As proposed by Taguchi, the objective of robust design is to make a product or process least sensitive to manufacturing variations, deterioration over time and environmental conditions. This is a cost-effective way to improve quality, because it builds quality into products and processes through design, simultaneously with little impact on cost. The experiments of a project were performed by a company three years ago, based on a L27 orthogonal array. The design factors were inappropriately assigned into the orthogonal array, including one 2-level factor and twelve 3-level factors. The experiments were expensive and can not be repeated now. In this paper, these data are sorted and utilized to perform robustness analysis, such that we can find significant main effects and significant interactions. Linear graphs developed from the interaction table are used to assign interactions and main effects, to perform multiple analysis of variances (ANOVAs). Through that, the significant effects and interactions can be identified so that the statistical model can be developed for product performance prediction. For confidential reasons, the data are normalized (coded) by a certain mechanism.
 
 

To be competitive in the market, industries must deal with the world-wide challenges in producing high-quality goods at low costs. Over the past few decades, to accommodate these increasing global challenges, manufacturers have made a significant progress toward quality and productivity improvement. This progress can be reflected in today's market by the increasing quality levels and decreasing prices of merchandises, especially for high-tech and electronic products. These advancements have brought in exciting and interesting reforms in quality design and philosophy, one of which is the application of robust design [1-4]. As proposed by Taguchi, the objective of robust design is to make a product or process least sensitive to manufacturing variations, deterioration over time and environmental conditions [5-7]. It is a cost-effective way to improve quality, because it builds quality into products and processes through design, simultaneously with little impact on cost. Philosophically, robust design consists of three basic steps: (1) System Design, (2) Parameter Design, and (3) Tolerance Design [8,9].

The objective of System Design is to obtain a workable prototype model of the product. The parameters of parts as well as components, except general dimensions of products, are not determined in this step. Much of the previous and current effort in industries is concentrated on this step. In addition to the development of a workable model, in this step, we also select materials, subsystems or parts based on general strength requirements, functional requirements and economical effects. The environmental factors or considerations for material selection and life cycle analysis can be integrated into this step.

Parameter Design, which is the most important and effective step in Robust Design, focuses on parameter setting selection for design factors. In this step, engineers intend to design a robust product by selecting the optimal parameter settings of the design factors. The goal of robustness is achieved by selecting the best parameter levels rather than using expensive parts or components. The design can be performed by using orthogonal arrays, analysis of variance (ANOVA) [10,11], statistical models and optimization. The controllable design factors are arranged into a special orthogonal array named Inner Array with appropriate level assignments.

The uncontrollable noise factors are assigned to another special orthogonal array called Outer Array to simulate the random variations associated with the design factors, such as those caused by manufacturing variations, deterioration over time, etc. Some user-adjustable factors such as signal factors are also assigned in the outer array. For each combination (each row of the inner array) of the levels of the design factors, we can conduct the experiments or simulations under the conditions specified in the outer array and collect data. By performing ANOVA on the design factors, we can find which factor has a significant effect on the variations of system performance or quality characteristics of the product and which factor has an insignificant effect. The significant factors will stay in the statistical model and the insignificant factors will be removed from the model. The best or robust parameters setting of design factors will be selected based on the statistical model and optimization techniques. The parameter settings selected through this procedure can make the performance of the product insensitive to the undesired variations of the parameters of design factors, such as undesired but uncontrollable manufacturing variations due to mass production. For details, one can refer to [12,13].

Tolerance Design, at the expense of higher cost, is usually used to tighten the tolerances to reduce the variations in product’s performances. In this step, designers balance quality and cost to meet or exceed the requirement of the design specification. By integrating environmental factors, we can design tolerances or allowances that can lead to less societal loss. The tolerance design is performed, based on quality loss function [14], economic effects of upgrading components and parts or materials. Traditionally, the design is aimed at minimizing the quality loss by selecting the best tolerances setting. References [15,16] give systematic studies on the details of the tolerance design.
 

2. EXPERIMENTS AND APPROACH

The experiments of an industrial project were performed based on a L27 orthogonal array [17] three years ago by a company who had limited knowledge on interpreting the use of confounding techniques for fractional factorial design at that time. The design factors were inappropriately assigned into the orthogonal array, which may lead to confounding some important main effects and important interactions. This design includes one 2-level factor and twelve 3-level factors, leaving only one degree of freedom for errors. Since the experiments were expensive and can not be repeated now, to utilize these expensive data we will discuss how they can be sorted for robustness analysis such that significant main effects and significant interactions can be identified for the interest of the company. Several linear graphs are used to perform multiple analysis of variances (ANOVAs). Through that, we can identify the significant effects and interactions. Based on these significant effects and interactions, the statistical model can be developed for product performance prediction. For confidential reasons, the data are normalized (coded) by a certain mechanism.

