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Optimal design of an accelerated degradation experiment with reciprocal Weibull degradation rate
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by Indira Polavarapu.
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Thesis (M.S.I.E.)University of South Florida, 2004.
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ABSTRACT: To meet increasing competition, get products to market in the shortest possible time, and satisfy heightened customer expectations, products must be made more robust and fewer failures must be observed in a short development period. In this circumstance, assessing product reliability based on degradation data at high stress levels becomes necessary. This assessment is accomplished through accelerated degradation tests (ADT). These tests involve over stress testing in which instead of life product performance is measured as it degrades over time. Due to the role these tests play in determining proper reliability estimates for the product, it is necessary to scientifically design these test plans so as to save time and expense and provide more accurate estimates of reliability for a given number of test units and test time. In ADTs, several decision variables such as inspection frequency,the sample size, and the termination time at each stress level are important.In this research, an optimal plan is developed for the design of accelerated degradation test with a reciprocal Weibull degradation data using the mean time to failure (MTTF) as the minimizing criteria. A non linear integer programming problem is developed under the constraint that the total experimental cost does not exceed a predetermined budget. The optimal combination of sample size, inspection frequency and the termination time at each stress level is found. A case example based on Light Emitting Diode (LED) example is used to illustrate the proposed method. Sensitivity analyses on the cost parameters and the parameters of the underlying probability distribution are performed to assess the robustness of the proposed method.
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Adviser: Okogbaa, Geoffrey.
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degradation failure.
reciprocal Weibull distribution.
degradation testing.
life testing.
highly reliable products.
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Optimal Design of An Accelerated Degrada tion Experiment with Reciprocal Weibull Degradation Rate by Indira Polavarapu A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering Department of Industrial and Ma nagement Systems Engineering College of Engineering University of South Florida Major Professor: Okogbaa Geoffrey, Ph.D. Rao A.N.V, Ph.D. Qiang Huang, Ph.D. Date of Approval: September 1, 2004 Keywords: highly reliable produc ts, life testing, degradation testing, reciprocal weibull distribution, degradation failure Copyright 2004, Indira Polavarapu
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ACKNOWLEDGMENT I would like to express my gratitude to all those who gave me the possibility to complete this thesis. I am deeply indebted to my advisor Dr. O. Geoffrey Okogbaa whose help, stimulating suggestions and encouragement helped me in all the time of research and simplified my task in completion of th is thesis. His understanding, encouraging and personal guidance have provided a good basis for this work Special thanks to Dr.A.N.V.Rao for hi s wide knowledge and expertise and his important support as a committee member throughout this work. I warmly thank my other committee member, Dr.Qiang Huang for his valuable advice and friendly support. During this work I have collaborated w ith many friends and colleagues for whom I have great regard, and I wish to extend my warmest thanks to all those who have helped me with my work in the Department of I ndustrial & Management Systems Engineering. I would like to thank all the staff at The Inst itute on Black Life for their help, support and friendship. Grateful thanks to my family for being there from the very start standing the whole way through storms and sunshine. W ithout their encouragement and understanding it would have been impossible for me to finish this work.
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i TABLE OF CONTENTS LIST OF TABLES iii LIST OF FIGURES iv ABSTRACT v CHAPTER 1 INTRODUCTION 1 1.1 Background 1 1.2 ADTs Versus Other Testing Methods 2 1.3 Reliability Analysis 3 1.4 Applications 4 1.4.1 Light Emitting Diodes 5 1.4.2 Logic Integrated Circuits 5 1.4.3 Power Circuits 5 1.5 Motivation for This Research 6 1.6 Organization of This Thesis 7 CHAPTER 2 LITERATURE REVIEW 8 2.1 Introduction 8 2.2 Degradation Models 9 2.2.1 Linear Degradation 9 2.2.2 Convex Degradation 9 2.2.3 Concave Degradation 10 2.3 General Degradation Path Model 11 2.4 Degradation and Failure Types 13 2.4.1 Soft Failures: Specified Degradation Level 13 2.4.2 Hard Failures: Joint Distribution of Degradation and Failure Level 13 2.5 Constant Stress Degradation Models 14 2.5.1 The Arrhenius Rate Degradation Model 14 2.5.2 Inverse Power Relationship 15 2.5.3 Eyring Relationship 15 2.6 Acceleration Model 15 2.6.1 Elevated Temperature Acceleration 16 2.6.2 Nonlinear Degradation Path Reactionrate Acceleration 16
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ii 2.6.3 Linear Degradation Path Reactionrate Acceleration 17 2.6.4 Degradation with Parallel Reactions 18 2.7 Estimation of Accelerated Degradation Model Parameters 18 CHAPTER 3 PROBLEM STATEMENT 23 3.3 The Optimization Problem 23 CHAPTER 4 METHODOLOGY 25 4.1 Degradation Model with Random Coefficient 25 4.2 Assumptions 25 4.3 The MeanTimeToFailure 26 4.4 The Computation of 0 0 27 4.5 The Computation of 0 0 E 32 4.6 The Cost Function n l f TCm i i i, ,1 34 4.7 The Optimization Model 34 4.8 Algorithm 35 CHAPTER 5 EXAMPLE 37 5.1 Simulation Experiment 38 5.2 Optimal Test Plan Based on the LED Data 48 5.2.1 Optimal Parameters Based on the ADT Experiment 49 5.3 Sensitivity Analysis 50 5.3.1 Test Plans under a Variety of Cb 50 5.3.2 Test Plans for Different Values of m and a Variety of Combinations of m i iS1 51 5.3.3 Sensitivity Analysis of Misspecifying u, b, and2 53 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH 56 6.1 Future Research Directions 57 REFERENCES 59
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iii LIST OF TABLES Table 5.1 The Standardized Sample Degradation Paths at Stress Level S1 40 Table 5.2 The Standardized Sample Degradation Paths at Stress Level S2 41 Table 5.3 The Standardized Sample Degradation Paths at Stress Level S3 42 Table 5.4 Optimal Degradation Test Pl ans under Various Cost Conditions (Cs, Cp, Cm, Cd, Cb) 48 Table 5.5 The Optimal Test Plans for Some Values of Cb 50 Table 5.6 The Optimal Test Plans for Vari ous Values of m and Combinations of Stress Levels 52 Table 5.7 The Optimal Solution 3 1 *, n l fi i i for the Case that 3 1 i iare Changed over the Ranges% 5 2 53 Table 5.8 The Optimal Solution 3 1 *, n l fi i i for the Case that 3 1 i iare Changed over the Ranges% 5 54 Table 5.9 The Optimal Solution 3 1 *, n l fi i i for the Case that 3 1 i iare Changed over the Ranges% 10 54
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iv LIST OF FIGURES Figure 2.1 Possible Shapes of Degradation Curves 8 Figure 2.2 Propagation Delay Grow th Curve with a Plateau (Maximum Degradation) 19 Figure 2.3 Equal LogSpacing Plan for Measurement 20 Figure 5.1 The Standardized Sample Degradation Paths under (a) S1, (b) S2 and (c) S3 43 Figure 5.2 The Plots of ij (t) versus t0.5 under (a) S1, (b) S2 and (c) S3 44 Figure 5.3 The Weibull Pr obability Plot for 3 1 25 1 ij ij 46 Figure 5.4 The Normal Probability Plots for Residuals under (a) S1, (b) S2 (c) S3 47
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v OPTIMAL DESIGN OF AN ACCELERA TED DEGRADATION EXPERIMENT WITH RECIPROCAL WEIBULL DEGRADATION RATE INDIRA POLAVARAPU ABSTRACT To meet increasing competition, get products to market in the shortest possible time, and satisfy heightened customer expect ations, products must be made more robust and fewer failures must be observed in a shor t development period. In this circumstance, assessing product reliability based on degradation data at high stress levels becomes necessary. This assessment is accomplished through accelerated degradation tests (ADT). These tests involve over stress testing in which instead of life product performance is measured as it degrades over time. Due to the role these tests play in determining proper reliability estimates for the product, it is nece ssary to scientifically design these test plans so as to save time and expense and provide more accurate estimates of reliability for a given number of test units and test time. In ADTs, several decision variables such as inspection frequency, the sample size, and the termination time at each stress level are important. In this research, an optimal plan is developed for the design of accelerated degradation test with a recipr ocal Weibull degradation data using the mean time to failure (MTTF) as the minimizing criteria. A non linear integer programming problem is developed under the constraint that the to tal experimental cost does not exceed a pre
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vi determined budget. The optimal combination of sample size, inspection frequency and the termination time at each stress level is found. A case example based on Light Emitting Diode (LED) example is used to illustrate the proposed method. Sensitivity analyses on the cost parameters and the parameters of the und erlying probability distribution are performed to assess the robustness of the proposed method.
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1 CHAPTER 1 INTRODUCTION 1.1 Background Todays manufacturers are facing new pre ssures to develop hi ghly sophisticated products to match rapid advances in technology, intense global competition and increasing customer expectations. As a re sult manufacturers must produce components in record time, while improving productivity, reliability, and overall quality of the component. It is a significan t challenge to design, develop, test, and manufacture highly reliable products within s hort turn around times and remain within the stringent conditions, imposed by both internal and external circumstances. Estimating the timetofailure distribution or longterm performance of components of high reliability products is particularly difficult. Most m odern products are designed to operate without failure for several years. Thus few of such units will fail or degrade to a significant amount in a test of any practical length based on normal use conditions. For example, during the design and construction of a communication satellite, there may be only 6 months available to test the components which are meant to be in service for 15 to 20 years. The components used in submarine cables are often required to operate for 25 years under the sea. Very few test units are av ailable that will actua lly reflect the life profiles of these components. For these reasons Accelerated tests (A Ts) are used widely
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2 in manufacturing industries, part icularly to obtain timely information on the reliability of products. 1.2 ADTs Versus Other Testing Methods Traditionally, reliability assessment of new products has been based on accelerated life tests (ALTs) that record failure and censori ng times of products subjected to elevated stress. However, this approach may offer little help for highly reliable products which are not likely to fail during an expe riment of reasonable length. An alternative approach is to assess the reliability from the changes in performance (degradation) observed during the experiment, if there exists a quality characte ristic of the product whose degradation over time can be related to reliability. Accelerate d degradation tests compared to other tests have the advantage of analyzing performance be fore the material or the component fails. Degradation tests determine how much life there is left in a material or in components, and such knowledge enables life extensi on. Extrapolating performance degradation to estimate when it reaches failure level enable s analysis of degradation data. However, such analysis is correct only if a good model for extrapolation of performance degradation and a suitable performa nce failure have been established. Some of the general assumptions of accelerated degradation models are Degradation is not reversible. A model applies to a single degradation process mechanism or failure mode. If there are simultaneous degradation processe s and failure modes, each requires its own model. Degradation of specimen performance befo re the test starts is negligible. Performance is measured with negligible random error.
