Optimum Failure                       Optimum Failure                      Truncated Testing                       Truncat...
ASQ Reliability Division                  ASQ Reliability Division                 English Webinar Series                 ...
OPTIMUM FAILURE TRUNCATEDTESTING STRATEGIES                            ASQ RD Webinar SeriesOctober 11, 2012Erik E. Kostan...
8/16/2012                           E.E. Kostandyan     2• Corporations use testing laboratories to determine  durability ...
8/16/2012                                                    E.E. Kostandyan            3Cumulative probability and probab...
8/16/2012                          E.E. Kostandyan           4Failure Truncated Testing StrategiesUnder Investigation in t...
8/16/2012                               E.E. Kostandyan          5Assumptions1.     The components under consideration are...
8/16/2012                       E.E. Kostandyan      6Modified Sudden Death Test (MSDT)• Pascual and Meeker (1998) first d...
8/16/2012                              E.E. Kostandyan          7    Example of MSDT (k 2, r=4 n=10)                    (k...
8/16/2012                        E.E. Kostandyan    8Classified Sudden Death Test (CSDT)• Randomly divide components into ...
8/16/2012                              E.E. Kostandyan   9                Example of CSDT (k=2, r=4, n=5)                E...
8/16/2012                                       E.E. Kostandyan              10    Run Time    R Ti                       ...
8/16/2012                                            E.E. Kostandyan               11                     Total Accumulate...
8/16/2012                             E.E. Kostandyan         12Costs:C t           c1 i unit t ti ti               1 is i...
8/16/2012                                     E.E. Kostandyan       13   Run Time as a Random Variable in CSDT• The PDF an...
8/16/2012                         E.E. Kostandyan    14A technique to find Run Time Quantile inCSDT from any py parent dis...
8/16/2012                                          E.E. Kostandyan                  15 Run Time as a Random Variable for t...
8/16/2012                                       E.E. Kostandyan       16Example 1: 10% and 90% quantiles for MSDT (k 5, r ...
8/16/2012                                E.E. Kostandyan        17Example 2: 10% and 90% quantiles for MSDT (k 5, r n=20) ...
8/16/2012                                     E.E. Kostandyan                  18       Total Accumulated Time as a Random...
8/16/2012                    E.E. Kostandyan    19Run Time Comparison – MSDT v.s. CSDTSuppositions:• Number of testing fac...
8/16/2012                         E.E. Kostandyan         20Run Time Comparison – MSDT v.s. CSDT (cont.)CSDT possible conf...
8/16/2012                        E.E. Kostandyan     21Run Time Comparison – MSDT v.s. CSDT (cont.)MSDT possible configura...
8/16/2012               E.E. Kostandyan   22            Run Time Comparison            MSDT v.s. CSDT (cont.)
8/16/2012               E.E. Kostandyan   23            Run Time Comparison            MSDT v.s. CSDT (cont.)
8/16/2012               E.E. Kostandyan   24            Run Time Comparison            MSDT v.s. CSDT (cont.)
8/16/2012               E.E. Kostandyan   25            Run Time Comparison            MSDT v.s. CSDT (cont.)
8/16/2012                         E.E. Kostandyan                 26            Run Time – MSDT v s CSDT                  ...
8/16/2012                     E.E. Kostandyan   27            Total Accumulated Time Comparison                      MSDT ...
8/16/2012                     E.E. Kostandyan   28            Total Accumulated Time Comparison                   MSDT v.s...
8/16/2012                     E.E. Kostandyan   29            Total Accumulated Time Comparison                   MSDT v.s...
8/16/2012                     E.E. Kostandyan   30            Total Accumulated Time Comparison                   MSDT v.s...
8/16/2012                        E.E. Kostandyan          31    Total Accumulated Time – MSDT v s CSDT                    ...
8/16/2012                          E.E. Kostandyan   32                Cost Comparisons1. Total Components Cost: CSDT < MS...
8/16/2012                  E.E. Kostandyan   33Conclusion: Test StrategiesIf shape parameter >> 1:CSDT with k = 1 & r = R ...
