do their jobs more Acknowledgements small runs. International Statistical efficiently and effectively IT’s need at The authors would like to thank the Review. Vol 61, 131-146. least a familiarization with high-level Council of Scientific and Industrial Keats, J. Bert and Montogomery, D.C. quantitative tools such as discussed in Research for supporting this research. (1991). Statistical Process Control this paper to facilitate the analysis They would also like to thank Profes- in Manufacturing. New York: activity. Non-isomorphic interaction sors Jeff Wu and Vijayan Nair for the Marcel Dekker, Inc. Publishing. graphs represent an enhancement to the useful discussion and valuable sugges- Kim, Y.G. and Myers, R.H. (1992) IT’s tool box enabling IT’s to design tions. Finally, the authors would like to Response surface approach to data experiments that produce valuable data thank the review panel members for analysis in robust parameter design. and information from as few experi- providing constructive comments that Research report. Department of mental runs as possible – certainly with made it possible to enhance the quality Statistics, Virginia Polytechnic Inst. fewer runs than would be possible of the paper. and State University, Report 92-12. without using the graphs.

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Li, C.C., Washio, Y., Iida, T. and Figure 2. I = BCdeF = ACdeg Tanimoto, S. (1991) Linear graphs of resolution IV for orthogonal array L16(215). Technical Report, Department of Administration Engineering, Faculty of Engineer- ing, Keio University. Miller, A., Sitter, R.R., Wu, C.F.J., and Long, D. (1993). “Are large Taguchi-style experiments neces- sary? A reanalysis of Gear and Pinion Data.” Quality Engineering, Vol 6, 21-38. Montogomery, D.C. (1991). “Using fractional factorial design for robust process development.” Quality Engineering, Vol 3, 193-205. Robinson, G.K. (1993) “Improving Taguchi’s packaging of fractional factorial designs.” Journal of Quality Technology. Vol. 25, 1-11. Shoemaker, A.C., Tsui, K.L., and Wu, C.F.J. (1991). “Economical experi- mentation methods for robust design.” Journal of Quality Technol- ogy, Vol. 22, 15-22. Taguchi, G. (1987) System of experimental design, 2 volumes, Translated and published in English by UNIPUB / Kraus International Publications, White Plains, New York. The National Association of Industrial Technology (1997). Definition of Industrial Technology. [Online]. Available: http://www.nait.org/ main.html [2002, April 16]. Welch, W.J., Yu, Tl.K., Kang, S.M., and Sacks, J. (1990). “Computer experiments for Quality control by parameter design.” Journal of Quality Technology, Vol. 22, 15-22. Wu, C.F.J., and Chen, Y. (1992). “A graph aided method for planning two-level experiment when certain interactions are important.” Technometrics. Vol. 34, 162-175. Wu, J., and Hamada, M., (2000). Experiments: Planning, Analysis, and Parameter Design Optimization. New York: John Wiley and Sons Publishing.

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Appendix

Table 1. Number Of Non-Isomorphic Interaction Graphs For Various 2 Level Fractional Factorial Designs With Different Number Of Control Factors And Noise Factors.

Design Defining Relation No. of No. of No. of Control Factors Noise Factors Non-Isomorphic Interaction Graphs