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3 The failure process at high stress levels ar e the same as at the design or use stress levels. Accelerated degradation tests are usually ve ry expensive and thus it is essential to plan them carefully. Good test plans yiel d better results for a given cost and time parameters, on the other hand poor test plans waste time and resources, and may not even yield the desired information. When conducti ng an ADT, the following issues are usually of interest. How long should the test be run? or How many units s hould be tested at each stress level? Thus to address the issues, a scientific plan is need ed to make the most efficient use of test resources and ultimately to obtain an accurate estimate of the life profile of an entity unde r the normal use conditions. 1.3 Reliability Analysis Most things have a life spa n, defined in some form or another. These life times when measured, present us with data sets that are used for scientific or other purposes. It is natural to study the life time distribution of an entity through a set of measured data. An area of research, which is still vary much ac tive, is the theory of reliability. A generic definition of reliability is: Reliability is the probability that a product or a system will perform its intended function without failure for a specified pe riod of time under specific operating conditions. To express this relationship mathematically we define the continuous random variable to be the time to failure of the component or system. Thus reliability at time t can be expressed as: 0 1 0 0 : 0 ; Pr t R Lim R t R Where T t T t Rt
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4 In reliability analysis, the major issue is the probability distribution of the life times of the entity under study. For this purpose, the standard method is to take a set of observed life times T1, T2... Tn, or censored sometimes, where we assume that the observed life time is a function of an unknown parameter which can be expressed as Ti ~ F (.; ). Where F (.; ) is the probability density function of an unknown parameter From the likelihood function constructed from this sample, we can make an inference with respect to the unknown parameter When the form of F (.; ) is known and the complete distribution of F is determined by a finite dimensional parameter then we have a classical parametric model; if F is completely unknown except for some qualitative descriptions such as continuity or smoothness, then we ha ve a nonparametric model; finally, if F is unknown but the parameter has some structure to explore, then we have a semiparametric model. When the parameter exhibits some structure, we will naturally embed our inference problem into traditional, and timete sted models for statistical analysis. These include techniques such as experimental design, regression, l ogistic regression, accelerated life testing, etc. These methods incorporate various situations that may be encountered in practice. Ther e is no need, however, to restrict the inference to the classical frequentists parame tric setup. We can, if the situation requires, use the Bayesian method or even the empirical Bayes techniques. 1.4 Applications Applications of accelerated degradation te sts include light emitting diodes, logic integrated circuits, power supplies, etc.
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5 1.4.1 Light Emitting Diodes (LED) Light emitting diodes are widely used in many fields ranging from consumer electronics to optical fiber transmission syst ems. The LED has many features such as less power consumption, small volume, good visual effects and long life. Nowadays they are used as electronic boards on highways and street s, and as smoke censors on ceilings, etc. Because of their high reliabilit y, it is difficult to obtain the product life time information under normal stress levels in a relatively shor t time. Thus, ADTs are used to obtain the reliability information of LED products [6, 7]. 1.4.2 Logic Integrated Circuits Some logic integrated circu its are used as components of submarine cables. The important parameter in determining the reliabil ity information of logi c integrated circuits is propagation delay [8]. The logic integr ated circuits might not function if the propagation delay of a logic gate increases (d egrades). For example, a logic circuit which is designed to have a maximum propagation delay of 10 nsec from input to the output requires that the combined propagation delay of the individual logic gates in the critical path does not exceed 10 nsec. These logic integrat ed circuits are required to operate for at least 25 years under the sea w ithout failure. So, accelerated degradation tests are employed to predict the life of logic inte grated circuit and to study the associated propagation delay. 1.4.3 Power Circuits For power supplies, failures are common due to low DC output. Power supply units show downward drift in their DC output Accelerated degradation tests are used to measure the DC output and to monitor the device for reliability information.
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6 Nelson [3, pp 521548] lists other applica tions of accelerated degradation tests which include: metals, semi conductors and mi cro electronics, dielec trics and insulations, food and drugs, and plastics and polymers. 1.5 Motivation for This Research An Accelerated Degradation test is a m echanism designed to shorten the life of products by subjecting the test uni ts to higher levels of stresse s that are more severe than the normal use stress levels. The information from high stress leve ls is extrapolated through a reasonable statistical method to obtain estimates of life, or longterm performance, at the normal use stress level. Traditional approaches are based on life tests that record only timetofailure. Such an alyses have been extensively studied and developed over the past few decades and many articles have been published in this area. Due to the fact that traditi onal life testing of hi ghly reliable products does not give good reliability estimates, reliability assessment using degradation data has become increasingly important. In the literature, most degradation mode ls assumes that the degradation paths or transformed degradation paths are linear, and are developed for only the normal use stress level [38]. Most of the literature focuses on estimating the parameters in the linear degradation model and the life distribution. Re search about accelerated test planning is also reported. But, when carry ing out the accelerated degrad ation tests several decision variables such as inspection frequency, the sa mple size, and the termination time at each stress level are important. The primary objective for this research is to determine the optimal parameters of an ADT with respect to products whose degrad ation rates follow the reciprocal weibull
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7 distribution. This is accomplis hed by taking the MLEs of th e variance of the confidence interval of the MTTF under the constraint that the total experimental cost does not exceed a predetermined budget. A non linear program ming problem is developed to determine the optimal value of the decision variables su ch as sample size, inspection frequency and the termination time at each stress level. 1.6 Organization of the Thesis In Chapter 2, a review of the literature is discussed. Chapter 3 discusses the problem statement, assumptions made and notations used. An optimization problem is proposed. In Chapter 4, an optimal plan for solvi ng the optimization problem is presented. To illustrate the optimal plan, an example of LED degradation is presented in chapter 5.Finally, the conclusions of th e study and suggestions for fu rther research are presented in chapter 6.
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8 CHAPTER 2 LITERATURE REVIEW 2.1 Introduction During the 1990s Nelson[3], (chapter 11) provided a fair ly thorough survey on ADT, which included areas of applications, st atistical models, describes basic ideas on ADT models, and, using a specific example, s hows how to analyze a type of degradation data. Carey and Koenig[4] (1991) have descri bed a dataanalysis strategy and a modelfitting method to extract reliability informa tion from observations on the degradation of integrated logic devices that are components in a new generation of submarine cables. Most failures can be traced to an underlying degradation process. Meeker and Escobar (1998) gave some examples of three general shapes for degradation curves in arbitrary units of degradation and tim e: linear, convex, and concave which are shown in fig .2.1. The dashed horizontal line represents the degr adation level at which failure would occur. Figure 2.1 Possible Shapes of Degradation Curves
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9 2.2 Degradation Models 2.2.1 Linear Degradation Meeker and Hamada [17] uses linear degr adation in some simple wear processes like automobile tire wear. Let D (t) represents the amount of auto mobile tire tread at time t, and wear rate dt t dD ) ( = C, then D (t) = D (0) + C t. The parameters D (0) and C could be taken as constant for indi vidual units, but random from unittounit. 2.2.2 Convex Degradation The convex degradation approach is used in models for which the degradation rate increases with the level of degradation such as in modeling the growth of fatigue cracks. Let a (t) denote the size of a crack at time t. The Paris model[18] is given as ma k C dt t da )] ( [ ) ( (2.21) Where a = size of the crack, C and m = material properties, and ) ( a kstressintensity factor,
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10 is often used to describe the growth of fati gue cracks. Lu and Meeker [18] use this model in which a a k ) ( for describing the growth rate of fatigue cracks within a certain size range. Then, } ) 1 ( )] 0 ( [ 1 { ) 0 ( ) () 1 /( 1 1 m mt m C a a t a (2.22) where a (0) = 0.90 inches is the initial crack length at t = 0. Dividing both sides of Eq. (2.2) by a (0) yields a (t)/a (0) = 1/) 1 /( 1 1} ) 1 ( )] 0 ( [ 1 { m mt m C a (2.22) 2.2.3 Concave Degradation Meeker and LuValle [19] describe m odels for growth of failurecausing conducting filaments of chlorinecopper com pounds in printedcircuit boards. They consider degradation from a firstorder chem ical reaction. These filaments cause failure when they reach from one copperplated throughhole to another. Let A1(t) be the amount of chlorine ava ilable for reaction at time t, and A2(t) be the amount of failurecausing chlorinecopp er compound at time t. Under appropriate conditions, copper combines with chlorine A1 to produce the chlorinecopper compound A2 with a constant rate k. The equations for the rate for this process are 1 1kA dt dA and 2 2kA dt dA Let c and A2(0) be the initial amounts. Assuming A2(0) = 0 we get
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11 A1(t) = A2( )[1exp(kt)]. (2.23) This function is illustrated by the concav e curve in Fig.2.1. The asymptote at A2( ) = A1(0) reflects the amount of chlorine avai lable for the reaction producing compounds LuValle and Meeker [19] also suggest othe r more elaborate models for this failure process. Carey and Koenig [4] use similar m odels to describe degradation of electronic components. 2.3 General Degradation Path Model Lu & Meeker (1993) use the following model [Eq 2.31] for the analysis of degradation data at a fixed le vel of stress (i.e., no acceleratio n) and to estimate a timetofailure distribution. They denote the true degr adation path of a particular unit(a function of time) by D(t), t > 0. In applications, values of D(t) are sampled at discrete points in time,t1,t2,.The observed sample degradat ion path for unit i at time tij is a units actual degradation path plus er ror and is given by ij ij ijD y i = 1,.., n j = 1, 2,,mi (2.31) Where Dij = D (tij i ) is the actual path of the ith unit at time tij(the times need not be the same for all units), 2, 0 ~Nij is the deviation from the assumed model for unit i at time tij i = ( 1i ,. ki) is a vector of k unknown parameters for unit i. The deviations are used to describe th e measurement error. The total number of inspections on unit i is denoted by mi. Time t could be realtime, operating time, or some surrogate like miles for automobile tires or lo ading cycles in fatigue tests. Typically small paths are described with a model that has up to four points (i.e., k=1, 2, 3, 4). Some of the
