臺灣博碩士論文加值系統

本論文研究探討Xbar管制圖的經濟統計性設計，我們假設品質特性測量值(也就是觀察值)服從非常態分配且系統在製程管制內時間服從韋伯分配。當我們設計一個管制圖時，必須決定三個設計參數：樣本數n、抽樣間隔時間h及管制上下限的距離因素k，若使用經濟統計性設計，這三個參數必須滿足在型一與型二誤差限制下使得期望單位時間成本最低，其中成本函數是依據McWilliams成本模式計算得來的。

雖然文獻上不乏探討非常態品質特性測量值的研究，但是皆假設樣本平均數服從某非常態分配。然而樣本平均數的分配會受到樣本數的影響，在未計算出樣本數前，難以進行樣本平均數的機率模式化。因此我們假設樣本觀察值是獨立從一組Johnson分配抽樣出來。Johnson分配家族可涵蓋所有可能的偏度與峰度值，在機率模式化的應用上範圍很廣，然而缺點是型一與型二誤差必須藉由模擬實驗求得，所以這是一個隨機最佳化的問題。因此我們發展一套隨機最佳化的演算法來求解。

我們提出一種隸屬於隨機最佳化演算法的修正回溯彈性容忍法來求解包含一個離散和兩個連續決策變數X={n, h, k}的最佳化問題。測試問題的實驗結果告訴我們修正回溯彈性容忍法找出來的解是非常接近真正解。最後，我們執行非常態性和韋伯參數對最佳解{n, h, k}的影響之敏感度分析，結論如下所示。

當偏度值固定而且等於0，峰度值遞增的時候，會造成樣本數n遞增、抽樣間隔時間h遞減及管制上下限的距離因素k遞增。當偏度值固定而且不只大於0且接近0(也可以是遠離0)，峰度值遞增的時候，會造成n遞增、h遞減及k遞增。當峰度值固定而且接近常態峰度值(也可以是稍微遠離常態峰度值)，偏度值遞增的時候，會造成n遞減、h遞增及k遞減。當峰度值固定而且遠離常態峰度值，偏度值遞增的時候，會對n無顯著影響、h遞增及k遞減。當偏度值和峰度值都遞增的時候，會造成n、h、k都遞增。

韋伯的形狀和規模參數只有在形狀參數小於1時才會對決策變數中的抽樣間隔時間h敏感。當形狀參數遞增的時候，h也會遞增。

關鍵詞：經濟統計性設計；期望單位時間成本；修正回溯彈性容忍法；非常態性； Xbar管制圖。

This thesis considers the economical-statistical design of Xbar-control chart assuming that the quality characteristic measurement (i.e. observations) are nonnormal and the in-control time is Weibull. When designing a control chart, three parameters-the sample size n, time h between successive samples, and control-limit factor k-must be determined. In economic-statistical design, the three parameters are chosen so that the expected cost per hour is minimized under constraints on Type I and Type II error probabilities. The cost function is computed by the McWilliams cost model.

Although some nonnormal literature exists, the assumption is made on the distribution of sample average, which depends on the unknown sample size. We assume that the quality characteristic measurement are sampled independently from a Johnson distribution. The Johnson distribution is general in that it can be modeled to fit all possible values of the skewness and kurtosis. This is a stochastic optimization problem because Type I and Type II error probabilities is difficult to compute and need to be estimated via simulation experiment. Hence, a stochastic optimization algorithm is needed.

We propose an algorithm of stochastic optimization, revised retrospective optimization flexible tolerance method, to solve our optimization problem including one discrete and two continuous decision variable vector X ={n, h, k}. Empirical results show that the solution of Revised RO-FT is very close to the true optimum in our testing problems. Finally, we perform the sensitivity analysis of the nonnormality and Weibull effect on the optimal values of {n, h, k}. Conclusion of sensitivity analysis are as follows.

When skewness is constant and equal to zero, an increase on kurtosis leads to increases on sample size n, decreases on time h between successive samples, as well as a wider control-limit factor k. When skewness is constant and not only greater than but also close to zero (as well as far away zero), an increase on kurtosis leads to increases on n, decreases on h, as well as a wider k. When kurtosis is constant and close to normal (as well as a little far away normal), an increase on skewness leads to decreases on n, increases on h, as well as a narrower k. When kurtosis is constant and far away normal, an increase on skewness leads to no signidicant effect on n, increases on h, as well as a narrower k. When an increase both on skewness and kurtosis leads to increases on n and h, as well as a wider k.

Weibull shape parameter heta and scale parameter lambda are sensitive when shape parameter heta<=1 on the decision variable time h between successive samples. When an increase on shape parameter heta leads to increases on h.