24-1 I=ABCd 3 1 3 24-1 I=Abcd 2 2 2 25-1 I=ABCe 4 1 3 25-1 I=ABce 3 2 3 26-2 I=ABCE=BCDf 5 1 27 26-2 I=ABCE=BCdf 4 2 14 26-2 I=ABCe=BCDf 4 2 26 26-2 I=ABce=BcDf 3 3 4 26-2 I=ABCe=BCdf 3 3 12 27-2 I=BCDeF=ACDeG 6 1 2 27-2 I=BCDEF=ACDEg 6 1 3 27-2 I=BcdeF=AcdeG 5 2 2 27-2 I=BCDeF=ACDeg 5 2 3 27-2 I=BCDEf=ACDEg 5 2 2 27-2 I=BCdeF=ACdeg 4 3 3 27-2 I=BcdeF=AcdeG 4 3 2 27-2 I=BCDef=ACDeg 4 3 2 27-2 I=BcDEf=AcDEg 4 3 1 27-3 I=ABCE=BCDF=ACDg 6 1 1 27-3 I=ABCE=BCDf=ACDg 5 2 1 27-3 I=ABCe=BCDf=ACDg 4 3 84 27-3 I=ABCE=BCdf=ACdg 4 3 28 28-3 I=BCDEF=ACDEG=ABDEh 7 1 27 28-3 I=BCDeF=ACDeG=ABDeH 7 1 7 28-3 I=BCDEF=ACDEg=ABDEh 6 2 26 28-3 I=BcDEF=AcDEG=ABDEh 6 2 14 28-3 I=BCDeF=ACDeG=ABDeh 6 2 30 28-3 I=BCdeF=ACdeG=ABdeH 6 2 7 28-3 I=BCDEf=ACDEg=ABDEh 5 3 3 28-3 I=BcDEF=AcDEg=ABDEh 5 3 12 28-3 I=BCDeF=ACDeg=ABDeh 5 3 26 28-3 I=BcDeF=AcDeG=ABDeh 5 3 14 28-3 I=BCdeF=ACdeG=Abdeh 5 3 30 28-3 I=bcDEF=AcDEg=AbDEh 4 4 4 28-3 I=BcDEf=AcDEg=ABDEh 4 4 2 28-3 I=BCDef=ACDeg=ABDeh 4 4 4

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Design Defining Relation No. of No. of No. of Control Factors Noise Factors Non-Isomorphic Interaction Graphs

28-3 I=BcDeF=AcDeg=ABDeh 4 4 12 28-3 I=BCdeF=ACdeg=ABdeh 4 4 26 28-3 I=BcdeF=AcdeG=ABdeh 4 4 14 28-4 I=ABCE=BCDF=ACDG=ABDh 7 1 125 28-4 I=ABCE=BCDF=ACDg=ABDh 6 2 241 28-4 I=ABCE=BCDf=ACDg=ABDh 5 3 253 28-4 I=ABCE=BCdf=ACdg=ABdh 4 4 38 28-4 I=ABCe=BCDf=ACDg=ABDh 4 4 125 29-3 I=ABCDg=ACEFH=CDEFJ 8 1 6 29-3 I=ABCDG=ACEFH=CDEFj 8 1 8 29-3 I=ABCDg=ACEfH=CDEfJ 7 2 6 29-3 I=ABCDg=ACEFH=CDEFj 7 2 8 29-3 I=ABCDG=ACEFh=CDEFj 7 2 3 29-3 I=ABCDg=ACefH=CDefJ 6 3 2 29-3 I=ABCDg=ACEfH=CDEfj 6 3 4 29-3 I=ABCDg=ACEFh=CDEFj 6 3 2 29-3 I=ABCdG=ACEFh=CdEFj 6 3 1 29-3 I=ABcDg=AcefH=cDefJ 5 4 2 29-3 I=ABCDg=ACefH=CDefj 5 4 4 29-3 I=ABCDg=ACEfh=CDEfj 5 4 4 29-3 I=ABCdg=ACEFh=CdEFj 5 4 1 29-3 I=aBCdG=aCEFh=CdEFj 5 4 1 29-4 I=BCDEF=ACDEG=ABDEH=ABCEj 8 1 395 29-4 I=BCDeF=ACDeG=ABDeH=ABCeJ 8 1 66 29-4 I=BCDEF=ACDEG=ABDEh=ABCEj 7 2 971 29-4 I=BcdEF=ACdEG=ABdEH=ABCEj 7 2 310 29-4 I=BCDeF=ACDeG=ABDeH=ABCej 7 2 442 29-4 I=BCDEF=ACDEg=ABDEh=ABCEj 6 3 184 29-4 I=BcdEF=ACdEG=ABdEh=ABCEj 6 3 55 29-4 I=BCDeF=ACDeG=ABDeh=ABCej 6 3 593 29-4 I=BcdeF=ACdeG=ABdeH=ABCej 6 3 190 29-4 I=BcdEf=ACdEg=AbdEh=ABCEJ 5 4 12 29-4 I=BcdEF=ACdEg=AbdEh=ABCEj 5 4 74 29-4 I=BCDeF=ACDeg=ABDeh=ABCej 5 4 186 29-4 I=BcdEF=AcdEG=ABdEh=ABcEj 5 4 70 29-4 I=BcdeF=ACdeG=ABdeh=ABCej 5 4 197 210-4 I=ABCDG=ACEFH=CDEFJ=ABCEk 9 1 32 210-4 I=ABCDG=ACEFH=CDEFj=ABCEK 9 1 107 210-4 I=ABcDG=AcEFH=cDEFJ=ABcEK 9 1 90 210-4 I=ABCDg=ACEFH=CDEFj=ABCEK 8 2 150