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12 parameters in could be random from unittounit a nd some of them could be modeled as constant across all units. The scales of y and t can be chosen to simplify the form of D(t, ). The choice of a degradation model requires not only th e specification of the form of D(t, ) function, but also the specificati on of which of the parameters in are random and which are fixed as well as the joint di stribution of the random components in Lu & Meeker (1993) describe the use of a general family of transformations to a multivariate normal distribution with mean vector and covariance matrix It is generally reasonable to assume that the random components of the vector are independent of the ij. We also assume that ij are independent and identically distributed. Because the yij are taken serially on a unit, however, there is potential for autocorrelation among the ij. Especially if there are many closely spaced readings. In many practical applications involving infe rence on the degradation of units from a population or process, however, if the mode l fit is adequate an d if the testing and measurement processes are in control, then the autocorrelation is typically weak and moreover, it is dominated by the unittounit variability in the values and thus can be ignored. Also, it is well known th at point estimates of regres sion models are not seriously affected by autocorrelation, but ignoring autocorr elation can result in standard errors that are seriously biased. This however is not a problem when confidence intervals are constructed by using an a ppropriate simulationbased bootstrap method. In more complicated situations it may also happen that will depend on the level of the acceleration variable. Often, how ever, appropriate modeling (f or variance stratification,
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13 e.g., transformation of the degradation res ponse) will allow the use of a simpler model based on constant 2.4 Degradation and Failure Types 2.4.1 Soft Failures: Specified Degradation Level For some products there is a gradual lo ss of performance (e.g., decreasing light output from a fluorescent light bulb). Then failure would be defined in an arbitrary manner at a specified level of degradation su ch as 60 % of initial output. Tseng, Hamada, and Chiao (1995) explain this with an example and defined this as soft failure. A fixed value of Df is used to denote the critical level for the degradation path above (or below) which failure is assumed to have occurred. The failure time T is defined as the time when the actual path D(t) cr osses the critical degradation level c and tc is used to denote the planned stopping time in the experiment. Inferences are made on the failuretime distribution of a particular produc t or material. For soft failures it may be possible to continue observation beyond Df 2.4.2 Hard Failures: Joint Distribution of Degradation and Failure Level For some products, a failure event is de fined as when the product stops working (e.g., when the resistance of a resistor devi ates too much from its nominal value, causing the oscillator in an electronic circuit to stop oscillating or wh en an incandescent light bulb burns out). These are called hard failures. In general with hard failures, failure times correspond to a particular level of degradation. But, the le vel of degradation at which failure occurs is random from unit to unit or from time to time. This could be modeled by using a distribution to describe unittounit variability in Df or, more generally, the joint distribution of and the stochastic behavior of Df
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142.5 Constant Stress Degradation Models Nelson (1990) briefly describes some ba sic degradation models for constant stress. The following are the most widely used constant stress degradation models. 2.5.1 The Arrhenius Rate Model The Arrhenius rate relationship is wide ly used for temperatureaccelerated degradation. This model is mostly used in pharmaceuticals, insulations, dielectrics, plastics, polymers, Adhesives, battery cells, and incandescent lamp filaments. Arrhenius law: According to the Arrhenius rate law, the rate of a simple (firstorder) chemical reaction depends on temperature as follows )] /( exp[kT E A rate (2.51) where: E is the activation energy of the reaction, usually in electronvolts. k is Boltzmanns constant, 8.6715 electronvolts per 0C. T is the absolute Kelvin temp erature; it equals the temperat ure in Centigrade plus 273.16 degrees; the absolute Rankine temperature e quals the Fahrenheit temperature plus 459.7 Fahrenheit degrees. A is a constant that is characteristic of the product failure mechanism and the test conditions. The product is assumed to fail when some critic al amount of the chemical has reacted (or diffused); Critical amount = (rate) (time to failure) or, Time to failure = (critical amount) / (rate)
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15 Therefore, the nominal time to failure (life) is inversel y proportional to the rate. This yields the Arrhenius life relationship )] /( exp[kT E A 2.5.2 Inverse Power Relationship The inverse power relationship is widely us ed to model product life as a function of an accelerating stress. This is mostly us ed in electrical insulations, dielectrics in voltageendurance tests, ball and roller bearings, incandes cent lamps and flash lamps etc. The relationship is sometimes called the inverse power law or simply the power law. Suppose that the accelera ting stress variable V is positive. The inverse power relationship between nominal life of a product and V is V A V/; (2.52) Here A and are parameters characteristic of the product, specimen geometry and fabrication, the test method, etc., The parameter is called the power or exponent. 2.5.3 Eyring Relationship An alterative to the Arrhenius relations hip for temperature acceleration is the Eyring relationship. The Eyring relationship for nominal life as a function of absolute temperature T is )] /( exp[ ) / ( kT B T A ; (2.53) here A and B are constants that are characteri stic of the product ad test method, and k is the Boltzmanns constant. 2.6 Acceleration Model To obtain timely information from laborator y tests, sometimes it is required to use some form of acceleration. In some failure mechanisms such as the weakening of an adhesive mechanical bond or the growth of a conducting filament through an insulator,
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16 the chemical or other degradation process can be accelerated by increasing the level of acceleration variables like temperature, humidity voltage, or pressure. If an adequate physicallybased statistical model is availa ble to relate failure time to levels of accelerating variables, the model can be used to estimate lifetime or degradation rates at product use conditions. Lu, Meeker & Escoba r [5] mentioned the following acceleration models. 2.6.1 Elevated Temperature Acceleration The Arrhenius model, which describes the effect of temperature on the rate of a simple firstorder chemical reaction is 15 273 11605 exp ) 15 273 ( exp ) (0 0temp E temp k E temp Ra B a (2.61) Where temp is temperature in 0C and kB= 1/11605 is the Boltzmanns constant in units of electron volts per 0C. The preexpon ential factor 0 and the reaction activation energy Ea in units of electron volts are characteristics of the particular chemical reaction. Taking the ratio of the reaction rates at temperatures temp and Utemp cancels 0 giving an Acceleration Factor AF(temp, Utemp ,Ea) = ) ( ) (Utemp R temp R 2.6.2 Nonlinear Degradation Path and Reactionrate Acceleration The simple chemical degradation path model with a temperature acceleration factor affecting the rate of reaction is given by ]} ) ( exp[ 1 { ) ; ( t temp AF R D temp t DU (2.62)
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17 Here RU is the rate reaction at use temperature Utemp RU AF (temp) is the rate reaction at temperature temp, and D is the asymptote. When degradation is measured on a scale decreasing from zero, D < 0 then the failure occurs at the smallest t such that D(t) Df Equating D(T ; temp) to Df and solving for T gives the fa ilure time at temperature temp as ) ( ) ( ) ( 1 log 1 ) ( temp AF temp T temp AF D D R temp TU f U Where T (Utemp ) = (1/RU) log (1Df/D) is failure time at use conditions. Here, the life/temperature model induced by the simple degradation process and the Arrheniusacceleration model results in a Scal e Accelerated Failure Time (SAFT) model. Under the SAFT model, the degr adation path of a unit at any temperature can be used to determine the degradation path that the same unit would have had at any other specified temperature by scaling the time axis by the acceleration factor AF (temp) 2.6.3 Linear Degradation Path Reactionrate Acceleration Consider the model with non linear degradation path and reaction rate acceleration along with the critical level Df. When D(t) is small relative to D ]} ) ( exp[ 1 { ) ; ( t temp AF R D temp t DU (2.63) t temp AF R t temp AF R DU U ) ( ) ( If failure occurs when D(T) Df then D(T;temp)= Df and the failure time is given as ) ( ) ( ) ( 1 ) (temp AF temp T temp AF R D temp TU U f where T(tempU) = Df / URis failure time at use conditions. This is al so an SAFT model.
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182.6.4 Degradation with Parallel Reactions A more complicated degradation path m odel with two parallel onestep failure causing chemical reactions is given by : ]} ) ( exp[ 1 { ]} ) ( exp[ 1 { ) ; (2 2 2 1 1 1t temp AF R D t temp AF R D temp t DU U (2.64) Where R1U and R2U are the usecondition rates of the two parallel reacti ons contributing to failure. This degradation model does not lead to an SAFT model because the temperature affects the degradation processe s differently, inducing nonlinearity into the acceleration function rela ting times at two different temperatures. 2.7 Estimation of Accelerated Degradation Model Parameters Lu and Meeker (1993) use a twostage method to estimate the parameters of the mixedeffects accelerate d degradation model. Stage 1For each unit, fit the degradation model to the sample path and obtain the estimate of the model parameters of each unit. Stage 2Combine the estimate of the model pa rameters of each unit in the first stage to produce estimates of the population parameters. In another research Lu, Meeker & Esc obar [5] suggest that, in some cases, an approximate maximum likelihood (ML) is faster than n nonlinear least squares estimations required for the twostage method. ML estimation also has the advantages of desirable largesimple properties and easy to use sample paths for which all of the parameters cannot be estimated. The twostag e estimation is useful for obtaining starting values for the ML approach for modeling, esp ecially when another distribution other than a joint normal distribution for the random effects is being considered.
