Studies on a New Class of Shock Models for Deteriorating Systems

1. Studies on a New Class of Shock Models for Deteriorating Systems Prof. Alagar RANGAN Ayşe Tansu TUNÇBİLEK Department of Industrial Engineering, Eastern Mediterranean University, North Cyprus

2. Outline • Introduction • - Shock Models • The Stochastic Model • The Statistical Characteristics • Specific Models and Discussions • Conjecture • Simulation and Estimation • Optimal Replacement Model for A Non-Repairable System

3. Outline • Optimal Replacement Model for A Repairable System • Optimal Replacement Model for A Non- Repairable System and Spare with Lead Time • Optimal Replacement Model for A Non- Repairable System with Emergency Replacement • Optimal Replacement Model for A Non-Repairable System and Spare with no Lead Time • Applications

4. Introduction • All real world systems are deteriorating in nature. Majority of such models aproach the problem through shock models. Abdel Hameed (1986), Feldman (1976), Shanthikumar (1983) • A system is subject to randomly occuring shocks, each of which adds a nonnegative random quantity to the accumulated damage process.

5. Cumulative damage threshold value series of randomly occurring shocks repair time Y X time Lethal shock working time Shock Models • The term shock refers any perturbation to the system. The system fails when accumulated damages crosses a threshold value

6. Yeh and Zhang (2004) in a refreshing departure introduced a new class of shock models and called them - shock models. • Earlier shock models concentrated solely on the magnitude of the damage caused by the shocks, Yeh and Zhang’s model paid attention to the frequency of the shocks.

7. 0 T Lethal shock -Shock Models • A system is likely to fail if two successive shocks occur within a short time, whereas it may not fail if they are separated by a longer duration. As an example • Think of an elastic material which is subjected to stretching. • If the second stretching occurs before the material recovers from the first stretching, the material breaks.

8. The Stochastic Model • This study attempts a comprehensive analysis of - shock models in which the shock counting process is generalized to a renewal process, and • Threshold times are considered as random variables. • Apart from deriving explicitly various statistical characteristics of the model we analyze optimal replacement problems of such a system.

9. Notations Used • Z: Random variable denoting the time between two successive shocks with pdf . • D: Random variable denoting the threshold value with pdf . • W: Random variable denoting time between two successive failures with pdf . • Counting variable denoting the number of failures in . • . • Laplace Transform of the density function f.

10. I.The Model Assumptions Assumption 1 The system is subject to shocks. The time between shocks, are assumed to be independently and identically distributed.

11. Cycle 1 Cycle 2 Cycle N Fails Fails > < Time between two shocks >nonlethal shock Time between two shocks <lethal shock

12. Assumption 3 • Threshold value is a random variable. Assumption 4 • The shock arrival times and the threshold value times are independent of each other.

13. The Statistical Characteristics • Probability density of W can be represented as the sum of a random number of random variables, • : a sequence of independently and identically distributed random variables, which are distributed as Z but conditional on Z>D. • : random variabledistributed as Z but conditional on Z≤D. It is assumed to be independent of the sequence ’s. time between two successive failures

14. N has the geometric distribution where • Define the conditional distributions of and as and

15. And taking Laplace Transform of ; • could alternatively be derived by writing the integral equation

16. t 0 0 t In the remaining interval of length there is no failure The first shock itself occurs only after t The first shock occurs at some instant The threshold time starting from t=0 is over by time

17. With simple differentiation; • Application of Laplace Transform we get the • Abate, J and Whitt W (1995),”Numerical Inversion of Laplace Transforms of Probability Distributions”, Informs Journal on Computing, 7, 36-43.

18. The Mean and Variance are obtained after simplifications;

19. Specific Models and Discussions • When the system is subjected to the same kind of shock each time, the threshold time of the system is likely to remain a constant, a case discussed by Yeh (2004,2006). • Under such a scenario, we have considered a few models for different shock arrival distributions: • Exponential • Gamma • Uniform

20. Conjecture • For constant threshold times, amongst all the nondegenerate shock arrival distributions with common mean, exponential shock arrivals lead to the minimum mean failure. • Amongst all the shock arrival distributions with common mean, the degenerate threshold time distribution leads to the minimum mean failure.

