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Joint statistical design of double sampling X-bar and s charts

This study presents the joint statistical design of X-bar and s charts using a genetic algorithm, showing improved efficiency compared to other control chart schemes. The optimization problem is solved, and the performance of the joint charts is evaluated.

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Joint statistical design of double sampling X-bar and s charts

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  1. Joint statistical design of double sampling X-bar and s charts 指導教授： 童超塵 老師 作者：David He *, Arsen Grigoryan 主講人：廖乃毅

  2. Contents • Introduction • The joint DS X-bar and s charts • Formulation of joint statistical design of the DS X-bar and s charts • Solving the optimization problem using genetic algorithm • Performance of the joint DS X-bar and s charts • Conclusions

  3. Introduction-Abstract • The statistical design of the joint DS X-bar and s charts is defined and formulated as an optimization problem and solved using a genetic algorithm. • the joint DS X-bar and s charts offer a better statistical efficiency in terms of ARL than combined EWMA and CUSUM schemes, omnibus EWMA scheme over certain shift ranges. • In comparison with the STD, TSS and VSS X-bar and R charts, the joint DS charts offer a better statistical efficiency for all ranges of the shifts.

  4. Introduction-EWMA(EEu) • What’s EEu? -EEU was obtained by running a two-sided EWMA mean chart and a high-side EWMA variance chart simultaneously. -two-sided EWMA mean chart： against sample number t for t=1, 2, . . . (Crowder, 1987a,b, 1989; Lucas and Saccuci, 1990) -high-side EWMA variance chart： against sample number t for t=1, 2, . . . • (Crowder andHamilton, 1992) the sample variance.

  5. Introduction-EWMA(EE) • What’s EE? -EE consists of a two-sided EWMA mean chart and a two-sided EWMA variance chart. -two-sided EWMA variance chart： against sample number t for t=1, 2, . . .. and (When process variance is equal to the target variance )

  6. Introduction-combined two-sided CUSUM(CC) • For the mean chart： St= and Tt= against sample number t • For the variance chart： Vt= and Kt= against sample number t

  7. Introduction-omnibus EWMA • The omnibus EWMA： for i=1,2…,where 0<λ<1 選擇上式中的一些α，其主要是當σ≧ σ0時，展現局　 　　部的敏感度，以及在分散中增加敏感度。而當σ≦σ0 時探索均數的小偏移是較有效的。

  8. Introduction-STD X-bar and R charts • X-bar control chart： warning limits: action limits : where 0<w<k • R control chart： warning limits: action limits : where wR(ni) and kR(ni)是標準差相關範圍的數目(R/σ)。

  9. Introduction-twos-tage sampling (TSS) • samples of size n0 are taken from the process at regular time intervals. • If one item’s X value of the sample is close to the target ,then the sampling is interrupted. Otherwise the sampling goes on to the second stage. • the X-bar and R values are computed based on the whole sample size n0.

  10. Introduction-DS X-bar & DS S chart • DS X-bar chart was developed to improve the statistical efficiency (in terms of ARL) without increased sampling, or alternatively, to reduce the sampling without reducing the statistical efficiency. Daudin (1992) and He and Grigoryan (2002, 2003) • the DS s charts result in a significant reduction in average sample sizewithout decreasing the out-of-control ARL in comparison withthe traditional s charts. He and Grigoryan (2002, 2003)

  11. The joint DS X-bar and s charts 取n2得Y-bar 取n2得S12

  12. Formulation of joint statistical design of the DS X-bar and s charts(一) the out-of-control ARL of the joint DS X-bar and s charts. 發出警報的機率 發出警報的機率

  13. Formulation of joint statistical design of the DS X-bar and s charts(二) • 解釋上述最佳模式： -the probability of taking the 2’nd sample: -取n2的集合: -P(A∪B∪C)= -α=在製程均數及變異數沒有偏移時，卻下在管制外的結論。 -β=在製程均數及變異數有偏移時，卻下在管制內的結論。

  14. Formulation of joint statistical design of the DS X-bar and s charts(三) • Note that since ARL1= the optimization model (1)–(4) becomes: where Pa1 ,Pa2 ,Pa1s,Pa2s :第一、二階層在管制內的機率。

  15. Solving the optimization problem using genetic algorithm(一) • optimization problem formulated by model (5)–(9) is characterized by mixed continuous-discrete variables, and discontinuous and non-convex solution space. • The operation of the genetic algorithm : (a) create a random initial solution; (b) evaluate fitness, i.e., the objective function that min ARL;(c) reproduction and mutation; (d) generate new solutions. • GA find a global optimum solution with a high probability.

  16. Solving the optimization problem using genetic algorithm(二) • Crossover is made up in hope that new chromosomes will have good parts of old chromosomes and maybe the new chromosomes will be better. However it is good to leave some part of population survive to next generation. • Mutation is made to prevent the search falling into local extremes, but it should not occur very often, because then GA will in fact change to random search. • In this paper the population size=1000, crossover probability=0.5 and mutation probability=0.06.

  17. Performance of the joint DS X-bar and s charts • Comparison with combined EWMA chart, combined CUSUM chart, and omnibus EWMA chart : - The ARL was calculated using computer simulation.(10000 independence runs) - Joint DS-1 was optimized for detecting the shift with δ=0.169 and λ=1.188. - All tabulated data for joint DS X-bar and s charts were confirmed by using Monte Carlo simulation with MATLAB

  18. Performance of the joint DS X-bar and s charts for detecting the shift with δ=0.169 and λ=1.188

  19. Performance of the joint DS X-bar and s charts Joint DS-1較佳 Joint DS-1較佳

  20. Performance of the joint DS X-bar and s charts • in process mean with δ≧0.75 and shifts in process standard deviation with λ≧1.3 the joint DS scheme is better. • the combined EWMA and CUSUM outperform the DS for shifts with δ≦ 0.5 and λ ≦1.2. • EWMA,CUSUM在反應偏移上可能有延遲出現，這種延遲稱為”inertia problem”(慣性問題)。因此EWMA,CUSUM在探索小偏移勝過DS的優點必須忽略掉此問題(inertia problem)才可。

  21. Performance of the joint DS X-bar and s charts • Comparison with joint STD, VSS, and TSS charts : -The design parameters of all the schemes were chosen such that the in-control ARL = 433, and the average sample size=5 -Joint DS-2 was optimized for detecting the shift with δ=0.5 and λ=1.1. -Joint DS-3 was optimized for detecting the shift with δ=0.75 and λ=1.5. • joint DS X-bar and s charts result in a better statistical performance than the rest of the charts.

  22. Performance of the joint DS X-bar and s charts

  23. Performance of the joint DS X-bar and s charts

  24. Conclusions • The results of the comparison with the combined EWMA and CUSUM, the omnibus EWMA show that in process mean with δ≧0.75 and shifts in process standard deviation with λ≧1.3 the joint DS scheme is better. • In comparison with the joint STD, TSS and VSS X and R charts,the results show the proposed joint DS X-bar and s chart scheme outperforms these schemes for all shifts in process mean with 0<δ≦1.0 and shifts in process standard deviation with 1.0<λ≦2.0.

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