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19 Meeker and Escobar (1993) have give n an updated literature survey on various approaches used to assess reliability inform ation from degradation data. Boulanger and Escobar[7] address the problem of designing a class of ADTs. They assume that each unit is subjected to an elevated constant stress level over the duration of the experiment. The performance degradation of each test unit at a stress level could be described by a growth curve which levels off to a plateau (maximum degradation) after a pe riod of time. Figure 2.2 shows the degradation amount over time. The model is give by: y(t) = [1exp(( t))] + (t), Where y(t) = observed change of propagation delay up to time t, = plateau where the degradation will level off, = random coefficient, = a constant, which is equal to 0.5, and (t) = measurement error. Figure 2.2 Propagation Delay Growth Curve with a Plateau (Maximum Degradation)
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20 The authors consider as lognormally distributed and stress dependent. The design problem they consider is to minimize th e variance of the estimate of the mean of the logarithm, of the plateau, ln( ) at use condition. Their obj ective here is to provide some guidelines in designing a useful plan for a special class of accelerated degradation tests. They first determine the optimal stre ss levels and the proportion of units allocated to each stress level, and then determine optimal times to measure the performance degradation of units at each stress level. The test stress levels are chosen to be 4480K and 3730K, which are the highest temperatures th e plastic package can withstand, and the minimum temperature the measurement equipmen t can detect any degradation at the end of the experiment, respectively. Equal logsp acing plan, shown in Fig.2.3., is used for measurement because the process shows a great deal of degradation at the early stage and then stabilizes. Figure 2.3 Equal LogSpacing Plan for Measurement Although the result is interesting, it is not practical since an appropr iate termination time for an experiment is usually not known in advance. In the literature most of the degradati on models are linear, or can be transformed to linear models. Also most of the literature focuses on estimating the parameters in the degradation model and the life distribution. Yu and Tseng[8] proposed an intuitively online and realtime rule to determine an appropriate termination time for an ADT. Park and Yum[9] develop optimal accelerated degradation test plans under the assumptions of
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21 destructive testing and the si mple constant rate relations hip between stress and product performance. The authors determine stress leve ls, the proportion of test units allocated to each stress level and measurement times such that the asymptotic variance of the maximum likelihood estimator of the MTTF at the use condition is minimized. Yu and Tseng [1, 2, 1214] describe a method for conducting a degradation experiment efficiently considering several factors, such as the inspecti on frequency, termination time and the sample size. They consider a typica l degradation path of an LED, which is ln(ln(y(t))) = ln( ) + ln(t) + (t), Where y (t) = standardized light intens ity of an LED device at time t, = parameters of the degradation path, and (t) = measurement error of the device at time t. Based on data (ti,k yi,j(ti,k)), i is the for stress level, k = 1,2,,m, where k is the index that represents the mth measurement for unit j the degradation parameters for unit j can be obtained. The failure time of unit j can be found using 1) ln( D, assuming D is the critical value for the sta ndardized light intensity when an LED fails. In these papers, they deal with the optimal design for a degradation experiment under the constraint that the total experimental co st does not exceed a predetermined budget. The optimal decision variables are obtained by minimizing the variance of the estimated 100pth percentile of the lifetim e distribution. But, these th ree decision variables have a
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22 great deal of influence upon the experimental cost and the precision of selecting the most reliable product using degradation data. A nonlinear integer programming problem is developed to determine the optimal combinati on of sample size, inspection frequency and termination time. As is evident from the above review, an extensive literature on the design of degradation tests and accelerated degrad ation tests exists. But when designing a degradation test, the distribut ion of the degradation rate of the product/component at which it degrades is very important. The Weibull and lognormal distributions are two most popular lifetime models in re liability analysis that have been used for this purpose. An incorrect choice of the distribu tion may lead to serious bias.
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23 CHAPTER 3 PROBLEM STATEMENT With regard to highly reliable products, it is important to consider the issues of how to plan tests that provide the most efficient use of resources especially as it relates to conducting an accelerated degradation test. Th e problem is to minimize the expected value of the MeanTimeToFailu re of a product subject to th e constraint that the total cost of the test does not exceed a predetermined test budget. 3.1 The Optimization Problem The following decision variables are important in conducting an ADT efficiently (Yu & Chiao [1, 2]). These variables not only affect the experimental cost but also affect the precision of estimating the MTTF (0 ), which can be defined as the expected or the mean value of the failure time. The pertinent questions regarding the test plan are How is an appropriate inspection frequency (fi ) determined? How many times (li ) should the products performa nce be inspected at each stress level? How many devices (sample si ze, n) should be taken fo r testing at each stress level? In order to measure the precision of estima ting the MTTF, the expected width of the confidence interval values of the MeanTimeToFailure 0 is computed. The expected value of a realvalued random variable gi ves the mean or central tendency of the
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24 distribution of the variable. The unbiased maximum likelihood estimator of the MTTF can be achieved by estimating the asymptotic variance. The asymptotic variance can be obtained by minimizing either the mean square e rror or the expected va lue of the range of MTTF (MTTFmaxMTTFmin) Since we do not have a prior estimate of the mean square error we will estimate the asymptotic variance by minimizing the MTTF. Thus, the optimal decision problem based on the expected range of the MTTF is formulated as follows: Minimize 0 0 E (3.11) Subject to b m i i iC n l f TC ,1 (3.12) i il f ,, n N = {1, 2, 3} (3.13) i = 1, 2, 3 .m Where as, n l f TCm i i i, ,1 denote the total cost of conducting an ADT. n l fm i i i, 1 0 denote an estimator of 0 based on a test plan n l fm i i i, ,1 0 0, denote a 100(1p) % CI of 0 from the test plan n l fm i i i, ,1 0 0 E denotes the expected width of the )% 1 ( 100 p CI of 0 Cb is the total cost of the budget. p is the percentile of the lif e time distribution of the product at normal use condition.
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25 CHAPTER 4 METHODOLOGY 4.1 Degradation Model with Random Coefficient Let t denote the quality characteristic (degra dation path) of the product at time t. Assume that there exis ts a suitable function (.) such that 0 )) ( ( t t t (4.11) Where > 0 is a fixed and known constant; >0 is a random coefficient that varies from unit to unit. 4.2 Assumptions The ADT uses m stress levels, S0, S1, S2 Sm, satisfying (S0 ) S1 S2 Sm, where S0 is the use condition. Due to the measurement erro rs, the actual degradation path cannot be observed directly. Let yij ( ti, k) denote the sample degradation path of the jth device at time ti,k under the stress level Si .The path can be expressed as follows: ) ( )) ( (, , k i ij k i ij k i ijt t t y (4.21) The units put into test are randomly sele cted from the samples, and are randomly assigned to test stress levels At each stress level, n devices are randomly selected for testing.
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26 Suppose that, under stress level Si the inspections are made li times for every fi units of time (e.g. fi hours or fi days) until time ti li = fi li tu ,where li is a positive integer and tu is a unit of time. Assume that ij follows a reciprocal wei bulldistribution then 1 follows a weibull distribution with scale parameter and shape parameter (which is denoted by 1 ~ Weibull ( ) The shape parameter does not depend on the stress level and the scale parameter or the characteristic life is a function of transf ormed levels of stress: 0 1 0ln X (4.22) Where 0 and 1 are unknown parameters to be estimated from the data, Xi =X (Si) and X (.) is a suitable transformation. Two familiar examples for X (.) are as follows: X (Si) = 1/ Si, if an Arrhenius model is assumed = ln Si, if an inversepower model is assumed Some other relationships which are commonly used are mentione d in the literature review in chapter 2. In order to solve the optimization proble m the MTTF has to be computed first. 4.3 The MeanTimeToFailure The product life time ( ) is suitably defined as the time when crosses the critical level D. From Eq (3.11), can be expressed as / 1) ( D Taking natural logarithm on both sides ln )) ( ln( 1 ) ln( D (4.31)
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27 Since follows reciprocal we ibull distribution, ln follows the extreme value distribution with scale parame ter u, and location parameter b (which is denoted by ln ~ Extreme (u, b) and in equation (4.11) with fixed, it can be shown that follows the Weibull distribution with scale parameter / 1))) ( ( ( D and shape parameter Let 0 denote the products lifetime under S0. Thus, we have )) ( ( */ 1 0D Weibull The MTTF, 0 of the products lifet ime distribution under S0 is 1 1/ 1 0 0D b u D 1 exp0 / 1 0 (4.32) Where ) (0 1 0 0S X u Here, the problem is to design an efficient ADT such that 0 can be estimated precisely. The optimization problem can be solved by using the following steps. 4.4 The Computation of 0 0 For 1 j n and 1 i m, based on the observations il k k i ij k it y t1 ,) ( ,, the leastsquares estimator (LSE) ij ofij conditional on ij can be computed by minimizing 2 1 ,) ( il k k i ij k i ij ijt t y LS Thus, we obtain i il k k i l k k i k i ij ijt t t y1 2 1 , (4.41)