While considering process or product improvement, quality control engineers are concerned with variation caused by uncontrollable factors. Experiments are therefore designed to test and identify those parameters in a process/product that are less sensitive to these uncontrollable factors. The controllable factors can be quantified, while the uncontrollable factors are difficult to quantify and are referred as noise factors. Since late 1950s Dr. Taguchi has introduced several new statistical tools and concepts of quality improvement that depend on the statistical theory for Design of Experiments. These methods of design optimization are also refereed to as Robust Design. To make a process robust, we seek to make the system less sensitive to external variations by designing the parameters to meet specifications.

The orthogonal array is an important tool for robust design – a table of integers whose column elements (1,2, and 3) represent the low, medium and high levels of the column factors. Examples of orthogonal arrays are L8, L9, L27 etc.; In this project, L27 is used. Due to the cost involved in running all the experimental data required in a given analysis, appropriate resolutions are chosen. Considering our case of 12 factors at three levels and 1 factor at 2 levels, a total of 1,594,323 experiments would be required. This is too costly and the benefits may not justify the costs involved. That is why tables of resolutions and orthogonal arrays are useful in solving such complex engineering analysis. Analyses of process/product parameters using Signal - Noise ratio so as to determine the factors less sensitive to noise. The method used in this approach is experimental parameter design using SNL for the system is optimized when the response is large i.e. Larger the better. The normalized test data are given in Table 1.
 
 
 

3. DATA ANALYSIS USING ANOVA

The first ANOVA is used to preliminarily screen all main effects with no interactions (Table 2). This was done with resources available at Morgan State University. Statgraphics Plus software is used. Since the errors have only one degree of freedom, the significant level (p-level) may not be very accurate. To obtain a better result, we pooled the terms with a F-ratio less than 1.00 into errors. Those terms include Factors B, G and J. The new ANOVA is given in Table 3
 
 

4. ANOVAs WITH INTERACTIONS

We have run the ANOVA table several times, each time with 7 considered main factors and different combinations of additional two insignificant factors. Once some important main factors have been identified, we are then interested in identifying possible significant interactions. For this case, only 2 factor interactions are considered, while assuming 3 and more factors interactions are believed to be insignificant and could be pooled to the error. Linear graphs are used to correctly match different combinations of significant effects i.e. assign factors to columns of the orthogonal array. Variables are assigned to the nodes first and using the table of interactions between any two columns, a line segment on the graph corresponds to an interaction between the nodes. We are trying to find the significant main effects and significant interactions, while it is not necessary to list all ANOVAs. As an example, one of the ANOVA design, with main factors M, E, L, D and G, as well as interactions MxE, MxL and MxD considered, is given below. The linear graph is given in Figure 1. The ANOVA table printed directly from Statgraphics is given by Table 4, together with the interaction plots in Figure 2. If the critical level of significance is 0.05, then main factors M, L, E and interaction MxL are significant.

Similarly, we can conduct a second ANOVA based on a linear graph as given in Figure 4. The ANOVA is printed directly using Statgraphics as Table 5. This ANOVA gives the significant factors: M, L, E as well as interaction MxL. To summarize the results from seven ANOVAs, we can develop the statistical model as given in next section.
 
 
 
 
 
 
 
 

D.O.F = 26 for (L27)

Main Factors: M, E, L, D, G 2*5 = 10

Interactions : MxE , MxL , MxD 4*3 = 12

Error 26-10-12 = 2
 
 

D.O.F = 26 for (L27)

Main Factors : A , E, D , M 2*4 = 8

Interactions : MxA , MxE , MxD 3*4 =12

Error : 26-8-12 = 6
 
 
 
 
 
 
 
 
 
 
 
 

D.O.F = 26 for (L27)

Main Factors : C, D ,K, M 2*4 = 8

Interactions : MxC , MxD , MxK 3*4 =12

Error : 26-8-12 = 6

 
 

D.O.F = 26 for (L27)

Main Factors : A , C , M ,H 2*4 = 8

Interactions : MxA , MxC , AxC 3*4 =12

Error : 26-8-12 = 6
 
 
 
 
 
 
 
 
 
 

D.O.F = 26 for (L27)

Main Factors : C, E ,L 2*2 + 1 = 5

Interactions : CxE , CxL ,ExL 4 + 2 + 2 =8

Error : 26-5-8 = 13
 
 
 
 
 
 
 
 
 
 

D.O.F = 26 for (L27)