8/16/2012                     E.E. Kostandyan    34  Conclusion: Test Strategies (cont.)If shape parameter = 1:•Time: MSDT...
8/16/2012                                                   E.E. Kostandyan                          35    Theoretical Sum...
8/16/2012                                                             E.E. Kostandyan                           36    Exce...
8/16/2012                            E.E. Kostandyan   37   Thank You Very Much for You Time                            • ...
8/16/2012                                E.E. Kostandyan              38References• Arizono, I., & Kawamura, Y. (2008). Re...
8/16/2012                                     E.E. Kostandyan         39References (cont.)R f        (   t)• Downton, F. (...
8/16/2012                                         E.E. Kostandyan                40References (cont.)           (cont )• J...
8/16/2012                                      E.E. Kostandyan               41References (cont.)• Lamberson, L., & Kapur,...
8/16/2012                                E.E. Kostandyan            42References (cont.)• Pascual, F., & Meeker, W. (1998)...
8/16/2012                        E.E. Kostandyan      43Supporting Slides / Additional Topics• Discussion, Further Researc...
8/16/2012                               E.E. Kostandyan    44Discussion & Further ResearchReliability levels from MSDT and...
8/16/2012                                                                            E.E. Kostandyan                     4...
8/16/2012                          E.E. Kostandyan   46Sudden Death Test• Sadden Death test is a special case of MSDT,  wh...
8/16/2012                       E.E. Kostandyan        47            For 5 Sudden Death Tests would be:                   ...
8/16/2012                     E.E. Kostandyan    48  Sudden Death Test (cont.)• For a Weibull distributed failure time wit...
8/16/2012                         E.E. Kostandyan   49      What is the Misleading part in Sudden                   Death ...
8/16/2012                                  E.E. Kostandyan   50       What is the Misleading Part in Sudden               ...
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Optimum failure truncated testing strategies

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Companies use specialized testing laboratories to perform the tests, where the capacity of the test stations is limited. This is a major restriction for reliability tests. Different testing strategies will vary in cost and time, so an optimum strategy for the reliability test would be desirable, to obtain the list expensive and fastest results. In this presentation, findings and summary are presented for the optimum testing strategy determination, assuming components under consideration are mechanical and non-repairable.

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Optimum failure truncated testing strategies

  1. 1. Optimum Failure  Optimum Failure Truncated Testing  Truncated Testing g Strategies Erik Kostandyan ©2012 ASQ & Presentation Erik Presented live on Oct 11th, 2012http://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_liability Calendar/Webinars ‐_English/Webinars_‐_English.html
  2. 2. ASQ Reliability Division  ASQ Reliability Division English Webinar Series English Webinar Series One of the monthly webinars  One of the monthly webinars on topics of interest to  reliability engineers. To view recorded webinar (available to ASQ Reliability  Division members only) visit asq.org/reliability ) / To sign up for the free and available to anyone live  webinars visit reliabilitycalendar.org and select English  Webinars to find links to register for upcoming eventshttp://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_liability Calendar/Webinars ‐_English/Webinars_‐_English.html
  3. 3. OPTIMUM FAILURE TRUNCATEDTESTING STRATEGIES ASQ RD Webinar SeriesOctober 11, 2012Erik E. Kostandyan, PhDE ik E K t derik.kostandyan@gmail.com
  4. 4. 8/16/2012 E.E. Kostandyan 2• Corporations use testing laboratories to determine durability of their p y products. operation.• Testing is a time and money consuming operation• Diff Different testing strategies may be different in duration of t t ti t t i b diff t i d ti f the test or the cost of the test, or both.• Durability tests can be failure or time truncated.• Studying optimal failure truncated tests is the essence of this presentation.
