Review on Energy Resilience

Reliability Engineering and System Safety 150 (2016) 126–135

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Reliability Engineering and System Safety

http://d 0951-83

n Corr E-m

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journal homepage: www.elsevier.com/locate/ress

Measurement errors in degradation-based burn-in

Qingqing Zhai a, Zhi-Sheng Ye a,n, Jun Yang b, Yu Zhao b

a Department of Industrial and Systems Engineering, National University of Singapore, Singapore b School of Reliability and Systems Engineering, Beihang University, Beijing, China

a r t i c l e i n f o

Article history: Received 5 May 2015 Received in revised form 19 January 2016 Accepted 27 January 2016 Available online 18 February 2016

Keywords: Burn-in test Degradation Wiener process Measurement error Cost model

x.doi.org/10.1016/j.ress.2016.01.015 20/& 2016 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ65 6601 2303; fax: ail addresses: [email protected] (Q. Zhai), [email protected] (J. Yang), [email protected]

a b s t r a c t

Burn-in is an effective tool to improve product reliability and reduce field failure costs before a product is sold to customers. As many products are becoming highly reliable, traditional burn-in that tests a batch of a product until most weak units fail requires an unaffordable testing duration. If the product failure can be associated with an underlying degradation process and a weak unit degrades faster than a normal one, then degradation-based burn-in can be implemented. Due to such various factors as human errors and limited precision of the measurement device, measurement errors are often inevitable. Ignoring measurement errors in the degradation observations would lead to inferior burn-in decisions. This study uses the Wiener process to model the underlying degradation and considers Gaussian measurement errors in the observations. Two burn-in models with different cost structures are studied and the optimal cutoff level for each model is obtained analytically. The relation between the two models is discussed, leading to a new cost model.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

With the increasing competition in the global market, compa- nies are urged to design and manufacture products with high quality. Nevertheless, it is inevitable that some defects are intro- duced by the inherent variability of the materials used and the manufacturing process itself [1], leading to a small portion of weak units with high failure rate or fast degradation. The weak units should be identified and scrapped before delivered to customers. Otherwise, losses including warranty costs and reputation will be incurred. Burn-in has been proven to be efficient in identifying and eliminating these weak units. It is also widely adopted in real production.

Many burn-in studies focus on the lifetime distribution of the product and assume a short period of decreasing failure rate of the product [2–8]. With burn-in, the product is operated in conditions similar to field environments so that most early failures will be induced in the test. Accelerated burn-in may be used by elevating the test conditions, such as temperature and voltage, to shorten the test [9,10]. The shock is another method of burn-in to elim- inate the weak units [11]. After the test, a useful life period with stable failure rate is reached and the product can function con- sistently for a relatively long time. From this point of view, burn-in

þ65 6777 1434. [email protected] (Z.-S. Ye), n (Y. Zhao).

test is a pre-field operation to have the company suffer some disposal costs of weak units instead of administrative and war- ranty cost from early field failures.

However, as argued by Tseng et al. [12], it becomes more and more unrealistic for today’s highly reliable products to fail in a reasonable burn-in duration, even under accelerated test condi- tions. On the other hand, if a quality characteristic (QC) can properly reveal the status of the product and its degradation is correlated to the failure of the product, it is possible to identify and scrap the weak units by detecting the degradation trend. Based on this argument, some degradation-based burn-in models have been developed in the literature. Wiener processes (especially with linear drift) have been widely used in the product degradation modeling [13–20], and they have been introduced in degradation- based burn-in. For example, Tseng et al. [12] used the Wiener process to model the degradation path and they studied the optimal burn-in strategy based on this degradation model. Ye et al. [21] studied the optimal burn-in strategy based on the Wiener process by considering preventive maintenance during the field operation. Motivated by the fact that the degradation may be not significant, Tseng and Peng [22] applied the integrated Wiener process to model the degradation and discussed the optimal burn- in strategy. Alternatively, some other studies used the Gamma process for the degradation modeling, e.g., [23,24].