Keywords: Economic-Statistical Design; Expected Cost Per Hour; Revised Retrospective Optimization Flexible Tolerance Method; Nonnormality; Xbar Control Chart.

LIST OF TABLES vii
LIST OF FIGURES x
ABSTRACT xi
I. INTRODUCTION 1
1.1 Problem Definition and Motivation 1
1.2 Expected Thesis Results 3
1.3 Organization of the Thesis 4
II. LITERATURE REVIEW 5
2.1 Economic Design of the Xbar Control Chart 5
2.1.1 Single Assignable Cause Models 5
2.1.2 Multiple Assignable Cause Models 11
2.1.3 Unified Approach 12
2.2 Statistical and Economic-Statistical Design of Xbar Control Chart 19
2.3 Assuming the Distribution of Quality Characteristic Measurement is Non-normal 19
2.4 Assuming the Distribution of the Process Failure Mechanism is Non-exponential 22
2.5 Economic Design of Dynamic Xbar Control Chart 26
III. THE ECONOMIC-STATISTICAL DESIGN OF XBAR CONTROL CHARTS ASSUMING NONNORMAL DATA AND WEIBULL IN-CONTROL TIME 27
3.1 Nonnormal Quality Characteristic Measurement: Johnson distribution 27
3.2 Cost Model and Constraints 31
IV. A RETROSPECTIVE OPTIMIZATION FLEXIBLE TOLERANCE METHOD 33
4.1 Optimization Problem 33
4.2 Simulation-Based Retrospective Optimization Method 36
4.2.1 The Sample-Path Approximation Approach of Problem P0 36
4.2.2 A General Retrospective Optimization Algorithm 36
4.3 Flexible-Tolerance Retrospective Optimization Algorithm 39
4.4 Empirical Results 47
4.4.1 Solution Quality of RO-FT Algorithm 47
4.4.2 Testing Problem 47
4.4.3 Result of Optimization Problem 49
V. THE REVISED RO-FT ALGORITHM AND SENSITIVITY ANALYSIS 53
5.1 The Revised RO-FT Algorithm 53
5.2 Empirical Results of Revised RO-FT Algorithm 57
5.2.1 Testing Problem 57
5.2.2 Result of Optimization Problem 60
5.3 Sensitivity Analysis 62
5.3.1 Sensitivity Analysis of Sample Size n 64
5.3.2 Sensitivity Analysis of Skewness alpha_3 69
5.3.3 Sensitivity Analysis of Kurtosis alpha_4 71
5.3.4 Sensitivity Analysis of Non-normality on the Optimal Design of Xbar Control Chart 72
5.3.5 Sensitivity Analysis of Weibull Shape and Scale Parameters on the Optimal Design of Xbar Control Chart 75
IV. SUMMARY, CONCLUSIONS, AND FUTURE RESEARCH 79
6.1 Summary of Results 79
6.2 Conclusions 80
6.3 Recommendations of Future Research 81
LIST OF REFERENCES 83
APPENDICES
Appendix A: The Flexible-Tolerance Method 86