21. Scale parameter Shape parameter • We have fitted a Power Law Process to our model by • estimating and from our simulated failure times. • Bayesian Estimation to estimate and for various • apriori distributions. II. Simulation and Estimation • Moment Estimators • Deteriorating Systems have been widely modeled using Power Law Process which is inhomogeneous Poisson Process with intensity function • Hypothesis Testing

22. C(T) 2C C T III. Optimal Replacement Model for ANon-Repairable System • The total cost of running the system increase in operating cost rate per unit with successive shocks Cost of periodic replacement Cost of operating the system/unit time Period of replacement of the system

23. The long run expected cost of operating the system per unit time; • The optimal value of the periodic replacement time always exists and is given by the unique solution of the integral equation where is the expected number of failure in .

24. All shock models in the literature have minimized the expected cost per unit time to obtain the optimal T*. • By controlling the second order characteristics of the failure process, the performance criterion of the system is controlled, so cost minimization emerges by considering variance of the number of shocks in (0,T).

25. be the optimal period for replacement using the cost function • We have obtained optimal and for vaious cases and compared the costs, and .

26. > IV. Optimal Replacement Model for A Repairable System Cycle 1 Cycle 2 Cycle N Fails Fails Fails 0 Replace the item repair time time for the first failure During the repair time, system is shut and arriving shocks have no effect on the system

27. For our model we have derived C(N), and the Optimal N* is obtained Expected successive repair times,form a geometric process with rate b Repair cost rate Replacement cost Expected Random Replacement time Expected operating time of the system following the (n-1)th repair in a cycle • We have derived optimal N* and the conditions which N* uniquely exists.

28. V. Optimal Replacement for A Non-Repairable System and Spare with Lead Time • This study considers a generalized age-replacement policy of a system subject to shocks with random lead time first considered of Sheu and Chien, 2004. Model: 1. In the age-replacement policy, planned (sheduled) replacements occurs whenever an operating unit reaches age T and a spare unit is available.

29. 2. System failure is governed by our model stated in I so that the distribution function of faliure time is W(t). 3. A spare unit for replacement can be delivered upon order and the Lead time L has distribution function H(t).

30. The time between system replacements form a cycle. The length of a cycle is calculated using the following 4 mutually exclusive cases. Case 1: T L<T<TF L TF 0 T If the ordered spare arrives before time T and no failures occurs before T, then delivered unit is put into stock and unit is replaced by that spare at age T, at a cost

31. Case 2: L L<T<TF 0 T TF L If the ordered spare arrives after time T and no failures occurs before the arrival of the ordered spare unit, then the unit is replaced by a spare as soon as the spare is delivered at a cost

32. Case 3: TF L<TF<T 0 TF L T If the ordered spare arrives before failure which occurs before the time T, then the delivered unit is put into stock and the unit replaced by the spare upon the failure at a cost

33. Case 4: L TF<L 0 TF L If failure occurs before the arrival of the ordered spare, then the system is shut downand replaced by the spare as soon as the spare is delivered at a cost -The cost rate for stocking is cs -The cost rate resulting from system down is cp

34. E[cost/cycle] E[cost/unit time]= E[ length of the cycle] Let and lead time density failure time density

35. We have established the following: • is monotonically increasing and

36. VI.Optimal Replacement for ANonRepairable System with Emergency Replacement • In system V we considered the case when the system could be shut down due to nonavailability of the spare. However systems such as power distribution cannot be shut down. • When the spare is unavailable, we replace the system on failure using emergency replacement. We assume the cost of emergency replacement to be CE>> Cp. In this case the expected cost rate is,

37. A similar optimality analysis has been carried out

38. VII. Non-Repairable System and Spare with no Lead Time Failure Unscheduled cost c1 T 0 Scheduled cost c2 0 T and

39. Optimality Conditions • If , is a decreasing function,then . • If , Let The optimal is given as the solution of If has no solution then Further if is increasing which is a natural assumption, it has been shown that is an increasing function, establishing the uniqueness of .

