臺灣博碩士論文加值系統

一般而言，產品的製造過程常與數個品質特性有關，隨著產品發展的複雜性與多樣性，品質最佳化必須同時評估數個反應變數的績效。在反應曲面法中，針對線上實務的問題是讓多個反應變數達到製程最佳化，而解決多反應值最佳化的基本方法為整合目標函數，同時各個反應值必須以數學方法結合形成單一目標函數，單一目標函數的建立需要考慮品質控制的限制範圍。本篇研究主要針對望大、望小與望目不同的品質特性，提供慾望方程式的設計概念以呈現多反應值的模式，並且應用最陡上升方向在慾望方程式的搜尋方法，而在實務問題應用上可針對不同品質特性的績效作調整，而慾望方程式中的函數轉換在本質上是屬於高度非線性。為了達到即時線上製程最佳化，本文中結合了最陡上升法與一階泰勒展開方程式來逼近非線性慾望方程式。最後，以最陡上升梯度方向搜尋法處理線上多反應值最佳化的問題，同時透過一個中央合成設計的問題來說明模擬的結果希望藉由此方法對於參數最佳化的調整有更進一步助益並且適用於更複雜的工業製程。

In practice, a manufactured product is often evaluated by several response variables simultaneously. One of the most practical problems in response surface methodology is how to proceed with on-line process optimization entailing more than single response variable. A common approach to solving multi-response optimization problem is a unifying objective function approach; that is, the individual responses are mathematically combined to form a single objective function. Unifying objective approaches are employed in the quality control area to optimize several responses at the same time. In this thesis, it is proposed to use the desirability function approach to perform the multi-response modeling, and then apply the method of steepest ascent direction to the resulting desirability function. Yet, the resultant desirability function is highly nonlinear in nature due to functional transformation. To facilitate on-line process optimization using the method of steepest ascent, the first-order Taylor series expansion will be employed to approximate the nonlinear desirability function. At last, the method of steepest ascent proceeds to conduct on-line multi-response process optimization. The proposed procedure for on-line process improvement is illustrated through a numerical example taken from the literature.

TABLE OF CONTENTS
摘要 i
ABSTRACT ii
TABLE OF CONTENTS iii
LIST OF FIGURES AND TABLES v
Chapter
1. INTRODUCTION 1
1.1 Introduction 1
1.2 Motivation 4
1.3 Description of the problem 5
1.4 Research Goal and Objectives 6
1.5 Organization of Thesis 6
2. LITERATURE REVIEW 9
2.1 Response Surface Analysis 9
2.2 The Method of Steepest Ascent in Process Improvement 11
2.3 Multiple Response Optimization 13
2.4 Desirability Function Approach 16
3. A MODIFIED STEEPEST ASCENT APPROACH TO PERFORM
MULTI-RESPONSE PROCESS IMPROVEMENT 20
3.1 Desirability Function Revisited 20
3.2 Determining the Direction of Steepest Ascent on Desirability Function Using
Taylor Approximation 24
3.3 Determining the Step Size 26
3.4 An Illustrative Example 29
4. INITIAL RESULTS AND FURTHER RESEARCH 32
4.1 Initial Simulation Optimization Study 32
4.2 Sensitivity Analysis by Changing Step Sizes 37
4.3 Directions for Further Research 39
4.4 Analysis of Fitting Second-order Response Surface Model 41
4.5 Another Case Study 64
5. CONCLUSION AND FUTURE RESEARCH 68
5.1 Conclusion 68
5.2 Future Research 68
REFERENCES 70