  5. 5. 8/16/2012 E.E. Kostandyan 3Cumulative probability and probability distributionfunctions of the r-th out of n ordered random variable(overview)( i )t1 ; t2 ...tn p , y yp • independent, identically distributed and continuous type random variables with f(t) probability density and F(t) cumulative distribution functions (parent distribution).t1:n ≤ t2:n ≤ ... ≤ tn:n • In increasing order t1:n ; t2:n ...tn:n • Random variables would have Gr:n(t) cumulative distribution and gr:n(t) probability density functions n ⎛n⎞ Gr:n (t ) = P (tr:n ≤ t ) = ∑ ⎜ ⎟F (t ) w (1 − F (t )) n − w w= r ⎝ w ⎠ ⎛ n − 1⎞ g r:n (t ) = n ⎜ ⎟ F (t ) r −1 (1 − F (t )) n − r f (t ) ( ⎝ r − 1⎠
  6. 6. 8/16/2012 E.E. Kostandyan 4Failure Truncated Testing StrategiesUnder Investigation in this Presentation g• Modified Sudden Death Test (MSDT).• Classified Sudden Death Test (CSDT).• Where time-to-failure (parent distribution) is Weibull time to failure distributed with shape parameter greater than unity. Shape parameter β ⎛t⎞ −⎜ ⎟ ⎝θ ⎠ F (t ) = 1 − e Scale parameter
  7. 7. 8/16/2012 E.E. Kostandyan 5Assumptions1. The components under consideration are mechanical and non-repairable.2. The component life will be modeled by a two-parameter Weibull distribution, with a shape parameter greater than or equal to one.3. For th F the sake of the comparison of MSDT and CSDT th k f th i f d CSDT, the available testing facilities utilization and the number of test results collected by either test strategies will be the same. y g4. The predetermined number of failures for each Group is the same for either test strategy. strategy5.5 Testing facility capacity is higher or equal to the desired test results.
  8. 8. 8/16/2012 E.E. Kostandyan 6Modified Sudden Death Test (MSDT)• Pascual and Meeker (1998) first described the g gy testing strategy called a “Modified Sudden Death” test.• Randomly divide components into Groups • Once there are a predetermined number of failures (r ) in a Group - the test is terminates for that Group.• Groups are tested in a series mode.
  9. 9. 8/16/2012 E.E. Kostandyan 7 Example of MSDT (k 2, r=4 n=10) (k=2 r 4, n 10) Group 1 Group 2 10 Components, 4th F il n=10 C Failure n=10 Components, 4th Failure Componentsk-groups, r-failures with n- components in each GroupTotal Components Used (N ) = k*n=20Total Results Collected (R )=k*r=8
  10. 10. 8/16/2012 E.E. Kostandyan 8Classified Sudden Death Test (CSDT)• Randomly divide components into Groups. Groups – Once there are a predetermined number of failures (r ) in a Group, the test is terminates for that Group. p• Groups are tested in parallel mode.
  11. 11. 8/16/2012 E.E. Kostandyan 9 Example of CSDT (k=2, r=4, n=5) E l f (k 2 4 5) 10 Components 8th Failure Components,k-groups, r-failures with n- components in each GroupTotal Components Used (N ) = k*n=10Total Results Collected (R )=k*r=8
  12. 12. 8/16/2012 E.E. Kostandyan 10 Run Time R Ti •Time to finish a t t Ti t fi i h testParallel mode (CSDT) => Tp = max{tr:n ,1...tr:n ,i ...tr:n ,k } tr:n ,1 tr:n ,2 kSeries mode (MSDT) => Ts = ∑t i=1 r :n ,i tr:n ,1 tr:n ,2 ,k-groups, r-failures with n- components in each Group
