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Warranty Prediction for Products with Random Stresses and Usages

Huairui Guo, PhD, ReliaSoft Corporation Arai Monteforte, ReliaSoft Corporation Adamantios Mettas, ReliaSoft Corporation Doug Ogden, ReliaSoft Corporation Key Words: warranty prediction, random stresses, random usages, stress-strength model, Monte Carlo simulation SUMMARY & CONCLUSIONS the effect of the random stresses on the product life can be quantitatively estimated. Making accurate warranty predictions is challenging. It In the following sections, the theory of life-stress becomes even more challenging when products are operated relationships in accelerated testing is discussed first. Then under random stresses and random usages. Traditional method methods for predicting warranty returns for two different only uses the average values of these random variables for situations are provided. In the first situation, the probability of warranty prediction. The randomness of the variables is failure during the warranty period is affected only by random ignored, which may result in inaccurate results. This paper stresses. In the second situation, the failure is a function of presents methods to solve this issue. Solutions for two both random stresses and random customer usages. different situations are discussed. In the first situation, only random stresses are the major concern. In the second situation, 2 THEORY ON ACCELERATED TESTING AND LIFE- both random stresses and random customer usages are STRESS RELATIONSHIPS considered. Two analytical solutions, the exact and the The life-stress relationship function explains how stresses approximated solution, are provided for each case. The affect product life. In this function, life is represented by a comparison shows that, although they are simple to use, approximated solutions can be very different from the exact percentile of the failure distribution, t p . In general, the results. In this paper, not only the mean value of the warranty function can be written in a log-linear form:     prediction, but also the variances and intervals of the t  e 0 1S1 1S2 ... (1) prediction are calculated. This is much better because interval p estimate provides more information than a simple point Si is the stress or a transformation of the stress. If we assume estimate. The proposed methods can be applied to many there is only one stress, based on different transformations, the industries such as electronic, automobile and home appliance life stress relationship in equation (1) can be [2]: companies. I B L  S Lt p Ae Arrhenius (2) J 1 INTRODUCTION 1 Lt  Inverse Power Law Warranty prediction is one of the most important issues in KL p K 4 S n reliability engineering. In the prediction of warranty returns, The percentile is selected according to the life characteristic of operating stresses and customer usage data must be accounted different life time distributions. Some typical life for [1]. Usually, nominal or average values of the operating characteristics are presented in Table 1. stresses are used. However, stresses are often not constant. Instead, they are random and can be described using pdf (Probability Density Life Distribution distributions. For example, not every user accumulates 12,000 Function) Characteristic miles per year on a vehicle, nor does every user print the same  C S 1 D t T Scale C t S D T number of pages per week on a printer. Therefore, using a Weibull D T e E U Parameter D T single constant value for a random variable in the calculation E U ( ) is not appropriate. The randomness of the stresses and usage  Mean Life Exponential  t must be considered. By considering the randomness, a e (1/  ) confidence interval rather than a single value of the warranty C  S2 1D ln(t) T 1   return can be calculated. In this paper, methods for obtaining Lognormal e 2E U Median ( e ) the confidence interval will be provided. t 2 To accurately predict the warranty returns, the life-stress Table 1 – Typical Life Characteristics relationship needs to be established first, mainly through accelerated tests. Once the life-stress relationship is obtained, The life-stress relationship can be integrated into a lifetime