The degradation should be measured by human or auto- matically by some measurement device. In any case, it is inevitable that measurement errors, e.g., due to the inaccuracy of the

Q. Zhai et al. / Reliability Engineering and System Safety 150 (2016) 126–135 127

measurement device, would be introduced [16]. Therefore, when designing the burn-in test, it should take measurement errors into consideration. Some researchers have considered measurement errors in studying the degradation data. Whitmore [25] first stu- died the statistical inference for degradation data by the Wiener process considering measurement errors. Peng and Tseng [13] used the Gaussian distribution to model the unit-to-unit hetero- geneity of the degradation rate and proposed a more general Wiener degradation model with measurement errors. Tang et al. [14] utilized the Wiener process with measurement errors in the remaining useful life prediction for the prognostics and health management of Li-ion batteries. Recently, Peng [26] used a mixed Wiener process to account for the heterogeneity in the population and proposed an optimal burn-in strategy for such a product.

The above studies focus much on the degradation data analysis with the Wiener process considering measurement errors, and some of them simultaneously considered the data analysis in the burn-in problem. Undoubtedly, the optimal burn-in strategy depends on the specific cost structure. Therefore, this paper focuses on the optimal burn-in strategies under different cost structures. We consider two different cost models in this paper, i.e., the misclassification cost model and the field failure cost model, and obtain the optimal cutoff levels under the two different cost models. The impact of measurement errors under the first model is further investigated. The relation of the two cost models is discussed and a new model is proposed based on the discussion. The new cost model compromises the two cost models, which approximates the second model but is more analytical tractable.

The remainder of the paper is organized as follows. Section 2 describes the model and the assumptions in the paper. Section 3 discusses the optimal burn-in strategy under the first cost model, i.e., the model considering the misclassification cost of burn-in, and investigates the impacts of measurement errors. Section 4 studies the optimal burn-in strategy under the second cost model, i.e., the model considering the field failure cost. In Section 5, we demonstrate that the two cost models are consistent, and a new cost model is proposed based on the comparison. An example is studied in Section 6 to illustrate the analysis in this paper. Con- cluding remarks are given in the end.

2. Model description

2.1. Wiener process with linear drift for degradation modeling

Considering a specific QC of the product, the underlying degradation of which, X tð Þ, follows a Wiener process with positive linear drift (probably after some time-scale transformation, as in [12]), i.e.,

X tð Þ ¼ μtþσB tð Þ;

where μ40 is the drift rate, σ40 is the diffusion coefficient, and B tð Þ is the standard Brownian motion. As the process has inde- pendent and Gaussian distributed increments, i.e., X tð Þ�X uð Þ is independent of X uð Þ for t4u and it follows the Gaussian dis- tribution N μ t�uð Þ; σ2 t�uð Þ

� � , the distribution of the degradation

X tð Þ at fixed time t follows the Gaussian distribution N μt; σ2t � �

. If the pre-determined threshold for X tð Þ is Xf 40, then the first

passage time

T ¼ inf tZ0 XðtÞZXf �� ��

follows an inverse Gaussian distribution with a probability density function (PDF) and a cumulative distribution function (CDF) given

by [27]

f IG t;β; λ � �

¼ λ 2πt3

� �1 2

exp �λ t�β � �2 2β2t

( ) ;

and

FIG t;β; λ � �

¼ Φ ffiffiffi λ t

r t β �1

� � ! þexp 2λ

β

� UΦ �

ffiffiffi λ t

r t β þ1

� � ! ;

respectively, where β ¼ Xf =μ, λ ¼ X2f =σ2, and Φ Uð Þ is the CDF of the standard Gaussian distribution. Apparently, T rt indicates that max

0rsrt X sð ÞZXf .

2.2. Two subpopulations assumption

Due to defects introduced during manufacturing, e.g., for Micro-Electro-Mechanical Systems (MEMS) [28,29], a fraction of the product would be defective and exhibit fast degradation. Hence, it is common to assume two classes in the product popu- lation, i.e., the weak and the normal classes, as in [2,3,11,21]. We assume that degradation of the normal class follows a Wiener process with drift rate μ1 40 and diffusion coefficient σ, while the weak class degrades following a Wiener process with the same σ but larger drift rate μ2 4μ1. The proportions of the normal and the weak classes are p1 40 and p2 40, respectively, where p1 þp2 ¼ 1.

