1. Economic Design of Charts Under Weibull Shock Models TECHNOMETIRCS, November 1988, VOL. 30, NO.4 P.K. Banerjee and M.A. Rahim 組員： 914011 陳致翔 914015 張泰宏 914016 張裕鳳 914024 胡政宏

2. 導 論 Introduction Notations and preliminary results Main results Numerical illustration Conclusion

3. Introduction Design of control chart  sample size  sample frequency or interval between samples  control limit for the chart Why use economic design of control charts ？

4. 目的 First proposed  “The economic design of - control charts to control normal process means.” -- Duncan (1965) The control limits : Sample size ： n （ Taken from the process every h hours. ） The process mean may shift from to The design parameters of control charts are n, L and h. The objective is to determine these parameters to minimize the expected total cost per unit time The objective is to determine these parameters to minimize the expected total cost per unit time

5. 背景 Duncan (1956) Gibra (1975), Montgomery (1980), and Vance (1983) Baker (1971), Heikes. Montgomery, and Yeung (1974) Montgomery and Heikes (1976) Hu (1984) Parkhideh, Behrooz; Case, Kenneth E. (1989) Chung, Kun-Jen, Lin, Chuan-Neng (1993)

6. Assumptions 1.The time that the process remains in the in-control state follows a Weibull distribution, its pdf is given by 2.Drawing random samples of size n at times h 1, ( h 1 + h 2 ), ( h 1 + h 2 + h 3 ) ……………... Duncan’s model assume h j =h for all j ( j=1,2,…. ). 3.The time to sample and chart one item is negligible. 4. Production ceases during the searches and repair.

7. 5.Make the following proposition ： h j are defined so as to keep the probability of a shift in an interval, given no shift up to its start constant for all intervals. This can be achieved by defining the length of the sampling intervals h j ( j=1,2,……) in the following fashion ： h j =[ j 1/k -(j-1) 1/k ]h 1 Note that h j satisfies the basic requirements; that is. (a) and (b) Further, for all j when k=1 Assumptions

8. Define  p j ( j = 1, 2, … ) ： The conditional probability that the unit used in the system will fail during the sampling interval j, given that it was in the operating state at the beginning of the interval j  q j ： The probability that the unit will fail during the sampling interval j Definitions

9. The expected duration of the in-control period within the sampling interval j, given that the shock occurred during this sampling interval

10. The expectation for the time in control during a sampling interval is defined as the weighted average of ’s with q j ’s as the respective weights

11. 兩種誤差 shift alarm Yes No Yes No

12. Notation as following n : sample size : the length of the jth sampling interval (j=1,2,…; =0) : expect search time associated with false alarm : expect time to discover assignable cause : expect time to repair process

13. The Expected Residual Times State E(residual cycle length) Probability Out of control and alarm Out of control but no alarm In control and no alarm In control and false alarm

14. State1. Out of control and alarm sampling 12 3 j-1 0 shift τ j …… alarm A cycle time Residual time

15. State2. Out of control but no alarm sampling 12 3 j-1 0 shift τ j …… alarm A cycle time Residual time

16. State3. In control and no alarm sampling 12 3 j-1 0 j ……

17. State4. In control and false alarm sampling 12 3 j-1 0 j …… alarm A cycle time

18. The Expected Residual Times State E(residual cycle length) Probability Out of control and alarm Out of control but no alarm In control and no alarm In control and false alarm

19. Expected length of the production cycle

20. Notation as following a : fixed sample cost b : cost per unit sampled L : control limit coefficient Y : cost per false alarm L : control limit coefficient W : cost to locate and repair the assignable cause : quality cost per hour while in control : quality cost per hour while out of control

21. The Expected Residual Costs State E(cost during the current period) E(residual cost) Out of control and alarm Out of control but no alarm In control and no alarm In control and false alarm