1-4244-2509-9/09/$20.00 ©2009 IEEE distribution. For example, assume there are two independent Equation (5) is an approximation of equation (6). This is from stresses. The life-stress relationship for one stress is Arrhenius the Taylor series expansion. According to the Taylor series and for the other stress it is the inverse power law. The expansion, the value of probability of failure can be combined life-stress relationship is: approximated by: B B  1 F(t0 , S1, S2 ) F(t0 , S1, S2 ) t  Ae S1  Ce S1 / S n (3) (8) p n 2 2 (n)F(t , S , S ) S  S KS 2  0 1 2  n 1 1  B (n) (Si Si ) ...    The above equation can be integrated into a life distribution, i 1 n! Si S1 S2 such as the Weibull distribution. For a Weibull distribution, If we assume the terms with order of 2 or higher can be  t p . The Weibull model is: ignored, the expected value of the probability of failure at t0 C  B S  D T 1 D tS ne S1 T  B C  B S  2 is: D C T n S1 D n S1 T D T  S e tS e E F(t0 ,S1,S2 ) F(t0 ,S1,S2 ) (9) f (t, S , S )  2 D 2 T e E U (4) 1 2 If all the stresses are independent, the variance of the C D C T E U probability of failure at t0 can be approximated by: In the above equations (1-4), stress S can be either a constant 2 2 FF(t ,S ,S )V  0 1 2 (10) or a random variable. Var F(t0 ,S1,S2 ) BG W Var(Si )  S In order to estimate the parameters in equation (4), failure i 1 H i X where is the variance of the ith stress. data are needed, so accelerated life tests are conducted to Var(Si ) obtain failure data first. Then, an estimation method such as If we assume only the terms with order of 3 or higher can the maximum likelihood estimation (MLE) method is utilized be ignored, the expected value of the probability of failure at for estimating the model parameters [3]. t is: In the following sections, we assume that the model 0 2 parameters have been correctly estimated. We will use the 2 C (2)F(t , S , S ) S F(t , S , S )  F(t , S , S )  B D 0 1 2 T Var(S ) (11) model to predict the probability of failures for a warranty 0 1 2 0 1 2 D (2) T i i1 E 2! Si U period when the stresses are random. and the variance is: 2 3 WARRANTY PREDICTION BASED ON RANDOM 2 C F(t , S , S ) S VarF(t , S , S )  B D 0 1 2 T Var(S ) OPERATING STRESSES 0 1 2 D  T i i1 E Si U (12) 3.1 Theory on Functions of Random Variables 2 2 C (2)F(t , S , S ) S  B D 0 1 2 T Var$%(S  S )2 Consider a product with two random stresses: temperature D (2) T i i i1 E 2! Si U and voltage. Its failure time distribution is given in equation All the derivatives in the above equations are calculated at the (4). In the traditional method for warranty prediction, the mean values of the stresses. In this paper, approximations in average stress values are used in the calculation. Then, the equation (9) and (10) are used. The first order derivatives for predicted probability of failure by the end of the warranty S1 and S2 are: period of t is: 0 B C t S   D T C S F C  B S V F CB S E U t  1 D T ; G D n S T W (5) e D T F(t S ,S )  1 exp  tS e 1 /C  n 2 0, 1 2 G D 2 T W S1 S 2 S1 E U G E U W H X C S  B D t T F nC D T C t S Instead of using the mean value of each stress, we can use  e S1 E U D T  n1 D T their distributions to obtain the expected probability of failure. S2 S2 E U Assume the distribution for S is g (s ) and all the stresses i i i 3.2 Example are independent. The expected probability of failure is: EF(t ,S ,S ) Consider a component under two stresses: temperature 0 1 2 and voltage. The temperature-nonthermal life-stress I Y (6) F C  B S V L L relationship of equation (3) is used. Accelerated tests are   G D n S1 T W OOJ1 exp tS2 e /C Zg1(S1)g2 (S2 )dS1dS2 00 G D T W L E U L conducted and a portion of the data is given in Table 2. K H X[ The life distribution is the Weibull distribution. The The variance of the probability of failure is: estimated parameters using MLE are:  2  $%2 Var F(t0 , S1, S2 ) E F(t0 , S1, S2 ) E F(t0 , S1, S2 ) =1.434; B =1986.038; C=125.962; n = 1.8763 2 I F B VY The Weibull probability plot at the test stresses is given in L C  S L (7)   G D n S1 T W Figure 1. OOJ1 exp tS2 e / C Z g1(S1)g2 (S2 )dS1dS2 00L G D T WL K H E U X[ From the design specifications, it is known that the operating temperature and voltage are random variables following  $%2 E F(t0 , S1, S2 ) normal distributions. They are assumed to be independent