40. Optimality Conditions • The condition for the existence of a finite optimal ,

41. Application Areas The developed model has several interesting applications in diverse areas of which we will mention a few: i. Neurophysiology ii. Stochastic Clearing Systems iii. Fatigue Failure of Material iv. Renewing Warranty Modeling

42. References 13. Blischke W.R. and Murthy D.N.P., (2000) StrategicWarranty Management: A Life Cycle Approach, IEEE Transactions on Engineering Management, 47, 40-54. 14. Blischke W.R. and Scheuer E.M., (1998) Calculation of the cost of warranty policies as a function of estimated life distribution, Naval Research Logistics Quarterly, 22 (4), 681-696. 15. Nguyen D.G. and Murthy D.N.P., (1984) Cost Analysis of Warranty Policies, Naval Research Logistic Quarterly, 31, 525-541. 16. Lam Yeh, Peggo Kwok W. and Lam (2001) An extended warranty policy with option open to consumers, European Journal of Operational Research 131, 514-529. • Alagar Rangan, Dimple Thyagarajan and Sarada Y. (2006) Optimal replacement of systems subject to shocks and random threshold failure, Interantional Journal of Quality and Reliability Management, Vol 23, 9, 1176-1191. • Lam Yeh and Yuan Lin Zhang (2004) A shock model for the maintenance problem of a repairable system, Computersand Operations Research, 31, 1807-1820. • Abate, J and Whitt W (1995). Numerical Inversion of Laplace Transforms of Probability Distributions, Informs Journal on Computing, 7, 36-43. • Abate, J and Whitt W (2006). A Unified Framework for Numerically Inverting Laplace Transforms, Informs Journal on Computing, 18, 408-421. 1.Block H.W., Borges W.S. and Savits T.H., (1985) Age dependent Minimal Repair, Journal of Applied Probability, 22, 370-385. • Brown M., Proschan F., (1983) Imperfect Repair, Journal of Applied Probability, 20, 851-859. • Kijima M., (1989) Some results for repairable systems with General Repair, Journal of Applied Probability, 26, 89-102. 4. Abdel Hameed M., (1986) Optimal replacement of systems subject to shocks, Journal of Applied Probability, 23, 107-114. 5.Feldman R.M., (1976) Optimal replacement with semi-Markov shock models, Journal of Applied Probability, 13, 108-117. 6. Shanthikumar J.G. and Sumita U. (1983) General shock model associated with correlated renewal sequences, Journal of Applied Probability, 20, 600-614. 7. Nakagawa T., (1976) On a replacement problem of a cumulative damage model, Operational Research, 27, No 4, 895-900. 8. Kijima M., Morimura H. and Suzuki Y., (1988) Periodical replacement problem without assuming minimal repair, European Journal of Operational Research, 37, 194-203. 9. Kijima M. and Nakagawa T., (1991) A cumulative damage shock model with imperfect preventive maintenance, Naval Research Logistics, 38, 145-156. 10.Stadje W., (1991) Optimal stopping in a cumulative damagemodel, Operations Research Spectrum, 13, 31-35. 11. Barlow R.E. and Hunter L., (1960) Optimum Preventive Maintenance Policies, Operations Research, 8, 90-100. • Lam Yeh and Yuan Lin Zhang, (2003) A Geometric –Process maintenance Model for a deteriorating system under a Random Environment, IEEE Transactions on Reliability, 52, 83-89. • Blischke W.R. and Murthy D.N.P., (1994) Warranty Cost Analysis, Marcel Dekker, New York.

43. “Some Results on a New Class of Shock Models” Asia-Pacific Journal of Operational Research (APJOR) (SCI Expanded)Accepted • “A new shock model for systems subject to random threshold failure”,A.Rangan, A.Tansu, International Journal of Computer and Information Science and Engineering 2;3,Summer 2008,p.203-208.

44. Thank You