The Economic Design of Integrated Model for Fuzzy Weibull Distribution control charts

1. The Economic Design of Integrated Model for Fuzzy Weibull Distribution control charts presented by Mr. PramoteCharongrattanasakul Advisor : Assoc.Prof. AdisakPongpullponsak

2. Presentation sequence • Introduction • Objectives • Scopes • Methodology • Numerical Example • Conclusion and Discussion

3. Introduction Lorenzen and Vance (1986) proposed a general method for determining the economic design of control charts. Alexander et al. (1995) combined Duncan’s cost model with the Taguchi loss function

4. Lindermanand Kathleen E. McKone-Sweet (2005) developed a generalized analytical model to determine the optimal policy by coordinating Statistical Process Control and Planned Maintenance to minimize total expected cost. Fig.1 Three monitoring - maintenance scenarios.

5. Wen-Hui Zhou and Gui-Long Zhu (2008) extendedto 4 Scenarios Fig. 2 Four monitoring - maintenance scenarios.

6. Karpisek , Stepanek and Jurak (2010) applied Fuzzy Logic with WeibullDistribution for reliability problem Hellendoorn and Thomas (2010) described 7 method for defuzzification fuzzy output function (membership function)

7. In this work, we develop the integrated for Fuzzy Weibull Distribution economic model by Control Chart. By optimizing these parameters, the fuzzy total cost per hour is expectedly minimized using genetic algorithm. The Centroid Method was used to find the answer of mean total hourly cost.

8. Objectives of research • To develop Fuzzy Weibull Distributed for economic model under integrated model by control chart. • Find four optimal variables by genetic algorithm. • Find the minimum fuzzy total hourly costs by genetic algorithm. 4. Find the answer of mean total hourly cost by Centroid Method. 5. To compare six model (Fuzzy) and six model (No Fuzzy)

9. Scope of research • Model assumptions : • The process characteristic monitored by the control chart follows a normal distribution with mean and standard deviation . • In the start of the process, the process is assumed to be in the in-control state; that is . • The process mean may be shifted to the out- of-control region; that is, . 4. Each scenario is independent.

10. Fuzzy Weibull Distributed model fuzzy random variable are the fuzzy number and where is the observed value of a crisp random variable and is a so-call vagueness coefficient. The vagueness coefficient is triangular fuzzy number with the main value and membership function Fig. 3 triangular fuzzy number. where , and boundary values they are given by an expert’s estimate. Fig. 3shows graph of .

11. Defuzzification Centroid method: This procedure (also called center of area or center of gravity) is the most prevalent and physically appealing of all the defuzzification methods (Sugeno, 1985; Lee, 1990); it is given by the algebraic expression 1

12. Methodology In this study Process Failure Mechanism that follow a Fuzzy Weibull Distribution(Karpisek, Stepanek and Jurak (2010)) are considered. As following, 2 3 where cuts of Fuzzy distribution function

13. Cycle Time : The expected time searching for a false alarm. : The expected time to identify maintenance requirement and to performa Planned Maintenance. : The expected time to determine occurrence of assignable causes. : The expected time to identify maintenance requirement and to perform a Reactive Maintenance. : The expected time to perform a Compensatory Maintenance.

14. : The mean elapse time from the last sample before the assignable cause to the occurrence of the assignable cause. : The average runs length during in- control period. : The average runs length during out-of- control period. : The expected time to sample and chart one item. : The indicator variable (If it equals 1 production continuous during Planned Maintenance (Reactive Maintenance, Compensatory Maintenance, validate assignable cause) or 0 otherwise).

15. : The probability that run length of control chart equals during in-control period. : The probability that run length of control chart equals during out-of- control period.

16. Cycle Costs : The cost of quality loss per unit time (the process is in an in-control state) often estimated by a Taguchi Loss function. : The cost of quality loss per unit time (the process is in an out-of-control state) often estimated by a Taguchi Loss function. : The cost of performing Planned Maintenance. : The cost of performing Reactive Maintenance. : The cost of performing Compensatory Maintenance.

17. : The fixed cost of sampling. : The variable cost of sampling. :The cost to investigate a false alarm.

18. Optimal Variable : The sample size (for optimal) : The interval between sampling ( for optimal) : The number of sample taken before Planned Maintenance( for optimal) : The width of control limit in units of standard deviation ( for optimal)

19. Scenario 1 (S1) the process in-control alert signal Fig. 4 Integrated model of Scenario 1. 4 5

20. Scenario 2 (S2) consider the process out-of-control alert signal 6 Fig. 5 Integrated model of Scenario 2. Since S2 assumes that the process shifts to an “out-of-control” state prior to the Planned Maintenance and process failure mechanism follows a Fuzzy Weibulldistribution, the in-control time follows a Fuzzy Weibulldistribution.

21. 7 8

22. Scenario 3 (S3) consider the process in-control chart no signal 9 10 Fig. 6 Integrated model of Scenario 3.

23. Scenario 4 (S4) consider the process out-of-control chart no signal 11 12 Fig. 7 Integrated model of Scenario 4.

24. Expected Hourly Cost (Lorenzenand Vance (1986)) where 13 14 15

25. Probability of scenario 1 16 17 Probability of scenario 2

26. Probability of scenario 3 18 19 Probability of scenario 4

27. Genetic Algorithm The solution procedure is carried out using genetic algorithms (GA) with MATLAB 7.6.0(R2008a) software to obtain the optimal values of that minimize The solution procedure for our example using the GA by MATLAB :

28. Numerical Example Input parameters in Economic model Table 1 Economic model input parameters

29. Table 2 The optimal four variables and total hourly cost in each (Lower, Upper)

30. Fig. 8Membership function of total hourly cost.

31. Optimal value for four variables and optimal value total hourly costs (mean) Table 3 Optimal value for four variables and hourly cost in Economic model

32. Conclusion

33. Integrated model by control chart (Fuzzy) Setup input parameters Genetic Algorithm Optimal Solution Create Membership function Fuzzy method Find the answer of mean total hourly cost Centroid Method

34. Discussion

35. Reference Lorenzen, T.J., and Vance, L.C.., 1986, “The economic design of control charts: a unified approach”, Technometrics, Vol.28, pp.3-10. Alexander, S.M., Dillman, J.S. Usher, M.A., Damodaran, B., 1995, “Economic design of control charts using the Taguchi loss function”, Computers and Industrial Engineering, Vol.28, pp.671–679. Linderman, K., K.E. McKone-Sweet, J.C., 2005, “An integrated systems approach to process control and maintenance”, European Journal of Operational Research, Vol.164, pp.324–340.

36. Wen-HuiZhou, Gui-Long Zhu, 2008, “Economic design of integrated model of control chart and maintenance”, Mathematical and Computer Modeling, Vol.47, pp.1389–1395. Karpisek, Z., Stepanek, P. and Jurak, P., 2010, "WEIBULL FUZZY PROBABILITY DISTRIBUTION FOR RELIABILITY OF CONCRETE STRUCTURES", Engineering MECHANICS, Vol. 17, pp. 363–372. Hellendoorn H., and Thomas C., 1993, “Defuzzification in fuzzy controllers”, J. Intell. Fuzzy Syst.,Vol. 1, pp. 109–123.

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