Main Factors : C, E ,L 2*2 + 1 = 5

Interactions : CxE , CxL ,ExL 4 + 2 + 2 =8

Error : 26-5-8 = 13
 
 
 
 
 
 
 
 

Upon using the seven different experimental design analysis (seven ANOVAs), it is found that Factors: A, E, L, and M have significant effects on the responses. The significant interactions are AxM, and MxL. The statistical model relates the different significant factors and interactions can be expressed as:
 

Where:
 
 

The final statistical model can be expressed as:
 
 

Where subscripts stand for the corresponding factors at their respective levels. For instance, subscript "a" stands for Factor A at level "a" (a=1, 2, or 3), subscript "m" for Factor M at level "m" (m=1, 2 or 3), while "am" for Factor A at level "a" (a=1, 2, or 3) and Factor M at level "m" (m=1, 2 or 3). Bar signs stand for the average. Thus, yam.-Bar stands for Factor A at level "a" (a=1, 2, or 3) and Factor M at level "m" (m=1, 2 or 3). This statistical model can be used to predict the product performance and find the optimal parameter settings for design factors. The insignificant factors will be set at the most economic levels.

6. CONCLUSION

Robust design is a cost-effective method to find the best design parameter settings to make the performance of the product least sensitive to the noise factors (the undesired variations in manufacturing, deterioration over time and environmental conditions). For an inappropriately assigned design problem that may lead to the confounding of the significant factors and important interactions, we can still sort the significant main factors and the important interactions by multiple use of ANOVAs. Through that, we can find the statistical model for the product performance prediction. Thus, our ultimate goal is to find the optimal parameter settings for the design factors.
 

REFERENCES

1. Kackar, R. N., "Off-line Quality Control, Parameter Design, and the Taguchi Method," Journal of Quality Technology, Vol.17, No. 4, 1985.
2. Taguchi, G., Introduction to Quality Engineering, Unipub/Kraus International Publications, White Plains, NY and American Supplier Institute Inc., Dearborn, MI, 1986.
3. Taguchi, G., "Quality engineering (Taguchi Methods) for the Development of Electronic Circuit Technology," IEEE Transactions on Reliability, Vol. 44, No. 2, 1995.
4. Bullington, R. G. and Lovin, S., "Improvement of an Industrial Thermostat Using Designed Experiments," Journal of Quality Technology, Vol. 25, No. 4, 1993.
5. Coleman, D. and Montgomery, D., "A Systematic Approach to Planning for a Designed Industrial Experiment," (with Discussion), Technometrics, Vol. 35, No.1, 1993.
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7. Chen, G., "Integrating Environmental Factors into Robust Design to Minimize Societal Loss", the Proceedings of the 4th International Conference on Manufacturing Technology in Hong Kong, Hong Kong, Nov. 30 - Dec. 3, 1997.
8. Chen, G. and Kapur, K. C., "A Two-step Robust Design Procedure of Linear Dynamic Systems for Reducing Performance Variations," International Journal of Reliability, Quality and Safety Engineering, Vol. 4, No. 2, 1997.
9. Chen, G., "Environmentally Conscious Robust Design and Manufacturing for Sustainable Development," a Chapter in Massie Book Special Elements of Environmental Engineering, to be published.
10. Montgomery, D. C., Design and Analysis of Experiments, 4th ed., John Wiley & Sons, Inc., New York, NY, 1997.
11. Hicks, C. and Turner, K., Fundamental Concepts in the Design of Experiments, 5th ed., Oxford, 1999.
12. Chen, G. and Kapur, K. C., "Tolerance Design by Break-even Analysis for Reducing Variation and Cost," International Journal of Reliability, Quality and Safety Engineering, Vol. 1, No. 4, 1994.
13. Khattree, R., "Robust Parameter Design: a Response Surface Approach," Journal of Quality Technology, Vol. 28, No. 2, 1996.
14. Chen, G. and Kapur, K. C., "Quality evaluation system using loss function", Proceedings of 1989 International Industrial Engineering Conference, Toronto, Canada, May 1989.
15. Kapur, K. C. and Cho, B., "Economic Design of the Specification Region for Multiple Quality Characteristics," IIE Transactions, Vol. 28, No. 3, 1996.
16. Chen, G., "Tolerance Design Based on Variation Transfer Function," Proceedings of the 3rd IASTED International Conference on Reliability, Quality Control and Risk Assessment, Washington, DC, Oct 3-5, 1994.
17. Taguchi, G., System of Experimental Design, Vol. 1 & Vol. 2, UNIPUB/Kraus International Publications, 1987.
 
 
 

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