22. State1. Out of control and alarm sampling 12 3 j-1 0 shift τ j …… alarm A cycle time Residual time

23. State2. Out of control but no alarm sampling 12 3 j-1 0 shift τ j …… alarm A cycle time Residual time

24. State3. In control and no alarm sampling 12 3 j-1 0 j ……

25. State4. In control and false alarm sampling 12 3 j-1 0 j …… alarm A cycle time

26. The Expected Residual Costs State E(cost during the current period) E(residual cost) Out of control and alarm Out of control but no alarm In control and no alarm In control and false alarm

27. Expected cost per unit of time

28. Numerical lustration 條件式

29. Z0Z0 找出錯誤警報的平均時間 0.25 hr Z1Z1 找出問題的平均時間 0.25 hr Z2Z2 修復製程的平均時間 0.75 hr a 固定抽樣成本 $20 b 檢驗一樣本的成本 $4.22 D0D0 製程 in control 時每小時的製造成本 $50 D1D1 製程 out control 時每小時的製造成本 $950 Y 錯誤警報時所需成本 $500 W 發生問題時所需的修復成本 $1100 製程偏移係數 0.5 製程 in control 時錯誤警報機率 L 與 的函數 製程 out control 沒有發出警報的機率

30. Example 1 當 抽樣時間為 Non-Uniform 時的最佳參數值如下 n23 h1h1 10.04 L1.56 0.118 0.2019 E(C) / E(T)$231.3

31. Example 2 當 Uniform 抽樣時間與 Non-Uniform 抽樣時間最大差異 UniformNon-Uniform

32. 當 抽樣時間為 Uniform 時的最佳參數值如下 n26 h2.43 hr L1.54 0.1228 0.1571 E(C) / E(T)$259.85

33. Rahim V.S Duncan Different Assumptions  Rahim’s assumption – Weibull shock model  Duncan’s assumption – Exponential shock model 、 Uniform Sampling 調整指數分配參數使得兩種模型的平均數相同可得 代入剛才所得之 E(C) 、 E(T) 中，可求出最佳參數值如下 n26 h2.36 hr L1.56 0.1185 0.1614 E(C) / E(T)$261.18 與 Uniform Sampling 中 Weibull 模型 並無明顯差異。

34. 小結 In Rahim’s Weibull shock model Various Sampling 所得的每小時平均花費 顯著的小於 Uniform Sampling 中的平均花費 在 Rahim’s Weibull shock model 與 Duncan’s exponential shock model 中，且採用 Uniform Sampling 抽樣方式，則 每小時平均花費兩者並無顯著差異

35. Sensitivity of the design and expected cost 何謂敏感性 ?  keep the mean time to failure the same  Change one of the parameter values at a time

36. TABLE1

37. TABLE2 TABLE3

38. Cost of Misspecification in Weibull parameters No misspecification: (j=1,2,….)are all equal Misspecification : E(T) 與 E(C) 變得很困難去求得 小結論： The decision variable (n, L) are not sensitive to a moderate degree of misspecification.

39. Conclusion Extensive computational experience indicates that the proposed non-uniform sampling scheme yields a lower cost than that of Hu(1984)uniform sampling scheme. A fixed-sampling-interval control rule is widely used in practice, mainly because of its administrative simplicity.

40. Extensive computational studies suggest that the lengths of the intervals can be rounded off to more convenient lengths. for example,when k is close to 2, one can redefine =, = /2, = /3,and = for j=4,5,… 優點 : 比 uniform sampling 有經濟利益 缺點 : 增加 the cost per hour

41. 研究與發展 在韋伯分配下，其他管制圖之經濟設計  Rahim MA, Costa AFB (2000) – Xbar and R charts  Yang SF, Rahim MA (2000) -- Xbar and S charts 時間序列模型  Ohta H, Kimura A, Rahim A (2002) – Time-varying control chart parameter 動態模型  Parkhideh and Case (1989) – Six decision variables in design methodology  Ohta H, Rahim MA (1997) – Reduce to three decision variables

42. References

43. THE END