To identify and eliminate weak units from the main population, a burn-in test is carried out. The duration of the test is denoted by b. After the test, the degradation of each unit is measured and the unit is discarded if the measured degradation is greater than a cutoff level ξb. Due to the inaccuracy of the measurement (resulting from human or equipment), it is assumed that the observed degradation is

Yi tð Þ ¼ Xi tð Þþϵ; i ¼ 1; 2 where ϵ � N 0; σ2ϵ

� � is the measurement error and Xi tð Þ ¼ μitþσB tð Þ

is the underlying degradation path of the normal (i ¼ 1) or the weak class (i ¼ 2). It is assumed that the measurement error ϵ is independent of the degradation Xi tð Þ. Therefore, the measured degradation Yi tð Þ at a fixed time t follows the Gaussian distribution N μit; σ

2tþσ2ϵ � �

. Burn-in can eliminate weak units and thus influence the life-

cycle cost. In the following, we consider two different cost models and study the optimal cutoff level ξb and the burn-in duration b under each model. The first cost model (Model 1) involves the cost of misclassification of different classes, while the second cost model (Model 2) considers the field failure cost. It should be mentioned that the related cost parameters are assumed known in this study. For the case where some cost parameters, e.g., the misclassification cost cα and cβ, are not specified, Wu and Xie [30] proposed a receiver operating characteristic based approach for burn-in decision-making. Interested readers are suggested to refer to Wu and Xie [30] for details.

3. Optimal burn-in test considering misclassification

For products suffering degradation, burn-in can be applied to identify and discard defective units. However, due to the ran- domness of the QC or measurement errors, normal units may be misclassified as weak while weak units may also muddle through burn-in. Such misclassification would introduce the so-called misclassification cost [12], which contributes to the overall burn- in cost that should be optimized. Such misclassification cost has been considered in burn-in models by e.g., Tseng et al. [12], Tseng and Peng [22] and Tsai et al. [1]. This section studies the optimal burn-in strategy considering the misclassification effect.

Q. Zhai et al. / Reliability Engineering and System Safety 150 (2016) 126–135128

3.1. Burn-in cost formulation

After the burn-in test, the degradation level of each unit is measured and compared with the cutoff level ξb. The probability that a unit passes the burn-in test, i.e., Y bð Þoξb, is

Pr Y bð Þoξb � �

¼ p1Φ ξb �μ1bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

þp2Φ ξb �μ2bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

: ð1Þ

An accepted unit may belong to the weak class or the normal class. As weak units tend to fail early in the field operation, it would introduce more cost (repair or warranty cost) if it is deliv- ered to the customer. On the other hand, normal units are possible to be discarded for exceeding the cutoff threshold ξb due to the inherent variability or measurement errors, though their long run performance is expected to outperform an accepted weak one. Hence, it is important to control the type-I and type-II errors simultaneously when designing the burn-in test. The type-I error is misclassifying normal units as weak, the probability of which is

Pr Y bð Þ4ξbjnormal unit � �

¼ Φ μ1b�ξbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

: ð2Þ

The type-II error is misclassifying weak units as normal, the probability of which is

Pr Y bð Þoξbjweak unit � �

¼ Φ ξb �μ2bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

: ð3Þ

Let cα and cβ be the cost of type-I and type-II errors, respec- tively. Then the misclassification cost is

cMC ξb; b � �

¼ p1cαΦ μ1b�ξbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

þp2cβΦ ξb �μ2bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

: ð4Þ

For each unit in the burn-in test, its burn-in implementation cost involves the setup cost cs, the burn-in operation cost c0 per unit time and the measurement cost cmea after the test. Thus, the overall per-unit cost is

C1 ξb; b � �

¼ cs þc0bþcmea þcMC ξb; b � �

: ð5Þ

3.2. Optimal burn-in strategy and impacts of measurement errors

The optimal burn-in strategy corresponds to the minimal total

cost C1, and may be found at the stationary point. With ∂C1 ξb;bð Þ

∂ξb ¼ 0,

we have the following proposition.

Proposition 3.1. Given a fixed burn-in duration b, the optimal cutoff level that minimizes the per-unit cost C1 is

ξ1�b ¼ μ1 þμ2 � �

b 2

þ σ 2bþσ2ϵ

μ2 �μ1 � �

b ln

cαp1 cβp2

! : ð6Þ

Apparently, the impact of measurement errors on the optimal cutoff level depends on the ratio cαp1=cβp2. If cαp1 is larger, indi- cating that the risk of type-I error is severer, then a larger cutoff level is preferred to mitigate the risk of discarding normal units due to measurement errors. Otherwise, a lower cutoff level could lead to smaller total cost. Generally, the proportion of the normal class in the population p1 is much larger than p2 while cα=cβ is not too smaller (e.g. p1 ¼ 0:945, p2 ¼ 0:055, cα ¼ 65 and cβ ¼ 90 as in [12]). Then, cαp1=cβp2 would be larger than 1 and the measure- ment error increases the cutoff level.