LIST OF TABLES
4.1 The true optimal values of h^* and k^* for Examples 1 and 2 with n = 2, ..., 6 50
4.2 The RO-FT estimates of n^*, h^*, and k^* for Examples 1 and 2 (the numbers in parentheses are standard errors of the estimates) 50
4.3 The true optimal values of h^* and k^* for Example 1 with n = 2,..., 6 51
4.4 The RO-FT estimates of n^*, h^*, and k^* for Example 1 (the numbers in parentheses are standard errors of the estimates) 51
5.1 The true optimal values of h^* and k^* for Examples 1 and 2 with n = 2, ..., 6 59
5.2 The Revised RO-FT estimates of n^*, h^*, and k^* for Examples 1 and 2 (the numbers in parentheses are standard errors of the estimates) 59
5.3 The true optimal values of h^* and k^* for Example 1 with n = 2,..., 6 61
5.4 The Revised RO-FT estimates of n^*, h^*, and k^* for Example 1 (the numbers in parentheses are standard errors of the estimates) 61
5.5 The list of skewness, skewness squared, and kurtosis for Examples 2 to 7 61
5.6 Comparisons of the true optimum and Revised RO-FT solutions for Examples 1 to 4 63
5.7 Comparisons of the true optimum and Revised RO-FT solutions for Examples 5 to 7 63
5.8 A list of table numbers for sensitivity analysis on Examples 1 to 5 with skewness alpha_3 = 0 and kurtosis alpha_4 = 3, 1.87, 2.63, 4, and 4.51 64
5.9 The values of Type I error probability alpha and Type II error probability eta for the normal population and n = 5, 30, 200 65
5.10 The values of estimated Type I error probability widehat{alpha}, Type II error probability widehat{ eta}, and estimated mean hourly cost for a Johnson bounded distribution with skewness 0 and kurtosis 1.87, and n = 3, 4, 5, 10, 30, 50, 100, 200 65
5.11 The values of estimated Type I error probability widehat{alpha}, Type II error probability widehat{ eta}, and estimated mean hourly cost for a Johnson bounded distribution with skewness 0 and kurtosis 2.63, and n = 4, 5, 7, 10, 30, 50, 100, 200 67
5.12 The values of estimated Type I error probability widehat{alpha}, Type II error probability widehat{ eta}, and estimated mean hourly cost for a Johnson unbounded distribution with skewness 0 and kurtosis 4, and n = 4, 5, 7, 10, 30, 50, 100, 200 67
5.13 The values of estimated Type I error probability widehat{alpha}, Type II error probability widehat{ eta}, and estimated mean hourly cost for a Johnson unbounded distribution with skewness 0 and kurtosis 4.51, and n = 4, 5, 7, 10, 30, 50, 100, 200 68
5.14 The values of estimated Type I error probability widehat{alpha}, Type II error probability widehat{ eta}, and estimated mean hourly cost for Johnson bounded distributions with skewness = 0, 0.1, ..., 0.8 and kurtosis 1.87, and n = 5 69
5.15 The values of estimated Type I error probability widehat{alpha}, Type II error probability widehat{ eta}, and estimated mean hourly cost for Johnson unbounded distributions with skewness = 0, 0.1, ..., 0.9 and kurtosis 4.51, and n = 5 70
5.16 The values of estimated Type I error probability widehat{alpha}, Type II error probability widehat{ eta}, and estimated mean hourly cost for Johnson bounded distributions with skewness = 0 and kurtosis = 1.87, 2.63, 3, 4, 4.51, and n = 5 71
5.17 The sensitivity analysis of non-normality on the optimal values n^*, h^*, and k^* as well as estimated Type I and II error probabilities widehat{alpha} and widehat{ eta} 74
5.18 The sensitivity analysis of Weibull shape heta and scale lambda on the optimal values n^*, h^*, and k^*, as well as estimated Type I and II error probabilities widehat{alpha} and widehat{ eta}, for a Johnson bounded distribution with skewness 0 and kurtosis 2 76
5.19 The sensitivity analysis of Weibull shape heta and scale lambda on the optimal values n^*, h^*, and k^*, as well as estimated Type I and II error probabilities widehat{alpha} and widehat{ eta}, for a Johnson lognormal distribution with skewness 5 76
5.20 The sensitivity analysis of Weibull shape heta and scale lambda on the optimal values n^*, h^*, and k^*, as well as estimated Type I and II error probabilities widehat{alpha} and widehat{ eta}, for a Johnson unbounded distribution with skewness 0 and kurtosis 6 77


LIST OF FIGURES
2.1 Diagram of the quality cycle by Duncan 6
2.2 Diagram of the quality cycle by Lorenzen anf Vance 14
2.3 Plot of the Burr probability density function 21
3.1 Plot of the Johson S_B density function 29
3.2 Plot of the Johson S_U density function 30
5.1 Johnson system 73