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28 and 2can be estimated by ij m i n j iLS l mn ) 1 ( 111 2 (4.42) By considering the firstorder Tayl or series expansion about 1 of ij ij ln,we can obtain the following approximate formula for ij ln: 1 ln lnij ij ij ij (4.43) where conditional on ij 0 1 ij ijE and 2 1 2 2/ ij l k k i ij ijit Var .Hence, it is seen that il k k i ij ijt as1 2 ,, 0 1 (4.44) From equation (4.44), it is seen that th e asymptotic distribution of unconditional ij lnfollows an extreme value distribution with i ib uThus i ib u , the conventional maximum likelihood estimators (MLEs) of i ib u,, can be obtained directly (Lawless,1982) by: i ib n j i ij ub x n e 1 exp 1 and
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29 0 1 exp exp1 11 11 n j ij i m i n j i ij m i n j i ij ijx n b b x b x x where 1 1 ln n j m i xij ij Here iu and ib can be solved by using some numerical methods( e.g., Newtons met hod) with an iterative procedure. Based on the asymptotically efficient propert y of maximum likelihood estimate (Lawless, 1982), the joint density of iu and ib follows an asymptotically bivariate normal distribution as follows. , ~ i i i ib u N b u (4.45) where i i i i i ib Var b u Cov b u Cov u Var , = I1 (ui bi) denotes the covariance matrix of iu and ib The fisher information matrix I(ui bi) can be expressed as follows: , ln ln ln ln ,2 2 2 2 2 2 i i i i i i i i i i i i i i i ib b u L E b u b u L E b u b u L E u b u L E b u I where i i j i i j n j i i ib u x b u x b b u Lexp exp 1 ,1
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30 By using the technique of integration by parts, Var [iu ], Cov [iu ,ib ], and Var [ib ] can be obtained as follows. 2 2 2 21 6 6 n b u Vari i, (4.46) 1 6 2 2 n b b u Covi i i, (4.47) 2 26 n b b Vari i (4.48) Where (x) is the gamma function and = 0.5772 is the Eulers constant. In a real situation, the experiment is onl y conducted up to time tl .Thus, the parameters ) (i ib ucan be slightly calibrated by the conditional expectation technique. Assuming li lib u, denotes the parameters after re fined calibration, the approximate relations between li lib u, and ) (i ib ucan be expressed (Hong Fwu Yu, [20]) as follows: where: 2 1 1 2 2 2 2 2 2 22 1 6 il k k i i u i i i lit f t b b To assure that i il k k it1 2 ,is sufficiently large, it is reasonable to set 1 0 1 2 1 62 / 1 1 2 2 2 2 2 2 s m i s t f til k k i i u i i (4.49) ) (i li i lib b u u
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31 This equation indicates that the slower the quality characteristic of a product degrades, the longer the degradation test should last. Thus, b can be further estimated as follows: m i ib m b1 1 (4.410) Based on these estimators, 1 m i iuthe LSEs ) (1 0 of ) (1 0 in Equation (4.22) can be obtained as follo ws (Lawless [16]): Y X X XT T 1 1 0 where XT = ) ( ... ) ( ) ( 1 ... 1 12 1 mS X S X S X and m Tu u u Y ,........ 2 1 .Thus, u0 can be estimated by ) ( 0 1 0 0S X u (4.411) The approximate distribution of 0 u and b is as follows G H n b u N u ) 2 ( 6 6 ~ 2 2 2 0 0 (4.412) where ) ( ) ( 2 ) ( ) (1 0 2 0 2 1 m i i m i iS X S X S mX S X H and 2 1 2 1) ( ) ( m i i m i iS X S X m G 2 26 ~ n b b N b (4.413) The approximate ) 1 ( 1001p% and ) 1 ( 1002p % confidence interval (CI) of u0 and b can be obtained as follows:
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32 0 0 2 / 1 0 0 0 2 / 1 0 0 0 ) var( exp ) var( exp ,1 1u u u u u u u up p and b b b b b b b bp p var exp var exp ,2 / 1 2 / 12 2 Where: 2 / 11p is theth p ) 1 ( 1001percentile of standard normal distribution and 2 / 12p is theth p ) 1 ( 1002percentile of standard normal distribution. p1 and p2 are the percentile values for scale(u0) and shape(b) parameters respectively. Now, substituting b u ,0 and b u ,0 into Equation (4.32) we obtain an approximate )% 1 )( 1 ( 1002 1p p CI for 0 as follows: b u D b u D 1 exp 1 exp0 1 0 1 0 0 (4.414) 4.5 The Computation of 0 0 E By taking the natural logarithm of bo th sides of Eq (4.32), we have b u D n l fi i1 ln ln 1 , ln0 0 (4.51)
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33 The asymptotic distribution of n l fi i, ln0 follows a normal distribution 2,m N where 1 6 1 ln 1 2 1 ln 1 ln var 1 1 ln ln 12 2 2 2 0 2 2 0 n b b E b E u and b E u D m Hence, the asymptotic mean of n l fm i i i, 1 0 can be expressed as follows: 2 exp , 2 1 0 m n l f Em i i i (4.52) Therefore, 2 exp2 2 2 1 2 1 0 0 m m E where n b n b b E b E b E b E u u and b E b E u u D m m2 2 2 2 2 2 2 0 0 2 2 2 2 1 0 0 2 11 ln 1 ln 1 6 1 2 1 ln 1 ln 1 ln 1 ln var var 1 1 ln 1 ln ln 1
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344.6 The Cost Function n l f TCm i i i, ,1 The total cost n l f TCm i i i, ,1 of conducting an ADT is divided into three parts (Yu and Tseng [1, 2]). The cost of conducting an ADT is m i i i p i i m i sl f C l f C1 1max, where Cs denotes the operators sala ry per unit of time and Cp denotes the unit cost of power of the testing equipment. The measurement cost is m i i ml n C1,where Cm denotes the unit cost of measurement. The cost to test the devices is mn Cd,where Cd denotes the unit cost per device. Therefore, the total cost of the experiment is mn C l n C l f l f n l f TCd m i i m m i i i i i m i i i 1 1 p m i 1 s 1, C + max C , (4.61) 4.7 The Optimization Model From the foregoing results, the optimizat ion problem can be expressed as follows m i n N n l f p p p where p p p s k f t C mn C l n C l f C l f C t s m m Mini i l k i u b d m i i m i m i i p i i m i si, ,......... 3 2 1 2 , , 0 : 1 1 1 ) 73 4 ( 2 1 6 ) 72 4 ( , max . ) 71 4 ( 2 exp2 1 2 1 2 2 2 1 2 2 2 1 1 1 2 2 1 2 1
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354.8 Algorithm Due to the complexity of the objective functio n, it is difficult to find an efficient method to solve the optimization model in equati on (4.71). The objective function can be expressed as a function of p1 and n l fm i i i, ,1. Hence, with simplicity structure of the constraint and the integer restriction on the decision variables, an approximate solution can be obtained by the following steps. Let 0 0 1 1, , E n l f pm i i i. Partition the interval 1, 0 pequally into l (say, l=100) subintervals. Set l p k k p ) (1, k = 1,2,.,( l1). For each ) (1k p, the corresponding optimal combination k n k l k fm i i i, ), (1 can be obtained as follows. Given m i if1. Determine the corresponding m i il1by Equation (4.49). Determine the corresponding n. Compute 0 0 1 E f Vm i ifrom the test plan n l fm i i i, ,1 The optimal solution k n k l k fm i i i ,1 can be determined if m i if1 satisfy ......1 1 1 1min min) ( ) 1 ( 1m i i f f f f m i if V f Vm b m b (4.81) Where 1 2 1 6 12 2 2 2m i s t fG u i b
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36 and Gx denotes the largest inte ger not greater than x. In the minimization process of (4.61), any n l fm i i i, ,1 that does not satisfy the cost constraint would not be taken in to consideration. Finally, an approximate optimal solution 1 * 1, , n l f pm i i i can be determined if ) ( , , ,* 1 * 1 1 1 1 * 1mink n k l k f k p n l f pm i i i l k m i i i
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37 CHAPTER 5 EXAMPLE The reliability of electronic devices is of a critical concern especially for military, aerospace and communication applications. LEDs (light emitting diodes) are considered a good light source for optical links with good temperature dependence, small power consumption and high reliability. Since LEDs ar e designed to be in service for several years without failure, it is hard to observe failures under normal operating conditions in a short time. The reliability performance of LEDs (Light Emitting Diodes) has nearly always been superior to that of incandescen t, neon and other type lamps. In addition, todays LEDs have much higher reliabi lities than early LED devices. Improved assembly, growth methods and structures al ong with new materials have allowed for the development and mass production of extremel y reliable high brightness LEDs in all colors including white. The expected useable lifetime of an LED is usually estimated by the extrapolation of measured data or by estimating the value from accelerated testi ng. Accelerated testing involves subjecting the LED to more extreme conditions (i.e.: higher temperature and/or higher currents) than would be expected under normal operating conditions. This is necessary since it is often di fficult and impractical to actu ally test an LED for 100,000 hours or over 10 years. The main concern with accelerated te sting of LEDs is understanding how to accurately translate these results to normal operating conditions.
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38 The lifetime of an LED is defined as the time it takes for the light out put to reach 50% of its initial value. The average lifetime specified by LED manufacturers is 100,000 hours. This does not mean that the LED will cease to operate after 100K hours; in fact, most LEDs will function for thousands of hours beyon d the specified lifetime value. It means that after 100,000 hours, the LED w ill be half as bright as its initial luminosity level. In this chapter, the a pplicability of the proposed model is demonstrated by a numerical example. 5.1 Simulation Experiment The purpose of the simulation experiment is to generate the data that would be used to estimate the reliability of LEDs (type Ga AlAs) at normal operating condition with temperature S0 = 278 K (50 C), by using the degradation data obtained at the three accelerated stress levels, S1 = 298 K (250 C), S2 = 338 K (650 C), S3 = 378 K (10500 C). The data for twenty five LEDs were simulate d at each of these three temperatures. The duration of each cycle (Simulation run) is 336 hours and the total number of cycles is 29. Each cycle represents an inspection interval. Let ) ( tij denote the observed standardized light intensity of the jth LED at time t under Si. The data is simulated in Ma tlab by assuming the random variableij follows a reciprocal Weibull distributi on. By using the Arrhenius relationship between temperature and time, the degradation data was generated at the three stress levels S1, S2, and S3. The data represents the standardized light inte nsity of each component at a particular time 65 0t.The resulting data is given in tables 5.15.3. Figure 5.1 shows the simulated sample
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39 degradation paths of 25 LEDs. Figure 5.2 is the plots of ) ( tij versus 65 0t under S1 S2 and S3 It is seen from the figure that ther e exists a linear relationship between ) (tij and65 0tis given by: ) ( ) (65 0t t tij ij ij (5.1) Where ) (tij is the error term.