 . The upper and lower 90% two-sided bounds from each another. The distribution parameters are given in F F(t0,S1,S2 ) Table 3. for this example are [0.030, 0. 289]. To validate whether the approximation results are Time Failed (hrs) Temperature (K) Voltage (V) accurate enough, the exact analytical solutions of the expected 108 348 10 probability of failure and its variance are also calculated using 116 348 10 the integrals in equation (6) and (7). They are:  ;  27 348 15 E F(t0 ,S1,S2 ) 0.115 Var F(t0,S1,S2 ) 0.006 48 348 15 As expected, they are slightly larger than the approximated 51 378 10 solutions because the high order terms are ignored in equation 85 378 10 (9) and (10). The exact analytical solution for the confidence 138 378 15 bounds of the probability of failure is [0.036, 0.312]. For this example, the approximated and exact analytical solutions are 148 378 15 very close. … … … In the above analysis, the uncertainty (randomness) of the Table 2 – Failure Data for the Two Stresses Example operating stresses is considered in the calculation. The

ReliaSoft ALTA 7 - www.ReliaSoft.com uncertainly of the model parameters is not considered. The Probability - W eibull model parameters are estimated from the available failure

6.0 4.0 3.0 2.0 1.6 1.4 1.2 1.0 0.9 0.8 0.7 0.6 0.5 99.000 Probability data. Their uncertainty can be greatly reduced if there is a

90.000 Data 1 Temperature-NonThermal large enough sample size or enough history information. Weibull 348| 10 However, the uncertainty caused by the random stresses ( F=20 | S=0 Stress Level Points 50.000 Stress Level Line cannot be reduced. It is embedded in the product operation. If 348| 15 F=20 | S=0 Stress Level Points one wants to integrate the uncertainty of model parameters in Stress Level Line 378| 10 F=20 | S=0 the calculation, equation (8) can be expanded by treating Stress Level Points

Unreliability Stress Level Line model parameters as random variables. For details, please 10.000 378| 15 F=20 | S=0 Stress Level Points refer to [4, 5]. Stress Level Line 5.000 3.3 Simulation Results

6/ 25/ 2008 In section 3.2, the exact and approximated analytical 1.000 9:53:06 AM 5.000 10.000 100.000 1000.000 solutions are provided for the example. Both solutions require Time Beta=1.434; B=1986.038; C=125.962; n=1.876 intensive computation. In this section, simulation solutions are given. Using Monte Carlo simulation to solve problems with Figure 1 – Weibull Probability Plot of the Test Stresses random stresses is easy and straight forward. The simulation Stress Distribution Parameters procedure is:     Temperature Normal N( 300, 10)  Generate n set of random number for (S1, S2). N(  8,  2)  Voltage Normal Calculate F(t0 , S1, S2 ) for each set of (S1, S2).  Get the mean and variance for the n . Table 3 – Distribution for the Two Random Stresses F(t0 , S1, S2 ) Fortunately, it is not necessary to write new simulation Assume the warranty time is 400 hours. The expected code. Simulation software packages such as ReliaSoft’s probability of failure and its variance can be obtained using RENO can be used. For this example, 10,000 simulation runs equation (9) and (10). The expected value is: were conducted in RENO, taking about 1 minute to complete EF(t ,S ,S )  F(t ,S ,S ) 0 1 2 0 1 2 the simulation. The mean and variance for the probability of  0.101 failure at warranty time of 400 are 0.114 and 0.006. These and the variance is: values are very close to the exact analytical solutions and are 2 2 FF(t ,S ,S )V more accurate than the approximated solutions.  0 1 2 Var F(t0 ,S1,S2 ) BG W Var(S1 )  S i 1 H i X 4 WARRANTY PREDICTION BASED ON RANDOM STRESS  (0.003)2 100  (0.032)2 4 AND USAGE PROFILE  0.005 In section 3, we discussed how to make accurate warranty Once the mean and variance of the probability of failure are prediction by considering the randomness of random stresses obtained, its confidence bounds can be easily calculated by in the calculation. In some applications, such as washing assuming that the logarithm of is normally F (t 0 , S 1 , S 2 ) machines, the warranty return is not only related to a random distributed [5]: stress such as load, but is also affected by random customer F F usage. In this section, a stress-strength based method will be [ , ] (13) F  (1  F )w F  (1  F ) / w proposed to solve this complicated problem. where  ,   and F F(t0, S1 , S2 ) w exp{z1 / 2 Var(F) /[F (1 F )]}