[1]Arnold, B. F. and E. V. Collani.(1987). Economic Process Control. Statistica Neerlandica, 41, 89--97.
[2]Banerjee, P. K. and M. A. Rahim.(1988). Economic Design of Xbar Control Chart Under Weibull Shock Model. Technometrics 30, 407--414.
[3]Burr, I. W.(1942). Cumulative Frequency Functions. Annals of Mathematical Statistics 13, 215--232.
[4]Burr, I. W.(1967). The Effects of Non-normality on Constants for Xbar and R Charts. Industrial Quality Control 23, 563--568.
[5]Chen, H.(1994). Stochastic Root Finding in System Design. Ph.D. Dissertation, School of Industrial Engineering, Purdue University, West Lafayette, Indiana.
[6]Chiu, W. K. and G. B. Wetherill.(1974). A Simplified Scheme for the Economic Design of Xbar-Charts. Journal of Quality Technology 6, 63--69.
[7]Chou, C., C. Chen, and H. Liu.(2000). Economic-Statistical Design of Xbar Charts for Non-normal Data by Considering Quality Loss. Journal of Applied Statistics 27, 939--951.
[8]Collani, E. V.(1988). A Unified Approach to Optimal Process Control. Metrika 35, 145--159.
[9]Conte, S. D. , C. de Boor.(1980). Elementary Numerical Analysis: An Algorithmic Approach. McGraw-Hill, New York.
[10]DeBrota, D. J. , R. S. Dittus, J. J. Swain, S. D. Roberts, J.R. Wilson and S. Venkatraman.(1989).Modeling Input Processes with Johnson Distributions. Proceedings of the 1989 Winter Simulation Conference, ed. MacNair, E. A., Musselman, K. J., Heidelberger, P., 308--318. Institute of Electrical and Electronics Engineers, Piscataway, New Jersey.
[11]Duncan, A. J.(1956). The Economic Design of Xbar Charts Used to Maintain Current Control of a Process. Journal of the American Statistical Association 51, 228--242.
[12]Duncan, A. J.(1971). The Economic Design of Xbar Charts Where There Is a Multiplicity of Assignable Causes. Journal of the American Statistical Association 66, 107--121.
[13]Gibra, I. N.(1975). Recent Developments in Control Chart Techniques. Journal of Quality Technology 7, 183--192.
[14]Goel, A. L. , S. C. Jain and S. M. Wu.(1968). An Algorithm for the Determination of the Economic Design of Xbar-Charts Based on Duncan's Model. Journal of the American Statistical Association 63, 304--320.
[15]Hill, I. D. , R. Hill and R. L. Holder.(1976). Algorithm AS 99. Fitting Johnson Curves by Moments. Applied Statistics 25, 180--189.
[16]Ho, C. and K. E. Case.(1994). Economic Design of Control Charts: A Literature Review for 1981-1991. Journal of Quality Technology 26, 39--53.
[17]Hu, P. W.(1984). Economic Design of an Xbar Control Chart Under Non-Poisson Process Shift. Abstract, TIMS/ORSA Joint National Meeting, San Francisco May 14-16, 84.
[18]Hu, P. W.(1986). Economic Design of an Xbar Control Chart with Non-Exponential Times Between Process Shifts. IIE News, Quality Control and Reliability Engineering 21, 1--3.
[19]Jin, J.(1998). Simulation-Based Retrospective Optimization of Stochastic Systems. Ph.D. Dissertation, School of Industrial Engineering,
Purdue University, West Lafayette, Indiana.
[20]Johnson, N. L.(1949). Systems of Frequency Curves Generated by Methods of Translation. Biometrika 36, 149--176.
[21]Johnson, N. L. , S. Kotz , and N. Balakrishnan.(1994). Continuous Univariate Distributions, second edition. NY: John Wiley & sons.
[22]Lashkari, R. S. and M. A. Rahim.(1982). An Econmic Design of Cumulative Sum Charts to Control Non-Normal Process Means. Computers and Industrial Engineering 6, 118.
[23]Lorenzen, T. J. and L. C. Vance.(1986). The Economic Design of Control Charts: A Unified Approach. Technometrics 28, 3--10.
[24]McWilliams, T. P.(1989). Economic Control Chart Designs and the In-Control Time Distribution: A Sensitivity Study. Journal of Quality Technology 21, 103--110.
[25]McWilliams, T. P.(1994). Economic, Statistical, and Economic-Statistical Xbar Chart Designs. Journal of Quality Technology 26, 227--238.
[26]Montgomery, D. C.(1980). The Economic Design of Control Charts: A Review and Literature Survey. Journal of Quality Technology 12, 75--87.
[27]Montgomery, D. C. and W. H. Woodall.(1997). A Discussion on Statistically-Based Process Monitoring and Control. Journal of Quality Technology 29, 121--162.
[28]Parkhideh, B. and K. E. Case.(1989).The Economic Design of a Dynamic Xbar Control Chart. IIE Transactions 21, 313--323.
[29]Paviani, D. A. and D. M. Himmelblau.(1969). Constrained Nonlinear Optimization by Heuristic Programming. Operations Research 17, 872--882.
[30]Press, W. H. , S. A. Teukolsky., W. T. Vetterling., and B. P. Flannery.(1997). Numerical Recipes in C---The Art of Scientific Computing, second edition. New York, NY: Cambridge University Press.
[31]Rahim, M. A.(1985). Economic Model of Xbar-chart under Non-normality and Measurement Errors. Comput. & Ops. Res. 12, 291--299.
[32]Rahim, M. A.(1993). Economic Design of Xbar Control Charts Assuming Weibull In-control Times. Journal of Quality Technology 25, 296--305.
[33]Saniga, E. M.(1989). Economic Statistical of Control-Chart Designs with an Application to Xbar and R Charts. Technometrics 31, 313--320.
[34]Tagaras, G.(1989). Power Approximations in the Economic Design of Control Charts. Naval Research Logistic Quarterly 36, 639--654.
[35]Vance, L. C.(1983). A Bibliography of Statistical Quality Control Chart Techniques, 1970-1980. Journal of Quality Technology 15, 59--62.
[36]Woodall, W. H.(1985). The Statistical Design of Quality Control Charts. The Statistician 34, 155--160.
[37]Yourstone, S. A. and W. J. Zimmer.(1992). Non-Normality and the Design of Control charts for Averages. Decision Sciences 23, 1099--1113.