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40 Table 5.1: The Simulated Standardized Sa mple Degradation Paths at Stress Level S1 Time (hr) w11(t) w12(t) w13(t) w14(t) w15(t) w16(t) w17(t) w18(t) w19(t) w110(t) 18.33 03 1.07 935 1.106 374 1.117 466 1.133 543 1.125 094 1.10 0233 1.102 839 1.101 366 1.099 07 1.0 4425 25.92 296 1.126 22 1.060 427 1.066 732 1.073 545 1.050 061 1.06 5074 1.077 524 1.116 496 1.064 51 1.0803 37 31.74 902 1.055 46 1.047 119 1.050 312 1.053 935 1.055 861 1.07 712 1.054 351 1.058 936 1.053 86 1.054 77 36.66 061 1.047 18 1.053 95 1.085 407 1.057 926 1.050 733 1.05 7619 1.051 191 1.043 303 1.062 95 1.048 42 40.98 78 1.040 72 1.040 13 1.044 154 1.037 704 1.039 239 1.03 5047 1.039 479 1.049 797 1.057 07 1.0382 44 44.89 989 1.034 21 1.030 16 1.035 172 1.038 326 1.037 716 1.04 7078 1.038 366 1.029 211 1.039 72 1.0405 97 48.49 742 1.035 18 1.043 09 1.045 905 1.041 16 1.058 704 1.03 477 1.032 937 1.042 499 1.053 91 1.0378 75 51.84 593 1.038 25 1.038 73 1.037 903 1.035 722 1.031 493 1.03 3257 1.029 529 1.030 294 1.033 22 1.0364 45 54.99 091 1.035 45 1.037 78 1.039 389 1.029 118 1.032 946 1.02 45 1.028 547 1.028 461 1.022 74 1.033 16 57.96 551 1.027 26 1.044 42 1.026 739 1.028 678 1.027 194 1.02 7911 1.040 484 1.04 91 1.027 94 1.0347 01 60.79 474 1.026 56 1.027 71 1.029 465 1.030 429 1.024 977 1.02 6736 1.030 009 1.030 573 1.026 04 1.0253 27 63.49 803 1.034 37 1.023 95 1.027 438 1.026 053 1.045 125 1.02 6988 1.030 341 1.022 703 1.021 39 1.0336 62 66.09 085 1.021 36 1.020 01 1.029 822 1.028 299 1.023 832 1.02 9282 1.032 89 1.023 057 1.020 26 1.0555 58 68.58 571 1.026 11 1.032 15 1.042 356 1.025 861 1.022 144 1.01 9716 1.025 547 1.024 775 1.022 94 1.02 76 70.99 296 1.023 97 1.025 72 1.026 756 1.025 015 1.020 26 1.02 2398 1.023 347 1.028 618 1.049 82 1.0212 52 73.32 121 1.028 03 1.021 09 1.026 131 1.025 391 1.021 93 1.02 3118 1.023 271 1.020 993 1.024 96 1.0249 33 75.57 777 1.022 24 1.021 99 1.029 381 1.033 507 1.026 498 1.01 9545 1.019 913 1.018 471 1.026 39 1.0202 03 77.76 889 1.023 79 1.039 99 1.022 724 1.020 332 1.022 085 1.02 8844 1.022 826 1.027 802 1.016 34 1.0326 12 79.89 994 1.022 35 1.027 75 1.021 768 1.043 594 1.022 109 1.02 1503 1.034 442 1.020 03 1.024 71 1.0248 16 81.97 561 1.025 92 1.010 94 1.019 459 1.027 154 1.021 445 1.01 9957 1.020 114 1.020 628 1.018 07 1.0269 98 84 1.027 72 1.017 02 1.021 561 1.022 158 1.017 66 1.02 5087 1.019 705 1.022 86 1.024 41 1.018 34 85.97 674 1.012 59 1.029 28 1.023 346 1.018 301 1.031 948 1.02 2337 1.024 057 1.025 693 1.011 35 1.0202 77 87.90 904 1.023 93 1.029 74 1.018 872 1.026 758 1.019 456 1.02 0292 1.017 483 1.021 769 1.025 04 1.0188 83 89.79 978 1.021 29 1.022 05 1.019 215 1.01 69 1.017 946 1.01 7259 1.017 112 1.016 943 1.013 28 1.0181 43 91.65 151 1.021 26 1.019 31 1.019 346 1.029 531 1.026 679 1.02 4926 1.016 801 1.019 856 1.024 88 1.0155 01
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41 Table 5.2: The Simulated Standardized Sa mple Degradation Paths at Stress Level S2 Time (hr) w21(t) w22(t) w23(t) w24(t) w25(t) w26(t) w27(t) w28(t) w29(t) w210(t) 18.33 03 1.0830 64 1.0939 11 1.088 369 1.090 598 1.09 4499 1.092 493 1.079 407 1.096 592 1.158 085 1.0974 08 25.92 296 1.0634 51 1.0631 21 1.081 652 1.067 255 1.0 7079 1.06 906 1.061 419 1.080 809 1.069 296 1.0620 61 31.74 902 1.0557 11 1.0644 38 1.056 714 1.068 797 1.05 5978 1.049 084 1.052 435 1.053 248 1.060 183 1.0469 67 36.66 061 1.0542 08 1.0560 95 1.048 199 1.040 888 1.08 0672 1.046 832 1.050 334 1.047 707 1.048 236 1.0490 45 40.98 78 1.0479 02 1.0440 18 1.044 924 1.041 242 1.05 5953 1.04 147 1.041 537 1.047 809 1.051 648 1.0597 26 44.89 989 1.043 39 1.0431 06 1.04 219 1.048 251 1.04 1538 1.03 495 1.04 126 1.040 572 1.046 628 1.0339 62 48.49 742 1.035 19 1.0298 06 1.037 786 1.040 816 1.03 423 1.033 533 1.035 362 1.03 259 1.032 558 1.0369 29 51.84 593 1.0351 42 1.0314 49 1.045 936 1.038 761 1.03 1519 1.030 909 1.032 876 1.030 529 1.028 562 1.030 26 54.99 091 1.029 13 1.0260 32 1.032 026 1.028 897 1.04 1802 1.029 795 1.035 486 1.030 287 1.031 439 1.0330 77 57.96 551 1.0318 75 1.0400 95 1.036 833 1.034 542 1.02 5785 1.031 013 1.038 359 1.029 498 1.034 757 1.0276 13 60.79 474 1.0305 73 1.0266 84 1.038 962 1.026 103 1.02 6781 1.034 867 1.026 424 1.03 087 1.026 432 1.0309 75 63.49 803 1.0362 83 1.0344 69 1.029 039 1.027 316 1.03 4058 1.025 011 1.022 115 1.02 264 1.039 253 1.0210 25 66.09 085 1.0279 08 1.0255 32 1.032 858 1.021 379 1.02 9767 1.033 671 1.028 401 1.050 284 1.026 274 1.0205 01 68.58 571 1.051 73 1.025 11 1.022 872 1.025 943 1.02 736 1.029 152 1.030 734 1.024 586 1.025 578 1.0287 71 70.99 296 1.0333 09 1.0240 35 1.024 598 1.023 856 1.02 2682 1.023 191 1.028 628 1.022 958 1.027 683 1.0209 77 73.32 121 1.0179 47 1.0391 56 1.02 294 1.031 268 1.02 6811 1.021 873 1.023 791 1.02 178 1.031 946 1.0276 48 75.57 777 1.032 17 1.0243 48 1.022 624 1.020 449 1.03 0024 1.026 106 1.028 379 1.018 991 1.023 837 1.0262 98 77.76 889 1.0204 75 1.0201 05 1.02 263 1.017 911 1.02 2745 1.02 21 1.020 493 1.02 317 1.022 181 1.0196 69 79.89 994 1.0374 12 1.0227 64 1.018 497 1.022 932 1.02 2541 1.020 505 1.020 961 1.023 444 1.025 072 1.0171 31 81.97 561 1.0236 67 1.0328 42 1.02 225 1.022 134 1.02 7271 1.028 145 1.032 541 1.019 202 1.023 479 1.0175 26 84 1.0234 12 1.022 72 1.016 485 1.021 718 1.02 5872 1.022 958 1.020 715 1.018 847 1.021 911 1.0200 42 85.97 674 1.0264 91 1.0160 62 1.015 382 1.019 268 1.02 1383 1.023 284 1.02 077 1.016 924 1.017 539 1.0200 03 87.90 904 1.019 57 1.019 33 1.020 496 1.020 545 1.01 9898 1.018 306 1.016 381 1.020 434 1.019 204 1.0160 82 89.79 978 1.0330 29 1.020 61 1.016 696 1.018 849 1.01 5128 1.020 583 1.02 069 1.018 512 1.021 257 1.0237 01 91.65 151 1.0222 42 1.0175 28 1.021 078 1.027 367 1.01 7643 1.016 869 1.020 708 1.015 785 1.02 264 1.0222 33
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42 Table 5.3: The Simulated Standardized Sa mple Degradation Paths at Stress Level S3 Time (hr) w31(t) w32(t) w33(t) w34(t) w35(t) w36(t) w37(t) w38(t) w39(t) w310(t) 18.3 303 1.1167 13 1.116 424 1.07 611 1.08 5523 1.0803 13 1.108 905 1.0912 78 1.0914 49 1.1685 91 1.0930 37 25.92 296 1.0610 79 1.075 182 1.065 665 1.07 7985 1.0593 66 1.109 148 1.0566 61 1.052 38 1.0681 54 1.0631 44 31.74 902 1.0546 74 1.052 407 1.04 596 1.05 2782 1.0887 81 1.047 037 1.0449 86 1.0458 65 1.064 88 1.0527 65 36.66 061 1.0647 89 1.047 319 1.039 176 1.05 7227 1.0477 26 1.045 485 1.0477 74 1.0440 76 1.0511 49 1.0482 09 40.9 878 1.0460 79 1.05 674 1.044 139 1.04 0924 1.0623 97 1.048 351 1.0523 61 1.0412 78 1.0372 27 1.0352 44 44.89 989 1.0398 16 1.037 031 1.033 922 1.03 8509 1.0334 96 1.037 023 1.0358 39 1.0435 66 1.0743 23 1.0554 03 48.49 742 1.0321 78 1.036 642 1.037 947 1.03 4645 1.0391 46 1.037 357 1.0325 25 1.0412 78 1.0418 59 1.0367 32 51.84 593 1.0333 59 1.045 759 1.030 183 1.03 4775 1.0403 58 1.036 747 1.0308 82 1.0296 37 1.0471 75 1.0433 19 54.99 091 1.0337 93 1.039 719 1.027 402 1.02 9583 1.0309 77 1.038 164 1.0279 43 1.0342 62 1.0278 94 1.0358 69 57.96 551 1.0324 44 1.028 069 1.029 241 1.03 0035 1.0260 42 1.033 548 1.0284 64 1.029 35 1.0334 95 1.0254 66 60.79 474 1.027 56 1.02 863 1.023 949 1.03 0439 1.025 29 1.027 118 1.0340 97 1.0321 99 1.0270 59 1.0320 03 63.49 803 1.0288 33 1.023 803 1.045 537 1.04 7665 1.0261 24 1.029 232 1.0285 64 1.031 01 1.0290 18 1.0279 49 66.09 085 1.0267 27 1.032 092 1.027 556 1.02 5928 1.029 98 1.027 416 1.0291 07 1.0239 95 1.0240 48 1.0296 52 68.58 571 1.0322 06 1.030 197 1.019 919 1.03 2557 1.0242 54 1.032 864 1.028 94 1.0226 71 1.0265 92 1.0331 31 70.99 296 1.0218 81 1.024 141 1.027 005 1.0 2323 1.0211 14 1.027 875 1.0221 82 1.0311 82 1.0306 73 1.023 83 73.32 121 1.0267 63 1.023 206 1.028 572 1.03 4996 1.0205 36 1.024 918 1.0285 37 1.0389 58 1.0241 94 1.0301 05 75.57 777 1.0177 85 1.020 032 1.029 161 1.02 4041 1.0258 08 1.029 499 1.0217 35 1.0323 76 1.0185 48 1.0264 91 77.76 889 1.0238 26 1.022 353 1.021 229 1.02 1767 1.0256 63 1.023 262 1.0223 58 1.0191 12 1.022 78 1.0513 91 79.89 994 1.0460 34 1.022 948 1.022 963 1.01 9117 1.0187 48 1.020 358 1.0207 31 1.0199 56 1.0219 42 1.0202 62 81.97 561 1.029 18 1.024 729 1.027 092 1.0 1972 1.0313 39 1.01 666 1.0235 44 1.0335 03 1.0241 97 1.0234 29 84 1.0267 64 1.019 094 1.018 628 1.02 3794 1.0204 13 1.024 631 1.0341 46 1.0210 94 1.0208 07 1.0199 64 85.97 674 1.0239 56 1.01 79 1.019 372 1.02 0533 1.0216 07 1.019 568 1.0191 45 1.0178 28 1.0168 44 1.0175 77 87.90 904 1.0238 76 1.022 272 1.019 764 1.02 1695 1.0170 11 1.018 452 1.0175 67 1.0201 93 1.0199 67 1.0207 08 89.79 978 1.0190 46 1.020 171 1.01 824 1.01 7358 1.0165 58 1.036 252 1.0173 22 1.0228 77 1.0198 05 1.0189 59 91.65 151 1.0177 78 1.019 853 1.016 095 1.0 1622 1.0227 31 1.019 442 1.0228 78 1.0166 84 1.0237 61 1.0157 79