The impact of σϵ can be more clearly revealed when referring to the following ratio:

ξ1�b b

¼ μ1 þμ2 � �

2 þ σ

2bþσ2ϵ μ2 �μ1 � �

b2 ln

cαp1 cβp2

! :

The above ratio is an increasing function of σϵ for cαp1 4cβp2, a decreasing function of σϵ for cαp1 ocβp2, and a constant if cαp1 ¼ cβp2. To sum up, cαp1=cβp2 determines the dominating misclassification error and the optimal cutoff policy to mitigate this error, while measurement errors introduce more difficulty in efficient classification and push the cutoff level further from ðμ1 þμ2Þb=2.

Substituting ξ1�b in Eq. (6) into cMC, we have

cMC bð Þ ¼ cαp1Φ � μ2 �μ1 � �

b

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2bþσ2ϵp μ2 �μ1 � �

b ln

cαp1 cβp2

! !

þcβp2Φ � μ2 �μ1 � �

b

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2bþσ2ϵp μ2 �μ1 � �

b ln

cαp1 cβp2

! ! : ð7Þ

It can be proved that cMC bð Þ is monotonically decreasing with respect to (w.r.t.) b, indicating that a longer burn-in duration will lead to a smaller misclassification cost. Moreover, for a fixed misclassification cost, a larger measurement error σϵ will result in a longer burn-in duration. Hence, the classification efficiency would be reduced if the measured degradation is contaminated by measurement errors; and a longer burn-in is expected for a larger measurement error to achieve the same misclassification level.

For the optimal burn-in duration b1� that minimizes the overall cost C1, it can be proved that there exists one global optimal 0r b1� o þ1 that minimizes the overall cost C1 b; ξ1�b

� . Besides, C1

b; ξ1�b �

is always monotonically increasing w.r.t. b near b ¼ 0, indicating that a short burn-in is no better than none. Therefore, burn-in is not economic when the burn-in duration is relatively short, because the decrease rate of the misclassification cost w.r.t. b is small at the beginning. The optimal burn-in duration for a given parameter setting can be found by simple search method. As the optimal b1� cannot be analytically obtained, the impact of measurement errors on b1� is not obvious. However, numerical studies show that the optimal burn-in duration always increases when the measurement error increases (e.g., see Fig. 1 in Section 6). In real-world applications, the burn-in operation cost per unit time c0 is generally much smaller than the type-I or type- II cost. In this case, when the measurement error increases, the reduction in the misclassification cost by extending the burn-in duration is larger than the increase in the burn-in operational cost, and thus the measurement error prolongs the optimal burn-in duration. From the point of view of the total cost, the following proposition holds.

Proposition 3.2. The minimal per-unit cost C1 b 1� ; ξ1�b1�

� is non-

decreasing w.r.t. the measurement error σϵ. Particularly, C1 b 1� ; ξ1�b1�

� is strictly increasing w.r.t. σϵ if the optimal burn-in duration b

1� 40.

This proposition means, if there exists a nontrivial optimal burn-in duration b1� 40 that minimizes the overall cost C1, then improving the measurement accuracy can reduce the total cost. Related proofs for the preceding statements and Proposition 3.2 are given in Appendix. The analysis in this section show that measurement errors would impact the decision-making on the burn-in strategy and the overall cost. In practice, it is advised to prolong the burn-in duration and accordingly adjust the cutoff level according to cαp1=cβp2 when the measurement error increases. In turn, measurement devices with high accuracy could help to reduce the overall burn-in cost.

4. Optimal burn-in test considering field failure

The preceding section studies the optimal burn-in strategy under the misclassification cost structure, where the misclassification cost

Q. Zhai et al. / Reliability Engineering and System Safety 150 (2016) 126–135 129

may involve the cost of disposal and field failures. Sometimes, the type-I and type-II error cost cannot be easily determined, especially when these costs can vary with the burn-in strategy (i.e., burn-in duration and cutoff level). Therefore, instead of modeling the overall cost from the misclassification perspective, we directly model the disposal and the field failure costs in this section, and study the corresponding optimal burn-in strategy.