  13. 13. 8/16/2012 E.E. Kostandyan 11 Total Accumulated Time •Summation of all failed and survived component’s time during the test. k ⎛ r ⎞Parallel mode (CSDT) => T test _ ac = ∑ ⎜ ∑ ti:n , j + (n − r ) * tr:n , j ⎟Series mode (MSDT) => j =1 ⎝ i =1 ⎠ tr:n ,1 tr:n ,2 t1:n ,1 tr:n ,1 t1:n ,2 tr:n ,2 k-groups, r-failures with n- components in each Group
  14. 14. 8/16/2012 E.E. Kostandyan 12Costs:C t c1 i unit t ti ti 1 is it testing time cost t c2 is the cost for a unit component c3 is supervision / technician cost per unit testing timeTotal Accumulated Time Cost: CTTC = c1 * Ttest _ acTotal CT t l Component Cost: tC t CC = c2 * NRun Time Cost: • Parallel mode (CSDT): CS = c3 * Tp • Series mode (MSDT): CS = c3 * Ts
  15. 15. 8/16/2012 E.E. Kostandyan 13 Run Time as a Random Variable in CSDT• The PDF and CDF of run time for the CSDT test strategy (E.E.Kostandyan, et al., 2010, Int. J. of Mod.&Sim.) M r:n ,k (t ) = ( Gr:n (t ) ) k CDF is: mr:n ,k (t ) = k * g r:n (t ) * ( Gr:n (t ) ) k −1 PDF is: mr:n ,k (t ) Recursion Or:n ,k (t ) If we Or:n ,k (t ) = k= define: M r:n ,k (t ) function exist: Or:n ,1 (t ) Gr:n(t) and gr:n(t) are CDF and PDF of r-th out of n order random variable. CDF – Cumulative Distribution Function PDF – Probability Density Function
  16. 16. 8/16/2012 E.E. Kostandyan 14A technique to find Run Time Quantile inCSDT from any py parent distribution pc = Q −1 r:n ((1 − α ) ) 1k −1 tc = F ( pc ) Qr:n - is a Beta distribution with parameters Alpha = r & Beta = n-r+1 F - is the parent failure distribution
  17. 17. 8/16/2012 E.E. Kostandyan 15 Run Time as a Random Variable for the MSDT• Monte Carlo Simulation• An approximation method to find a Run Time Quantile in MSDT, (Kostandyan E.E. (2010), Optimum Failure Truncated Testing Strategies) p( r ,n,k ) = H −1 ( r ,n ,k ) (1 − α ) b( r, n, k ) = F ( p( r ,n ,k ) ) −1 ⎛ ( k (n + 1) − 1) ⎞ fb ( r ,n ,k ) (t ) = kr ⎜ ⎟ F (t ) kr −1 (1 − F (t )) k ( n − r +1) −1 f (t ) ⎝ kr ⎠ where t ≥ 0 s ( r , n , k ) ≈ k * b ( r , n, k )H(r,n,k)H(r n k) - is a Beta distribution with parameters Alpha = r*k &Beta = (n-r+1)*k
  18. 18. 8/16/2012 E.E. Kostandyan 16Example 1: 10% and 90% quantiles for MSDT (k 5, r n=20) r=1:20 (k=5 r, n 20), r 1:20 parent distribution is Weibull (shape 5 ; scale 10,000)
  19. 19. 8/16/2012 E.E. Kostandyan 17Example 2: 10% and 90% quantiles for MSDT (k 5, r n=20) r=1:20 (k=5 r, n 20), r 1:20 parent distribution is Uniform [8,500 – 11,500]
  20. 20. 8/16/2012 E.E. Kostandyan 18 Total Accumulated Time as a Random Variable in MSDT and CSDT• If shape parameter = 1 θ r*k = R Ttest _ ac ~ χ 2 E (Tttestt _ ac ) = Rθ 2 rk 2 (E.E.Kostandyan, 2010, J. of Mang.& Eng. Integration) Var (Ttest _ ac ) = Rθ 2 • If shape parameter > 1, h t 1 Simulation was used to investigate the behavior of Total Accumulated Time in MSDT and CSDT CSDT.
  21. 21. 8/16/2012 E.E. Kostandyan 19Run Time Comparison – MSDT v.s. CSDTSuppositions:• Number of testing facilities available is 50 (N=50).• Number of results required is 20 (R=20).• The shape parameter was increasing by 0 2 0.2 increments, from 1 to 6.• The scale parameter was set at 75,000.