LIST OF FIGURES AND TABLES
Figure
1. Generic model of an engineering process 3
2. Organization of thesis 8
3. First order response surface and path of steepest ascent 13
4. Desirability function of larger-the-better (LTB) 22
5. Desirability function of small-the-better (STB) 23
6. Desirability function of nominal-the-best (NTB) 23
7. The modified steepest ascent algorithm 29
8. Computational results of the proposed procedure using the step size of 0.1 35
9. Computational results of the proposed procedure using the step size of 0.2 35
10. Computational results of the proposed procedure using the step size of 0.3 36
11. Computational results of the proposed procedure using the step size of 0.5 36
12. Computational results of the proposed procedure using the step size of 0.7 37
13. Computational results of the proposed procedure using the step size of 0.8 37
14. Euclidean Distance by Computing in the Standard Error*10% Case 45
15. Comparing A with B in the Standard Error*10% Case 46
16. the Value of in the Standard Error*10% Case 46
17. Euclidean Distance by Computing in the Standard Error*30% Case 49
18. Comparing A with B in the Standard Error*30% Case 50
19. the Value of in the Standard Error*30% Case 50
20. Euclidean Distance by Computing in the Standard Error*50% Case 53
21. Comparing A with B in the Standard Error*50% Case 54
22. the Value of in the Standard Error*50% Case 54
23. Euclidean Distance by Computing in the Standard Error*70% Case 57
24. Comparing A with B in the Standard Error*70% Case 58
25. the Value of in the Standard Error*70% Case 58
26. Euclidean Distance by Computing in the Standard Error*100% Case 61
27. Comparing A with B in the Standard Error*100% Case 62
28. the Value of in the Standard Error*100% Case 62
29. Simulation Results by Computing Euclidean Distance in the five Different Standard Errors. 63
30. The Averages of Euclidean Distance in the five Different Standard Errors. 63

Table
1. Experimental Design 31
2. Deterministic Optimization Results by Using and for the Derringer and
Suich’s Example 33
3. Deterministic Optimization Results by Using the Design-Expert Software Package for the Derringer and Suich’s Example 33
4. Changing the Initial Points of Simulation results in the Original Case 34
5. Best Computational Results Obtained by Using the Proposed Procedure for the Derringer and Suich’s (1980) Example 34
6. Experiment Design with Linear model 40
7. Best Computational Results Obtained by Using Linear Model for the Derringer and Suich’s (1980) Example 40
8. Changing the Initial Points in Linear Model 41
9. Simulation results by Using Standard Error*10% 42
10. Changing the Initial Points of Simulation results by Using
Standard Error*10% 45
11. Simulation results by Using Standard Error*30% 46
12. Changing the Initial Points of Simulation results by Using
Standard Error*30% 49
13. Simulation results by Using Standard Error*50% 51
14. Changing the Initial Points of Simulation results by Using
Standard Error*50% 53
15. Simulation results by Using Standard Error*70% 55
16. Changing the Initial Points of Simulation results by Using
Standard Error*70% 57
17. Simulation results by Using Standard Error*100% 59
18. Changing the Initial Points of Simulation results by Using
Standard Error*100% 61
19. Central Composite Design with Three Responses 65
20. Deterministic Optimization Results by Using for the Montgomery’s (2001) Example 66
21. Deterministic Optimization Results by Using the Design-Expert Software Package for the Montgomery’s (2001) Example 66
22. Best Computational Results Obtained by Using the Proposed Procedure for the Montgomery’s (2001) Example 67

Ch’ng C. K., Quah S. H., and Low H. C., 2005, A New Approach for Multi-Response Optimization. Quality Engineering, 17: 621-626.

Derringer, G. and Suich, R., 1980, Simultaneous Optimization of Several Response Variables. Journal of Quality Technology, 12, 214-219.

Harrington, E. C. JR., The Desirability Function. Industrial Quality Control, Vol. 21, No. 10, 1965, pp. 494-498.

Antony J., 2000, Multi-response Optimization in Industrial Experiments Using Taguchi’s Quality Loss Function and Principal Component Analysis. Quality and Reliability Engineering International, 16: 3-8.

Khuri, A. and Conlon, M., 1981, Simultaneous Optimization of Multiple Responses Represented by Polynomial Regression Functions. Technometrics, 23, 363-375.

Kim, K. J and Dennis, K. J. Lin, 1999, Simultaneous Optimization of Mechanical Properties of Steel by Maximizing Exponential Desirability functions. Applied Statistics, 49, Part 3, 311-325.

Kros, J. F. and Mastrangelo, C. M., 2001, Comparing Methods for The Multi-response Design Problem. Quality and Reliability Engineering International, 17: 323-331.

Kros, J. F. and Mastrangelo, C. M., 2004, Comparing Multi-Response Design Methods with Mixed Responses. Quality and Reliability Engineering International, 20: 527-539.

Vining, G., 1998, A Compromise Approach to Multi-response Optimization. Journal of Quality Technology, 30(3), 309-313.
Wurl, RC and Albin, SL.,1999, A Comparison of Multi-response Optimization: Sensitivity to Parameter Selection. Quality Engineering, 1999; 11(3):405-415.