4.1. Cost model description

The setup cost cs, the burn-in operation cost per unit time c0 and the measurement cost cmea are the same as in Section 3. After the burn-in test of duration b, units with measured degradation smaller than a fixed cutoff level ξb will be put into field use with a gain K (e.g., sale price), while the others are disposed with a cost cd. Suppose there is a fixed threshold Xf for the degradation pro- cess and the product is deemed to fail if it exceeds this threshold. The product is expected to function properly for a mission time τ and some handling and administrative cost cf will be incurred if it fails before τ.

4.2. Lifecycle cost modeling and optimal burn-in strategy

As assumed, after a burn-in of duration b, the units with measured degradation smaller than ξb will be put into field use with a gain K and the others are disposed. The cost of this unit after burn-in is

cs þc0bþcmea þcdPr Y bð ÞZξb � �

�KPr Y bð Þoξb � �

; ð8Þ where Pr Y bð Þoξb

� � and Pr Y bð ÞZξb

� � ¼ 1�Pr Y bð Þoξb

� � can

be easily obtained according to Eq. (1). After burnt in, an accepted unit is put into field operation. The expected per-unit field failure cost is

cf Pr max 0rsrτ

X bþsð ÞZXf ; Y bð Þoξb �

¼ cf Pr X bð ÞZXf ; Y bð Þoξb � ��

þPr X bð ÞoXf ; max 0 os rτ

X bþsð ÞZXf ; Y bð Þoξb ��

: ð9Þ

Note that failure of a unit depends on that X tð Þ, rather than Y tð Þ, exceeds the threshold Xf . The reason is, the threshold is generally determined from a design perspective and the product is designed to fail if the underlying degradation rather than the measured degradation exceeds the threshold. In addition, there is still a probability that an accepted unit is an unqualified product due to the measurement error, i.e., Pr X bð ÞZXf ; Y bð Þoξb

� � . For the two-

subpopulation product, we have

Pr max 0 rs rτ

X bþsð ÞZXf ; Y bð Þoξb � ¼ X2 i ¼ 1

pi

� Pr Xi bð ÞZXf ; Yi bð Þoξb � �

þPr Xi bð ÞoXf ; max 0 osrτ

Xi bþsð ÞZXf ; Yi bð Þoξb ��

; ð10Þ

where

Pr Xi bð ÞZXf ; Yi bð Þoξb � �

¼ Z þ1 Xf

Pr ϵþxoξb � � 1

σ ffiffiffi b

p ϕ x�μib σ ffiffiffi b

p � �

dx

¼ Z þ1 Xf

Φ ξb �x σϵ

� � 1

σ ffiffiffi b

p ϕ x�μib σ ffiffiffi b

p � �

dx; ð11Þ

and

Pr Xi bð ÞoXf ; max 0osrτ

Xi bþsð ÞZXf ; Y bð Þoξb �

¼ Z Xf �1

Pr max 0osrτ

Xi bþsð Þ�Xi bð ÞZXf �x; ϵþxoξb �

1

σ ffiffiffi b

p ϕ x�μib σ ffiffiffi b

p � �

dx

¼ Z Xf �1

FIG τ; Xf �x � �

μi ; Xf �x � �2

σ2

! Φ ξb �x

σϵ

� � 1

σ ffiffiffi b

p ϕ x�μib σ ffiffiffi b

p � �

dx: ð12Þ

Here, ϕ Uð Þ is the PDF for the standard Gaussian distribution. Summing up the burn-in implementation cost and the field operation cost, the per-unit lifecycle cost is

C2 b; ξb � �

¼ cs þc0bþcmea þcdPr Y bð ÞZξb � �

�KPr Y bð Þoξb � �

þcf Pr max 0 rs rτ

X bþsð ÞZXf ; Y bð Þoξb �

¼ cs þcmea þcd þc0b� cd þKð ÞPr Y bð Þoξb � �

þcf Pr max 0 rs rτ

X bþsð ÞZXf ; Y bð Þoξb �

: ð13Þ

For a fixed burn-in duration b, the optimal cutoff level ξ2�b can be obtained by minimizing (13) over ξb. Let Λ ¼ 1�cd þKcf , which is always smaller than 1, we have the following proposition.