  22. 22. 8/16/2012 E.E. Kostandyan 20Run Time Comparison – MSDT v.s. CSDT (cont.)CSDT possible configurations Groups Component/ Results/ Total Total (k) group (n) group (r ) Results (R ) components 1 50 20 20 50 2 25 10 20 50 5 10 4 20 50 10 5 2 20 50
  23. 23. 8/16/2012 E.E. Kostandyan 21Run Time Comparison – MSDT v.s. CSDT (cont.)MSDT possible configurationsGroups Component/ Results/ Total Results Total (k) group ( ) (n) group ( ) (r (R ) components t 1 50 20 20 50 2 50 10 20 100 4 50 5 20 200 5 50 4 20 250 10 50 2 20 500 20 50 1 20 1000
  24. 24. 8/16/2012 E.E. Kostandyan 22 Run Time Comparison MSDT v.s. CSDT (cont.)
  25. 25. 8/16/2012 E.E. Kostandyan 23 Run Time Comparison MSDT v.s. CSDT (cont.)
  26. 26. 8/16/2012 E.E. Kostandyan 24 Run Time Comparison MSDT v.s. CSDT (cont.)
  27. 27. 8/16/2012 E.E. Kostandyan 25 Run Time Comparison MSDT v.s. CSDT (cont.)
  28. 28. 8/16/2012 E.E. Kostandyan 26 Run Time – MSDT v s CSDT v.s. Conclusions1- If shape parameter = 1:• Run Time CSDT > MSDT CSDT MSDT• Increasing k: Run Time in2- If shape parameter > 1:• Run Time CSDT < MSDT CSDT MSDT• Increasing k: Run Time in• Increasing k: Rate of Increase in CSDT MSDT3- If shape parameter > 1:• Run Time in CSDT is short when k = 1• MSDT with k = 1
  29. 29. 8/16/2012 E.E. Kostandyan 27 Total Accumulated Time Comparison MSDT v.s. CSDT
  30. 30. 8/16/2012 E.E. Kostandyan 28 Total Accumulated Time Comparison MSDT v.s. CSDT (cont.)
  31. 31. 8/16/2012 E.E. Kostandyan 29 Total Accumulated Time Comparison MSDT v.s. CSDT (cont.)
  32. 32. 8/16/2012 E.E. Kostandyan 30 Total Accumulated Time Comparison MSDT v.s. CSDT (cont.)
  33. 33. 8/16/2012 E.E. Kostandyan 31 Total Accumulated Time – MSDT v s CSDT v.s. Conclusions1- If shape parameter = 1:• Total Accumulated Time in CSDT = MSDT• Increasing k: Total Accumulated Time in CSDT = MSDT2- If shape parameter > 1:• Total Accumulated Time CSDT < MSDT• Increasing k: Total Accumulated Time in CSDT MSDT3- If shape parameter > 1:• Total Accumulated Time in all possible CSDT is the p same => with k = 1 might be chosen without an influence on the results• MSDT with k = 1
  34. 34. 8/16/2012 E.E. Kostandyan 32 Cost Comparisons1. Total Components Cost: CSDT < MSDT2. If shape parameter = 1: • Run Time Cost for CSDT > MSDT u e o CS S • Total Accumulated Time Cost for CSDT = MSDT3. If shape parameter > 1: • Run Time Cost for CSDT < MSDT • Total Accumulated Time Cost for CSDT < MSDT
  35. 35. 8/16/2012 E.E. Kostandyan 33Conclusion: Test StrategiesIf shape parameter >> 1:CSDT with k = 1 & r = R would be the best strategyThe same as MSDT with k=1If shape parameter > 1:• Run Time for CSDT < MSDT• Run Time Cost for CSDT < MSDT• T t l Components Cost for CSDT < MSDT Total C t C tf• Total Accumulated Time Cost for CSDT < MSDT
  36. 36. 8/16/2012 E.E. Kostandyan 34 Conclusion: Test Strategies (cont.)If shape parameter = 1:•Time: MSDT with k = R & r =1 would be the best strategygy•Cost: MSDT or CSDT: •Depends on Unit Component Cost ( 2) and p p (c Supervision / Technician Cost per unit testing time(c3) If shape parameter = 1: h t 1 • Run Time for CSDT > MSDT • Run Time Cost for CSDT > MSDT • Total Components Cost for CSDT < MSDT p • Total Accumulated Time Cost for CSDT = MSDT