9 Journal of Industrial Technology • Volume 18, Number 4 • August 2002 to October 2002 • www.nait.org

Design Defining Relation No. of No. of No. of Control Factors Noise Factors Non-Isomorphic Interaction Graphs

210-4 I=ABCDg=ACeFH=CdeFJ=ABCeK 8 2 39 210-4 I=ABcDG=AcEFH=cDEFj=ABcEK 8 2 110 210-4 I=ABCDG=ACEFh=CDEFJ=ABCEk 8 2 51 210-4 I=ABCDG=ACEFH=CDEFj=ABCEk 8 2 132 210-4 I=ABCDG=ACEFh=CDEFj=ABCEK 8 2 102 210-4 I=ABCDG=ACEFh=CDEFj=ABCEk 7 3 55 210-4 I=ABCDg=ACEFH=CDEFj=ABCEk 7 3 148 210-4 I=ABCDg=ACEfH=CDEfj=ABCEK 7 3 88 210-4 I=ABCDG=ACEfh=CDEfJ=ABCEk 7 3 17 210-4 I=ABCDG=ACEfh=CDEfj=ABCEK 7 3 34 210-4 I=ABCDg=ACeFH=CDeFj=ABCeK 7 3 99 210-4 I=ABcDg=AcEFH=cDEFj=ABcEK 7 3 150 210-4 I=ABCDg=ACeFH=CDeFJ=ABCek 7 3 20 210-4 I=ABcDg=AceFH=cDeFJ=ABceK 7 3 39 210-4 I=ABCdG=ACEFh=CdEFJ=ABCEk 7 3 7 210-4 I=ABCDg=ACEFh=CDEFj=ABCEk 6 4 19 210-4 I=ABCDg=ACEfH=CDEFj=ABCEK 6 4 52 210-4 I=ABCdG=ACEFh=CdEFj=ABCEk 6 4 5 210-4 I=ABCDg=ACeFH=CDeFj=ABCek 6 4 19 210-4 I=ABCDG=ACEFh=CDEfj=ABCEk 6 4 15 210-4 I=ABcDg=AcEfH=cDEfj=ABCEK 6 4 96 210-4 I=ABCDg=ACeFH=CDeFj=ABCek 6 4 19 210-4 I=ABcDg=AcEfh=cDEfJ=ABcEK 6 4 18 210-4 I=ABcDg=AceFH=cDeFj=ABceK 6 4 99 210-4 I=ABCDg=ACefH=CDefJ=ABCek 6 4 9 210-4 I=ABcDg=AceFH=cDeFJ=ACcek 6 4 20 210-4 I=ABCdg=ACeFH=CdeFJ=ABCEk 6 4 19 210-4 I=ABCDg=ACefh=CDefj=ABCeK 5 5 89 210-4 I=ABcDg=AcEfH=cDEfj=ABcEk 5 5 165 210-4 I=ABCDg=ACEfh=CDEfj=ABCEk 5 5 225 210-4 I=ABcDg=AcefH=cDefj=ABceK 5 5 211 210-4 I=ABCDg=ACEfH=CDefj=ABCek 5 5 285 210-4 I=ABCdg=ACEFh=CdEFj=ABCEk 5 5 209 210-4 I=ABcDg=AceFH=cDeFj=ABcek 5 5 128 210-4 I=ABCDg=ACefh=CDefJ=ABCek 5 5 4 210-4 I=ABCdg=ACeFh=CdeFJ=ABCek 5 5 5 210-4 I=ABcDg=AcefH=cDefJ=ABcek 5 5 9 210-4 I=ABCdg=ACeFH=CdeFj=ABCek 5 5 15 210-4 I=ABcdg=AceFH=cdeFJ=ABcek 5 5 19 210-4 I=AbCDg=ACefH=CDefJ=AbCek 5 5 2