4.1 Theory on Stress-Strength Model the usage profile of its customers. The average loads and average hours of using the machine were recorded for each The stress-strength model is widely used in structural user. Since this usage information was available, the company reliability calculation [6-8]. Assume the strength distribution wanted to use it to make more realistic estimates of the is F (X) and stress distribution is F (X). The expected 1 2 failures of the motors used in the washing machine for a 5 probability of failure is defined as:  year warranty period.  +  + E F1(X 2 ) P(X1 X 2 ) O P(X1 x) f2 (x)dx First, accelerated testing was conducted to establish the 0 (14)  life-stress (load) relationship. Failures were recorded in hours.  O F1(x) f2 (x)dx 0 A Weibull-IPL (inverse power law) model was used and the parameters were estimated from the failure data. They are: where Fi (x) is the CDF (cumulative distribution function) and =2.350; K=1.692E-5; n = 1.520 pdf (probability density function). Figure 2 compares a stress 1 The pdf of this failure time distribution is denoted as f(t). distribution and a strength distribution. From the survey data, the analyst found the average load

ReliaSoft Weibull++ 7 - www.ReliaSoft.com is 7.162 lbs and the average usage duration per week is 2.9 Probability Density Function 0.400 Pdf hours. Given a 5 year warranty, the total average operating

Strength\ Data 1 time is 754 hours. Applying these two average numbers to the Weibull-2P MLE SRM MED FM warranty prediction, the predicted probability of failure is: 0.320 F=0/ S=0 Pdf Line +     $%n 1  Stress\Data 1 p(x 754 | S 7.162) 1 exp tKS (15) Weibull-2P MLE SRM MED FM 1.52 2.35 0.240 F=0/ S=0  1 exp  (754 1.692E  5 7.162 )  0.039 Pdf Line However, because more information is available in the f( t)

0.160 customer usage data, instead of simply using the average values, the distribution of the load and the operating hour can be used. The analyst calculated the load distribution from the 0.080 survey data using a Weibull distribution. The parameters are:

ReliaSoft Corporation 2=5; 2 = 7.8 6/ 25/ 2008 0.000 1:46:47 PM 0.000 4.000 8.000 12.000 16.000 20.000 The pdf of the load distribution is denoted as l(s). Time, ( t) It was found that the average hours and average loads are Strength\ Data 1: 1-2../0 3-... Stress\Data 1: ,-.../0 ,-1.. correlated. For the customers with larger average loads, longer Figure 2 –Comparison of Stress and Strength Distributions operating hours are expected. In order to utilize this information, the analyst applied a Weibull distribution to the The larger the overlap area in Figure 2 is, the greater the operating hours and treated the scale parameter as a function probability of failure. of load. A general log-linear function was used for the -load The stress-strength method is also traditionally used in the relation, which is: automobile industry for warranty prediction [9]. For example,    e 0 1S the usage distribution in a 3 year warranty period can be 3 thought of as stress and the strength is the failure distribution where S is the load. The parameters in the model for the 5 year in terms of miles. The predicted probability of failure in the operating hour are: warranty period is calculated using equation (14). For the 3 =2; 0=6.094; 1= 0.0896 automobile industry, a typical warranty policy is 3 years and This pdf operating hour distribution is denoted as g(x). 36,000 miles. So equation (14) can be modified to consider So far, three distributions (the life distribution of the only vehicles with mileage less than 36,000. The modified motor, the load distribution and the usage hour distribution of equation is: customers) are available for the warranty return calculation. th 36,000 The expected probability of failure by the end of the 5 year + +  + P(X1 X 2 | X 2 36,000) O P(X1 x) f2 (x)dx 0 can be calculated using the stress-strength model with the  36,000 three distributions. O F1(x) f2 (x)dx 0 First, for a given load s, the expected probability of failure where Fi (x) is the probability of failure at mileage of x and is: EF(X | s)  p(T + X | s) f 2 (x) is the pdf of the usage mileage distribution at x.  For the automobiles, only the random usage is considered  O p(T + x | s)g(x | s)dx (16) in the calculation. For washing machines, the warranty returns 0  are affected not only by the random usage but also by the  O F(x | s)g(x | s)dx random load. In section 4.2, a method of solving problems like 0 the washing machine example will be provided. Then, treating load as a random variable, the overall probability of failure is: 4.2 Example A washing machine manufacturer conducted a survey on