  37. 37. 8/16/2012 E.E. Kostandyan 35 Theoretical Summary & Further Research MSDT Shape=1 Shape>1 # of Groups = 1 # of Groups > 1 # of Groups = 1 # of Groups > 1 Approximate Approximate method method proposed, proposed, Run Time Theory developed Theory developed Further research Further research available availableTotal Accumulated Further research Further research Time Theory developed Theory developed available available CSDT Shape=1 Shape>1 # of Groups = 1 # of Groups > 1 # of Groups = 1 # of Groups > 1 Run Time Theory developed Theory developed Theory developed Theory developedTotal Accumulated Further research Further research Time Theory developed Theory developed available available
  38. 38. 8/16/2012 E.E. Kostandyan 36 Excel Macro Input Cells, Changeable p g Column C Output Cells Column E Click the button to Calculate: Inputs Num of required results 6 Num of testing facilities available 12 Shape Parameter 2 Scale Parameter (hrs) 1,000 1st Quantile 10% 2nd Quantile 90% Supervision Cost /hr $ 5.0 Preliminary Estimated Data: Testing Time Cost /hr $ Component Cost /Unit $ 35.0 30.0 http://www.barringer1.com/wdbase.htm Outputs Test: 1 Test: 2 Test: 3 Test: 4 Num of Groups in Series 6 3 2 1 Num of Results per Group 1 2 3 6Num of Components per Group 12 12 12 12 Quantile 10% 90% 10% 90% 10% 90% 10% 90% Run Time (hrs) 1,121 1,964 874 1,492 740 1,277 583 1,008 Total Accumulated Time (hrs) 13,447 23,565 10,184 17,439 8,357 14,416 5,891 10,116 Run Time Cost $ 5,603 $ 9,819 $ 4,371 $ 7,462 $ 3,698 $ 6,386 $ 2,915 $ 5,038 Total Accumulated Time Cost $ 470,646 $ 824,758 $ 356,426 $ 610,356 $ 292,487 $ 504,571 $ 206,169 $ 354,052 Components Cost $ 2,160 $ 2,160 $ 1,080 $ 1,080 $ 720 $ 720 $ 360 $ 360 Total T t C t T t l Test Cost $ 478,409 478 409 $ 836 736 836,736 $ 361 876 $ 618 898 361,876 618,898 $ 296 905 $ 511 676 296,905 511,676 $ 209 445 $ 359 450 209,445 359,450
  39. 39. 8/16/2012 E.E. Kostandyan 37 Thank You Very Much for You Time • QuestionsErik E. Kostandyan, Ph DE ik E K t d Ph.D.Erik.Kostandyan@gmail.comPhD in Industrial EngineeringW tWestern Michigan U i Mi hi University, K l it Kalamazoo, Mi hi MichiganPost.Doc./PhD: Department of Civil Engineering,Aalborg University, Aalborg ,Denmark
  40. 40. 8/16/2012 E.E. Kostandyan 38References• Arizono, I., & Kawamura, Y. (2008). Reliability tests for weibull distribution with variational shape parameter based on sudden death data. Research, 189(2), 570. lifetime data European Journal of Operational Research 189(2) 570• Cohen, A. C. (1965). Maximum likelihood estimation in the weibull distribution based on complete and on censored samples samples. Technometrics, 7(4), 579-588.• Cousineau, D. (2009). Nearly unbiased estimators for the three- parameter weibull distribution with greater efficiency than the iterative likelihood method. The British Journal of Mathematical Statistical Psychology, 62(1), 167.• David H (1970) Order statistics. New York: Wiley. David, H. (1970). statistics Wiley• DeGroot, M. (1986). Probability and statistics. Reading Mass.: Addison-Wesley Pub. Co. Addison Wesley Pub Co