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43 1.4E23 1.2E23 1E23 8E24 6E24 4E24 2E24 033 6 1008 1 680 23 5 2 30 2 4 3 696 4 36 8 50 4 0 5 712 6 38 4 70 5 6 7 728 8 40 0 90 7 2time(hour)wij(t) (a) 1.4E20 1.2E20 1E20 8E21 6E21 4E21 2E21 0336 1008 1680 2352 3024 3696 4368 5040 5712 6384 7056 7728 8400 9072 9744time(hour) (b) 3E18 2.5E18 2E18 1.5E18 1E18 5E19 0336 1008 1680 2352 3024 3696 4368 5040 5712 6384 7056 7728 8400 9072 9744time(hour)w3j(t) (c) Figure 5.1. The Standardized Samp le Degradation Paths under (a) S1, (b) S2 and (c) S3
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44 1.4E23 1.2E23 1E23 8E24 6E24 4E24 2E24 043 8 6 52 42 1 08 .00 89 155 .3 94 19 5 .938 8 4 23 2.3 71 95 265 .9 4 91 29 7.3 79 05 327.111 09 35 5.4 52 47 382.624 92 t^0.65w 1 j(t ) (a) 1.4E20 1.2E20 1E20 8E21 6E21 4E21 2E21 043.865242 89. 5 87898 12 4. 86796 15 5.394 182. 9 6927 208.46146 232 37195 255. 023 29 276. 63 831 297. 3 7905 317.36795 336 70039 355. 452 47 373. 686 13 391. 4 5264t^0.65w2j(t) (b) 3E18 2.5E18 2E18 1.5E18 1E18 5E19 043.86 5 24 2 10 8.008 9 15 5.39 4 195 9 3 88 4 232.37 1 9 5 265.9491 297 3 7 90 5 327.11 1 0 9 35 5.45 24 7 382 6 2 49 2 t^0.65w3j(t) (c) Figure 5.2. The Plots of ij (t) versus t0.65 under (a) S1, (b) S2 and (c) S3
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45 Based on the observations 30 1 ,, k k i ij k it t, the LSEs ij and 2ij can be computed. To make sure of the appropriateness of the We ibulldistribution, Weibull probability plots were constructed for each higher stress level (Figure 5.3). All of the trends appear linear about the reference line. Figure 5.4 shows th e normal probability plots for the residuals under S1, S2, and S3. The plots indicate that the distribution assumptions for and ) (t are reasonable. From Equations 4.21, 4.22, 4.210 and 4.211, we have 2= 2.12683 16, b = 0.0268, and the Arrhenius re lationship is given by: 16 273 1 0 i iS u (5.2) Where 0 = 0.9977 and 1 = 62.3124. To obtain the optimal test plan for the ADT of LED, we need the actual values of2, b, and1 0, For convenience, these estimates are trea ted as the true values to evaluate the optimal test plan of LED data.
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46 (a) (b) (c) Figure 5.3 The Weibull Probability Plot for 3 1 25 1 ij ij
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47 (a) (b) (c) Figure 5.4 The Normal Probability Plots for Residuals under (a) S1, (b) S2 (c) S3
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485.2 Optimal Test Plan Based on the LED Data We now solve for the optimal parame ters of the objective function at 24ut hours and100 l. In solving the objective functi on we compute the following: Sample Size Inspection Frequency Termination Time and CIs of the parameters involve d in the MTTFs expression The optimal combination of the CIs of the pa rameters involved in the MTTFs expression is found such that the expected width of a 100(1p) % CI of the MTTF is minimized. The CI width of the MTTF is ) 1 )( 1 ( 100* 2 1 0 0p p E pU L .Where Land Uare the lower and upper limits of 0 0 E respectively. The optimal test plans for the p = 0.01, 0.05, 0.10, 0.20, 0.30, 0.40 and 0.50 under different cost conditions (Cs, Cp, Cm, Cd, Cb) are listed in Table 5.4. Here p is th e percentile of the lifetime distribution of the product at the norm al use condition. Table 5.4: Optimal Degradation Test Plans under Various Cost Conditions (Cs, Cp, Cm, Cd, Cb) p (Cs, Cp, Cm, Cd, Cb) 0 0 E * 3 2 1 3 2 1 2 1, , , , ,n l l l f f f p p 0.01 (16.5,4.05,0.65,35,10000) 697298.6(0.0087,0.00142,2,4,7,62,29,14,38) 0.05 (16.5,4.05,0.65,35,10000) 685465.5(0.0430,0.00731,1,5,8,100,42,10,44) 0.10 (16.5,4.05,0.65,35,10000) 575512.6(0.086,0.0153,2,4,7,62,30,14,38)
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49 (Table 5.4 continued) 0.20 (16.5,4.05,0.65,35,10000) 547801.8(0.086,0.0153,2,4,7,62,29,15,38) 0.30 (16.5,4.05,0.65,35,10000) 503478.3(0.087,0.0213,2,4,9,65,30,15,39) 0.40 (16.5,4.05,0.65,35,10000) 499734.1(0.091,0.0309,3,4,9,62,29,14,39) 0.50 (16.5,4.05,0.65,35,10000) 448923.4(0.085,0.0415,2,4,8,60,28,14,38) For example, with p=0.1 and (Cs, Cp, Cm, Cd, Cb) = (16.5, 4.05, 0.65,35, 10000), the optimal combination of the percentile values 2 1,p pis 2 1,p p= (0.086, 0.0153); the optimal inspection intervals equal 1f*24 = 2*24= 48 hours, 2f*24 = 4*24= 96 hours, and 3f*24= 7*24= 168 hours under S1, S2 and S3 respectively. The optimal sample size is n* = 38; and the optimal termination times are2976 24 62 2* 1* 1 lt 2* 2lt = 4*30*24 = 2880, 3* 3lt = 7*14*24 = 2352 hours under S1, S2 and S3 respectively. The CI width of the MTTF in this case is: U LE p0 0 100(10.0087) (10.00142) = 98.98 %. 5.2.1 Optimal Parameters Base d on the ADT Experiment In the table 5.4, as the percentile (p) va lue changes from 0.1 to 0.5, the expected width changes from 697298.6 to 448923.4. Corre spondingly, the percentage change in the width is 35 % whereas the percentage change in the percentile value p1 is 9 %. Thus, there is more significant reduction in the vari ation (CI width) than in the precision (p). Based on these results the optimal parameters for the design of this ADT experiment are: The percentile value p = 0.5 The optimal value of the expected width of the MTTF = 448923.4
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50 The optimal combination of the percen tile values for the expected width 2 1, p p = (0.085, 0.0415) The optimal inspection intervals 3 2 1, f f f = (2, 4, 8) The optimal termination times 3 2 1* 3 2 1, ,l l lt t t = (60, 28, 14) The optimal sample size n* = 38 The optimal CI width of the MTTF: U LE p0 0 100(10.085) (10.0415) = 87.70 %. 5.3 Sensitivity Analysis In practical situation, some parame ters of the LED problem (e.g., u, b,2, the cost parameters, etc) in the previous section are not well known. Thus, it is important to investigate the effects of these parameters on the optimal test plan. Considering the following cases for the above LED data, the e ffects of these parameters can be found. 5.3.1 Test Plans under a Variety of Cb The effect of Cb on the test plan can be assessed by computing 3 1 *, n l fi i i for various values of Cb with p = 0.1, and the cost condition ( Cs, Cp, Cm, Cd, ) = (16.5, 4.05, 0.65, 35). The results are given in Table 5. 5. It can be seen that all n* and 3 1 *i if and 3 1 *i il are sensitive to the moderate change of Cb. Table 5.5: The Optimal Test Plans for Some Values of Cb p Cb 0 0 E * 3 2 1 3 2 1, , , n l l l f f f 0.1 7000 55134.47 (1,2,2,92,44,30,21)
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51 (Table 5.5 Continued) 7500 8000 8500 9000 9500 10500 11000 11500 12000 52388.07 48945.37 46104.25 43707.46 41649.93 38785.14 37326.47 35615.43 34474.94 (1,2,2,92,44,30,23) (1,2,3,92,44,24,26) (2,4,4,62,29,20,29) (2,4,6,62,29,16,32) (2,4,7,62,29,14,35) (2,3,7,62,35,14,40) (2,3,7,62,35,14,43) (3,7,9,49,21,12,47) (3,6,9,49,23,12,50) 5.3.2 Test Plans for Different Values of m and a Variety of Combinations of m i iS1 The number of stress levels and the choice of m i iS1 in an ADT would affect the optimal test plan and the estimation precision. To assess the effects of the number of the stress levels and the choice of m i iS1 on the test plan, we assume that the scale parameter u satisfies Eq (5.2). Table 5.6 gives the optimal solutions for a variety of values of m and various combinations of m i iS1 with p = 0.1 and =100 under the cost condition ( Cs, Cp, Cm, Cd, Cb) = (16.5, 4.05, 0.65, 35, 10000). The results indicate that The estimation precision is better when th e stress levels are father away from each other and from the use condition S0
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52 There is only a moderate change in th e inspection frequency and termination time when the number of stress levels varies, but there is a drastic change in the sample size. Table 5.6: The Optimal Test Plans for Various Values of m and Combinations of Stress Levels m S1 S2 S3 S4 0 0 E * 4 3 2 1 4 3 2 1, , , , n l l l l f f f f 2 25 65 30 105 35 105 40 65 55 80 60 80 667891.74 889389.42 982861.30 872079.37 938174.21 1248977.80 2 4 62 14 92 2 10 59 19 89 4 12 33 9 54 1 3 32 5 44 2 8 46 9 32 2 8 27 14 34 3 25 65 105 30 65 100 35 65 95 40 65 90 55 80 105 60 80 100 685629.87 896120.78 907847.52 918733.29 948237.57 1187768.31 1 4 8 82 12 9 72 1 5 8 86 12 12 69 3 9 12 86 12 19 72 2 5 8 75 10 22 67 2 6 7 69 14 23 58 3 9 14 89 14 24 62 4 25 45 65 105 30 50 75 95 35 50 75 95 40 60 80 100 50 65 80 95 727863.67 896389.20 896782.11 947820.29 1092948.72 2 4 7 7 89 56 45 14 48 5 9 10 11 78 45 38 7 56 3 7 9 9 67 36 27 15 44 3 9 13 18 64 57 29 9 56 3 8 9 9 61 44 32 11 29