E$%F(X , S)  E$%EF(X | s) 4.3 Simulation Results  (17)  O p(T + X | s)l(s)ds The simulation procedure is: 0   Generate a random number for stress S  OOF(x | s)g(x | s)l(s)dxds  Use this S to generate a random number of usage X 00  The variance of the probability of failure is: Use S and X to calculate F(X ,S) . VarF(X, S)  EF(X, S)2  $%EF(X, S) 2  Repeat above steps for n times and get n F(X ,S)  (18) 2 2  Get the mean and variance for the n F(X ,S) .  OOF(x | s) g(x | s)l(s)dxds $%E F(X, S)  00 For this example, 10,000 simulation runs were conducted where: in RENO, taking about 1.5 minutes to complete the C S 1 D x T simulation. The mean and variance for the probability of D T F(x | s)  1 e E 1 U ; failure at a warranty time of 5 years are 0.075 and 0.014.

 C S 3 These results are very close to the exact analytical solutions 3 1 D x T C S D T 3 x E U and much better than the approximation results. g(x | s)  D T e 3 ; D T 3 E 3 U 5 CONCLUSION

 C S 2 2 1 s C S D T s E U In this paper, methods for warranty prediction of products l(s)  2 D T e 2 ; E U with random operating stresses and random customer usages 2 2 are discussed. Analytical solutions are provided for the case 1  ; studies. Because of the complexity of the procedure for 1 1.692E 5s1.52 obtaining the analytical solutions, the use of simulation to 6.0940.0896 s obtain the solutions is also illustrated. Unlike the traditional  e 3 method which can only calculate the mean value of the Equation (17) and (18) can be solved numerically. warranty failures and ignores the randomness of the stresses Commercial software packages such as Mathcad can be used. and usages, the proposed methods can calculate both mean For this example, the result for the mean value in equation and variance by integrating the randomness of stresses and (17) and the variance from equation (18) are: usage into the calculation. This is much better because interval EF(X ,S)  0.076 ; VarF(X ,S)  0.014 estimate provides more information than a simple point Using these two values, the 90% confidence interval for the estimate. Comparisons show that simulation results are very probability of failure is [0.005, 0.568]. accurate. Consequently, it should be preferred by engineers, Equation (17) can be shown to be similar to equation (6). given the complexity of obtaining the analytical solutions. Therefore, the approximation methods for the mean and REFERENCES variance in section 3 can be extended to use here. However, because of the complexity of the problem, the approximation 1. A. Mettas, “Reliability Predictions Based on Customer results are not close to the exact analytical solutions anymore. Usage Stress Profiles,” Proc. Ann. Reliability & To use the mean value of the load and usage to calculate the Maintainability Symp., (Jan.), 2005. probability of failure, we first need to obtain these two mean 2. ReliaSoft, Accelerated Life Testing Reference, ReliaSoft values. They are: Publication, 2007.  E(x)  OOxg(x, s)l(s)dxds  754 3. W. Nelson, Accelerated Testing: Statistical Models, Test 00 Plans, and Data Analyses, New York, John Wiley & and Sons, 1990.  E(s)  O sl(s)  7.162 4. ReliaSoft, Life Data Analysis Reference, ReliaSoft 0 Publication, 2007. Using these two values, the predicted probability of failure is 5. W. Q. Meeker and L. A. Escobar, Statistical Methods for 0.039, as given in equation (15). This value is very different Reliability Data, New York, John Wiley & Sons, 1998. from the exact analytical solution of 0.076 obtained by 6. J. Tang and J. Zhao, “A Practical Approach For equation (17). Therefore, the approximated analytical solution Predicting Fatigue Reliability Under Random Cyclic from the Taylor series expansion is not accurate for this Loading,” Reliability Engineering and System Safety, vol. example. 50, 1995, 7-15. From the above calculation procedure, it can be seen that 7. O. Gaudoin and J. L. Soler, “Failure Rate Behavior of obtaining an analytical solution for the probability of failure Components Subjected to Random Stresses,” Reliability becomes very challenging when multiple random variables are Engineering and System Safety, vol. 58, 1997, 19-30. involved. It becomes even more complicated when these 8. H. Chang and Y. Shi, “A Practical Reliability Analysis variables are correlated. Therefore, simulation becomes an Method for Engineers,” Reliability Engineering and attractive option. In section 4.3, the simulation solutions for System Safety, vol. 47, 1995, 93-95. the above example are provided. 9. M. W. Lu and R. J. Rudy, “Reliability Test Target