  41. 41. 8/16/2012 E.E. Kostandyan 39References (cont.)R f ( t)• Downton, F. (1966). Linear estimates of parameters in the extreme value distribution. Technometrics, 8(1), 3-17.• Estimation of weibull parameters from common percentiles.(2005). Journal of Applied Statistics, 32(1), 17.• Freudenthal, A. M., & Gumbel, E. J. (1953). On the statistical interpretation of fatigue test. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 216(1126), 309.• Haager P., Lundberg G., Palmgren A., Lee Eh., Spicacci A. (1949). Dynamic capacity of rolling bearings. Journal of Applied Mechanics, 16(4), 415.• Hirose, H. (1999). Bias correction for the maximum likelihood estimates in the two-parameter weibull distribution. Dielectrics and Electrical p ( ) Insulation, IEEE Transactions on, 6(1), 66-68.• Hogg, R., & Craig, A. (1970). Introduction to mathematical statistics. [New York]: M Y k] Macmillan. ill
  42. 42. 8/16/2012 E.E. Kostandyan 40References (cont.) (cont )• Johnson, L. (1964). The statistical treatment of fatigue experiments. New York: Elsevier Publishing Company. Company• Johnson, L. (1964). Theory and technique of variation research. New York: Elsevier Publishing Company.• Jong-Wuu Wu, Tzong-Ru Tsai, & Liang-Yuh Ouyang. (2001). Limited failure- censored life test for the weibull distribution. Reliability, IEEE Transactions on, 50(1), 107 111. 107-111.• Jun, C., & Balamurali, S. (2006). Variables sampling plans for Weibull distributed lifetimes under sudden death testing. IEEE Transactions on Reliability, 55(1), 53.• Kostandyan E.E., (2010). A Simulation Study for a Modified Sudden Death Test. Journal of Management and Engineering Integration, pp. 41-46, Vol. 3, Issue 1.• Kostandyan E.E., Lamberson L., Houshyar A., 2010, Time to Failure for a ‘k Parallel r-out-of-n’ System, International Journal of Modelling and Simulation, ACTA Press, pp. 479-482, Vol. 30, Issue 4.• Kostandyan E.E. (2010). Optimum Failure Truncated Testing Strategies. Monograph print ISBN: 978-3-8433-5593-3, LAP Lambert Academic Publishing AG & Co. KG, Germany.
  43. 43. 8/16/2012 E.E. Kostandyan 41References (cont.)• Lamberson, L., & Kapur, K. (1977). Reliability in engineering design. New York: Wiley.• Lawless, J. (1982). Statistical models and methods for lifetime data. New York: Wiley.• Lieblein J. (1955). On moments of order statistics from the weibull distribution Lieblein, J (1955) distribution. The Annals of Mathematical Statistics, 26(2), 330.• Lundberg G., P. A. (1949). Dynamic capacity of rolling bearings. Journal of Applied Mechanics, 16(2) 165. Mechanics 16(2), 165• Meeker, W., & Escobar, L. (1998). Statistical methods for reliability data. New York: Wiley.• Motyka, R. (2007). Sudden death testing versus traditional censored life testing. A monte-carlo study. Control and Cybernetics, 36(1), 241.• Nelson W. (1982). Applied life data analysis Hoboken: John Wiley Sons, Inc. Nelson, W (1982) analysis. Sons Inc• Niewiadomska-Bugaj, M., & Bartoszynski, R. (1996). Probability and statistical inference. New York: Wiley.