Proposition 4.1.

(1) If Λr0, then the optimal cutoff level is ξ2�b ¼ þ1. (2) If 0oΛo1, there exists a unique optimal cutoff level that

minimizes the overall cost C2. Specifically, ξ 2� b solves the following

equation:

P2 i ¼ 1piϕ

ξb � μibffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

Z þ1 0

SIG τ; y μi ; y2

σ2

� � 1 σc ϕ

y � Xf þμξb;i σc

� � dy � Λ

� � ¼ 0; ð14Þ

where σc ¼ σϵσ ffiffi b

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p , μξb;i ¼ μibσ

2 ϵ þξbσ2b

σ2bþσ2ϵ and SIG τ;

y μi ; y

2

σ2

� ¼ 1�

FIG τ; y μi ; y

2

σ2

� :

The ratio Λ represents the relative loss of selling a unit: if the unit is sold, then the company earns K and saves cd from disposal; at the same time, it may suffer a loss of cf due to the possible field failure. Hence, if Λr0, indicating that it is always profitable by selling a unit, then let all the units pass the burn-in is the best choice. In fact, no burn-in is required in this case. Note that even the company can actually earn some money by selling the low quality units, which means cd is negative, the price should be definitely lower than the normal units. Hence cd þK 40 and Λo1 always holds.

Since there are only two parameters, the optimal burn-in strategy can be numerically found for a given parameter setting. The involved integrals can be obtained by numerical integrations, such as Gaussian quadrature, which is quite efficient and available in most computing software. Because the analytical cutoff level cannot be explicitly obtained in this case, it is hard to study how the measurement error would influence the optimal burn-in strategy analytically. Nevertheless, Model 1 in Eq. (5) and Model 2 in Eq. (13) are underlying consistent, as will be shown, and Model 1 may be viewed as an approximation of Model 2. Thus, the analysis for Model 1 could provide some insight for Model 2.

5. Relations between Model 1 and Model 2

In this section, we will show that the two cost models in Sec- tions 3 and 4 are actually consistent, though they model the costs from different perspectives. The discrepancy of the two models are also pointed out, and a new cost model is proposed based on the relation between Model 1 and Model 2.

Q. Zhai et al. / Reliability Engineering and System Safety 150 (2016) 126–135130

5.1. Interpreting Model 2 from the misclassification perspective

For Model 2 in Eq. (13), we can derive the following expression after some rearrangement:

C2 b;ξb � �

¼ cs þcmeað Þþc0bþp1 ~cα b;ξb � �

Φ μ1b�ξbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

þp2 ~cβ b; ξb � �

Φ ξb �μ2bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

�Gp b; ξb � �

; ð15Þ

where

~cα b; ξb � �

¼ cd þK �cf Pr max 0rsrτ

X1 bþsð ÞZXf jY1 bð Þoξb �

;

~cβ b; ξb � �

¼ cf Pr max 0rsrτ

X2 bþsð ÞZXf jY2 bð Þoξb �

�cd �K;

Gp b;ξb � �

¼ K �cf Pr max 0 rs rτ

X1 bþsð ÞZXf jY1 bð Þoξb �� �

p1 �cdp2:

ð16Þ ~cα b; ξb � �

is the cost of misclassifying a normal unit as weak, which includes the additional disposal cost cd, the loss of gains from selling the good product K and the save from field failure cost cf Pr max0 rsrτX1 bþsð ÞZXf jY1 bð Þoξb

� � . Analogously, ~cβ b; ξb

� � is

the cost of misclassifying a weak unit as normal. Gp b; ξb � �

can be viewed as the gain of perfectly classifying the two subpopulations in the ideal situation.