Development,” Proc. Ann. Reliability & Maintainability University of Arizona. Her areas of research and interest Symp., (Jan.), 2000. include stochastic event simulation and system reliability and maintainability analysis. BIOGRAPHIES Adamantios Mettas Huairui Guo, PhD Vice President, Product Development Director, Theoretical Development ReliaSoft Corporation ReliaSoft Corporation 1450 S. Eastside LP 1450 S. Eastside LP Tucson, Arizona 85710, USA Tucson, Arizona 85710, USA e-mail: [email protected] e-mail: [email protected] Mr. Mettas is the Vice President of product development at Dr. Huairui Guo is the Director of Theoretical Development at ReliaSoft Corporation. He fills a critical role in the ReliaSoft Corporation. He received his PhD in Systems and advancement of ReliaSoft's theoretical research efforts and Industrial Engineering from the University of Arizona. His formulations in the subjects of life data analysis, accelerated dissertation focuses on process modeling, diagnosis and life testing, and system reliability and maintainability. He has control for complex manufacturing processes. He has played a key role in the development of ReliaSoft's software published papers in the areas of quality engineering including including Weibull++, ALTA and BlockSim, and has published SPC, ANOVA and DOE and reliability engineering. His numerous papers on various reliability methods. Mr. Mettas current research interests include repairable system modeling, holds a B.S degree in Mechanical Engineering and an M.S. accelerated life/degradation testing, warranty data analysis and degree in Reliability Engineering from the University of robust optimization. Dr. Guo is a member of SRE, IIE and Arizona. IEEE. Doug Ogden Arai Monteforte Vice President Manager, Simulation Group ReliaSoft Corporation ReliaSoft Corporation 1450 S. Eastside LP 1450 S. Eastside Loop Tucson, Arizona 85710, USA Tucson, Arizona 85710-6703 USA e-mail: [email protected] e-mail: Arai,[email protected] Mr Ogden joined ReliaSoft in 1997 and has served in a variety Ms. Arai Monteforte is the Manager of the Simulation Group of executive management roles. In these capacities he has at ReliaSoft Corporation. Over the years, she has played a key been instrumental in the growth and evolution of the sales role in the design and development of ReliaSoft's software, organization through the creation of sales plans, pipeline including extensive involvement in the BlockSim and RENO development, sales segmentation and growth strategies. Mr product families. Ms. Monteforte holds an M.S. degree in Ogden attended the University of Minnesota and is a member Reliability and Quality Engineering, a B.S. in Chemical of the Society of Reliability Engineers. Engineering and a B.S. in Computer Science, all from the