  44. 44. 8/16/2012 E.E. Kostandyan 42References (cont.)• Pascual, F., & Meeker, W. (1998). The modified sudden death test: Planning life tests with a limited number of test positions. Journal of Evaluation, 26(5), 434. Testing and Evaluation 26(5) 434• Rockette, H., Antle, C., & Klimko, L. A. (1974). Maximum likelihood estimation with the weibull model Journal of the American Statistical model. Association, 69(345), 246-249.• Vlcek, B. (2004). Monte carlo simulation of sudden death bearing testing. Tribology Transactions, 47(2), 188.• Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), 293.
  45. 45. 8/16/2012 E.E. Kostandyan 43Supporting Slides / Additional Topics• Discussion, Further Research• MLEs from Weibull distribution in “k” Groups with k censored data• Sudden Death Test – history logic and misleading points history,
  46. 46. 8/16/2012 E.E. Kostandyan 44Discussion & Further ResearchReliability levels from MSDT and CSDT• Parameters Estimates: • Linear estimates (Downton, 1966) • MLE (Cohen 1965) (Cohen, • From small sample sizes, MLE has biases • Cousineau suggested weights from complete Weibull data so MLE data, estimates are nearly unbiased (Cousineau, 2009)
  47. 47. 8/16/2012 E.E. Kostandyan 45MLEs from Weibull distribution in “k” kGroups with censored data p For Groups = 1, MLEs are r r 1∑ Ln( x ) ∑ x β Ln ( xi ) + ( n − r ) xr Ln ( xr ) β ⎛ r ⎞β ∑ xiβ + (n − r ) xrβ i i 1i =1 = i =1 − ⎜ ⎟ β r $ θ = ⎜ i =1 ⎟ ∑ x β + (n − r ) x β r i =1 i r ⎝ r ⎠ For Groups = k, MLEs are Kostandyan E.E. (2010), Optimum Failure Truncated Testing Strategies. rk rk 1∑ Ln( x ) ∑ x β Ln ( xi ) + k (n − r ) xr Ln ( xr ) β ⎛ rk ⎞β ⎜∑ xiβ + k (n − r ) xrβ i ii =1 = i =1 − 1 ⎟ rk β $ θ = ⎜ i =1 ⎟ ∑ rk xiβ + k (n − r ) xrβ ⎝ rk ⎠ i =1
  48. 48. 8/16/2012 E.E. Kostandyan 46Sudden Death Test• Sadden Death test is a special case of MSDT, where r=1 (1st failure out of n)• Leonard G. Johnson showed that the specimens characteristic life is equal to the Sudden Death characteristic life times k 1/ β For example, if the assembly consist of 8 specimens, k 8, k=8, then each 1st failure out of 8 is B8 3 of the 8.3 population. Let say we have 5 test results:
  49. 49. 8/16/2012 E.E. Kostandyan 47 For 5 Sudden Death Tests would be: Probability Plot of x Weibull W ib ll 99 90 80 70 60 50 40 30Percent t 20P 10 5 3 2 1 1 10
  50. 50. 8/16/2012 E.E. Kostandyan 48 Sudden Death Test (cont.)• For a Weibull distributed failure time with the scale parameter equal to unity the median time to fail 10 unity, specimens out of 20 is about 23.86% of the median time required t f il 10 out of 10 ti i d to fail t f 10.• If the scale parameter is equal to 2, then it is about 48.85%, 48 85% This means that running more specimens than one intends to fail, reduces testing time: fail This is True
  51. 51. 8/16/2012 E.E. Kostandyan 49 What is the Misleading part in Sudden Death Test• Running more components in a Group than one intends to fail, reduces testing time (run time)• This is True t >T t T
  52. 52. 8/16/2012 E.E. Kostandyan 50 What is the Misleading Part in Sudden Death Test (cont.) ( )• Running k Groups serially, does it reduce testing time (run time) ti ) ?• This is not always True• As it was shown, it depends on the shape parameter, see run time section above T > t1; T > t2 ; T > t3 but MSDT(k=3,r=1,n=5) T ? t1 + t2 + t3 t1 CSDT(k=1,r=3,n=5) CSDT(k=1 r=3 n=5) <= > MSDT (k=1,r=3,n=5) t2 T t3

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