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535.3.3 Sensitivity Analysis of Mis specifying u, b, and2 Let1 ,2 and 3 denote the predicted errors for u, b, and 2 respectively. Table 5.4 shows the optimal solution 3 1 *, n l fi i i for various combinations of 2 3 2 11 1 1 b u under the cost condition ( Cs, Cp, Cm, Cd, Cb) = (16.5, 4.05, 0.65, 35, 10000). Here the values of 3 1 i i are changed over the ranges% 10 % 5 %, 5 2 and. The optimal test plans are li sted in Table 5.7, 5.8 & 5.9 respectively. Table 5.7: The Optimal Solution 3 1 *, n l fi i i for the Case that 3 1 i iare Changed over the Ranges% 5 2 1 2 3 n l l l f f f , , ,* 3 2 1 3 2 1 2.5% 2.5% 2.5% 0 0 0 +2.5% +2.5% +2.5% 2.5% 0 +2.5% 2.5% 0 +2.5% 2.5% 0 +2.5% 2.5% 0 +2.5% 0 +2.5% 2.5% +2.5% 2.5% 0 1 3 5 38 19 12 44 1 3 6 39 24 14 43 2 3 6 40 24 14 42 2 4 7 39 19 14 42 1 3 8 38 18 9 42 1 3 8 38 19 10 42 2 4 7 40 18 9 43 2 4 7 39 19 9 40 3 5 7 38 19 10 40
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54 Table 5.8: The Optimal Solution 3 1 *, n l fi i i for the Case that 3 1 i iare Changed over the Ranges% 5 1 2 3 n l l l f f f , , ,* 3 2 1 3 2 1 5% 5% 5% 0 0 0 +5% +5% +5% 5% 0 +5% 5% 0 +5% 5% 0 +5% 5% 0 +5% 0 +5% 5% +5% 5% 0 4 6 8 68 32 14 45 4 6 8 67 30 14 44 3 6 9 62 29 14 39 3 6 8 62 30 14 39 3 6 9 62 30 11 38 4 7 9 59 31 14 39 3 6 8 60 32 11 44 4 6 8 59 34 12 40 2 7 9 58 32 16 41 Table 5.9: The Optimal Solution 3 1 *, n l fi i i for the Case that 3 1 i iare Changed over the Ranges% 10 1 2 3 n l l l f f f , , ,* 3 2 1 3 2 1 10% 10% 10% 0 0 10% 0 +10% 10% 0 10% 0 +10% 0 +10% 4 7 9 45 32 18 38 5 7 9 64 32 16 39 5 7 9 63 32 16 38 5 6 9 48 32 19 39 4 6 8 63 32 18 40
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55 (Table 5.9 Continued) 0 +10% +10% +10% +10% 10% 0 +10% 10% +10% 10% 0 4 5 7 62 33 18 39 4 7 8 63 32 17 39 4 6 9 63 32 17 38 4 6 9 62 31 16 39 From the results above, it is clear that the test plan is quite robust for a moderate deviation from the assumed values of b and 2.On the other hand, if the true value of u has a moderate change from the assumed value, then the test plan will also be changed moderately.
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56 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH In this thesis, we developed an optim al plan for the design of accelerated degradation test (ADT) usi ng reciprocal Weibull degrad ation data based on the minimization of the mean time to failure (MTTF) criterion. A nonlinear integer programming problem was developed under the constraint that the total experimental cost does not exceed a predetermined budget. The optimal combination of sample size, inspection freque ncy and the termination time at each stress level was found. An LED example was used to illustrate the proposed method by a simulation experiment to estimate the reliability of LEDs at normal operating condition with temperature S0 = 278 K (50 C), by using the degradation data obtained at three accelerated stress levels, S1 = 298 K (250 C), S2 = 338 K(650 C), S3 = 378 K ( 1050 C). By solving the optimization model developed in chapter 4, we determined the optimal values of the sample size, inspection frequency and the te rmination time. In addition the expected width of a 100(1p) % CI of the MTTF was al so minimized. From the results the optimal parameters for the design of ADT are: The percentile value p = 0.5 The optimal value of the exp ected width of the MTTF = 448923.4 The optimal combination of the percen tile values for the expected width 2 1, p p = (0.085, 0.0415)
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57 The optimal inspection intervals 3 2 1, f f f = (2, 4, 8) The optimal termination times 3 2 1* 3 2 1, ,l l lt t t = (60, 28, 14) The optimal sample size n* = 38 The CI width of the MTTF U LE p0 0 100(10.085) (10.0415) = 87.70 %. Sensitivity analysis was performed by va rying the cost and by varying the test plan with different combinations of stress leve ls to find the effect of different parameters on the optimal plan. The results from the sensitivity analysis indicate that the parameters sample size, inspection freque ncy and the termination time are sensitive to the moderate change in the cost. The estimation precision wa s better when the stress levels are father away from each other and from the use condition S0. There was only a moderate change in the inspection frequency and termination tim e when the number of stress levels varies, but there was a drastic change in the sample size. 6.1 Future Research Directions In this research, we developed the op timal test plan for an ADT under the assumption that the sample sizes for each stress level are equal. However this may not be practical because unequal alloca tion is much more common. Therefore research into the problem of unequal allocation is an important area that should be explored. Although ADT is an efficient life test met hod, it may not be applicable in some cases. For a newly developed product, it is very difficult to have many units to be put for testing at each stress level. In that case a StepStress Accelerated
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58 Degradation Test (SSADT), where a sample of tested devices is subjected to successively higher levels of stress can be used.
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59 REFERENCES 1. Yu, H.F., and Chiao, C.H., Designing an Accelerated Degradation Experiment by Optimizing the Interval Estimation of the MeanTimeToFailure, Journal of the Chinese Institute of Industrial Engineers, Vol.19, No.5, pp., 2333,2002. 2. Yu, H.F, Designing an Accelerated De gradation Experiment by Optimizing the Estimation of the Percentile, Quality And Reliability E ngineering International, 2003. 3. Nelson, Wayne, Accelerated Testing: Statistical Models, Test Plans and Data Analysis John Wiley & Sons, New York, 1990. 4. Carey, M.B., and Koenig, R.H., Reliabi lity assessment based on accelerated degradation, IEEE Transactions on Reliability, Vol.40, pp., 499506, 1991. 5. Meeker, W.Q., and Escobar, LA., A revi ew of research and current issues in accelerated testing, International Statistical Review Vol.61, No.1, pp., 147168,1993. 6. Shiau, J.H., and Lin, H.H., Analyzi ng Accelerated Degradation Data by Nonparametric regression, IEEE Transactions on Reliability Vol.48, No.2, pp., 149157, June 1999. 7. Boulanger, M., and Escobar, L.A., Experi mental design for a class of accelerated degradation tests, Techno metrics, Vol. 36, pp., 260272, 1994. 8. Tseng, S.T., and Yu, H.F., A Termination Rule for Degradation Experiments IEEE Transactions on Reliability, Vol.46, No.1, pp., 130133, March 1997. 9. Park, J.I., and Yum, B.J., Optimal de sign of accelerated degradation tests for estimating mean lifetime at the use condition, Engineering Optimization, Vol. 28, pp., 199230, 1997. 10. Tseng, ST., Hamada, M., and Chiao, CH., Using degradation data to improve fluorescent lamp reliability, Journal of Quality and Technology Vol. 27, pp., 363369, 1995. 11. Meeker, W.Q., and Escobar, LA., Statistical methods for reliability data, John Wiley & Sons, New York, 1998.
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60 12. Yu, H.F., and Tseng, S.T., Desi gning a degradation experiment, Naval Research Logistics Vol.46, pp., 689706, 1997. 13. Yu, H.F., and Tseng, S.T., Online pro cedure for terminating an accelerated degradation test, Statistica Sinica, Vol.8, No.1, pp., 207220, 1998. 14. Yu, H.F, Optimal selection of the most reliable product with degradation data Engineering Optimization Vol.34,pp., 579590, 2002. 15. Chao, M.T, Degradation Analysis and Related Topics: Some thoughts and a Review, Institute of Statistical Science, Vol.23, No.5, pp., 555566, 1999. 16. Lawless, J.F, Statistical models and methods for life time data, John Wiley & Sons, New York, 1982. 17. Meeker, W.Q., and Hamada, Michel, Sta tistical Tools for Rapid Development & Evaluation of HighlyReliable Products, IEEE transactions on reliability Vol.44, No.2, pp., 187198, 1995. 18. Lu, C.J., and Meeker, W.Q., Using Degr adation Data Measures to Estimate a TimeToFailure Distribution, Technometrics, Vol.37, No.2, pp.133146, May 1995. 19. Meeker, W.Q., and LuValle, M.J., An A ccelerated Life Test Model Based on Reliability Kinetics, Technometrics, Vol.37, No.2, pp., 133146, May 1995. 20. HongFwu Yu., Designing A Degradati on Experiment With A Reciprocal Weibull Degradation Rate, Journal of Quality Technology and Quantitative management, Jan 2003. 21. N.L.Johnson, S.Kotz, and N.Balakrishnan, Continuous Univariate Distributions John Wiley & Sons, Vol.1, 2nd Ed, 1995.