Comparing Eq. (15) and Model 1 in Eq. (5), it is apparently that the two models share the same cost structure, except that Eq. (15) takes the gain Gp b; ξb

� � into consideration, and that the type-I and

type-II error cost for Model 2 are functions of b and ξb instead of constants. If the variation of Pr max0rsrτXi bþsð ÞZXf jYi bð Þoξb

� � w.r.t. b is negligible, e.g.,

Pr max 0 rs rτ

Xi bþsð ÞZXf jYi bð Þoξb � � Pr max

0rsrτ Xi bþsð ÞZXf jYi bð Þoξb

� b ¼ 0;ξb-1

¼ FIG τ; Xf μi ; X2f σ2

! ;

Then ~cα , ~cβ and Gp are all constants and Model 2 degenerates to Model 1. From this perspective, Model 1 in Eq. (5) can be viewed as

an approximation of Model 2 in (13) when the variations of Pr

max 0rsrτ

Xi bþsð ÞZXf Yi bð Þoξb �� � w.r.t. b are negligible. In this case,

one can simply apply Model 1 in Eq. (5), and the results in Section 3.2 is applicable to analyze the impacts of measurement errors. On

the contrary, if the variations of Pr max 0 rs rτ

Xi bþsð ÞZXf Yi bð Þoξb �� �

are not negligible, the discrepancy between Model 1 and Model 2 would be significant (as for the example in Section 6). In order to facilitate the analysis of Model 2, we propose a new approximation for Model 2 to make the analytical analysis feasible.

5.2. A new cost model

If we assume that the condition Yi bð Þoξb has negligible impact on the event max

0rsrτ Xi bþsð ÞZXf , i.e., they are statistically inde-

pendent, then the following approximation holds:

Pr max 0rsrτ

Xi bþsð ÞZXf jYi bð Þoξb �

� Pr max 0rsrτ

Xi bþsð ÞZXf �

¼ FIG τþb; Xf μi ; Xf σ2

� � : ð17Þ

The rationale of this approximation is given as follows. In practice, the burn-in duration is generally much shorter than the field usage time. Then, given the class a unit belongs to, the long- term performance of the unit (i.e., the unit fails in the field

operation or not) would exhibit little dependence on the short- term performance (i.e., the unit passes the burn-in or not), because the long-term performance depends much on the degradation trend, i.e., the class it belongs to.

By applying this approximation to Model 2 in Eq. (15), we have the following new model:

C3 b; ξb � �

¼ cs þc0bþcmea þcMC b; ξb � �

¼ cs þcmeað Þþc0b

þp1ccα bð ÞΦ μ1b�ξbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

þp2ccβ bð ÞΦ ξb �μ2bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

�cGp bð Þ; ð18Þ

with

ccα bð Þ ¼ cd þK �cf FIG τþb; Xfμ1; Xf σ2

� � ;

ccβ bð Þ ¼ cf FIG τþb; Xfμ2; Xf σ2

� � �K �cd;

cGp bð Þ ¼ K �cf FIG τþb; Xfμ1; Xf σ2

� �� � p1 �cdp2: ð19Þ

This approximation relieves the functional dependence of the type-I and type-II error cost and the gain on the cutoff level ξb, which enables us to derive the analytical expression for the opti- mal cutoff level.

Proposition 5.1. Given a fixed burn-in duration b, the optimal cutoff level that minimizes the per-unit cost C3 is

ξ3�b ¼ μ1 þμ2 � �

b 2

þ σ 2bþσ2ϵ

μ2 �μ1 � �

b ln

p1ccα bð Þ p2ccβ bð Þ

!

¼ μ1 þμ2 � �

b 2

þ σ 2bþσ2ϵ

μ2 �μ1 � �

b ln

p1 cd þK �cf FIG τþb; Xf μ1 ; Xf σ2

� � p2 cf FIG τþb;

Xf μ2 ; Xf σ2

� �K �cd

� 0 @

1 A: ð20Þ

With Eq. (20), the impact of measurement errors on the cutoff level is obvious, which depends on the ratio of p1ccα bð Þ=p2ccβ ðbÞ, as in Model 1. Besides, cd þKð Þ=cf A 0; 1ð Þ is a necessary condition for the existence of the optimal ξ3�b , as in Model 2. Substituting Eq. (20) into Eq. (18), the optimal burn-in duration can be easily obtained. With this model, it is possible to analytically study the impact of measurement errors on the optimal burn-in strategy and the lifecycle cost, and illuminate the impact of measurement errors under Model 2.

6. Illustrative example

We use a MEMS device example [31] to illustrate the results in this paper. The micro-engine inside the MEMS suffers degradation type failure. Its population consists of two subpopulations, the proportions of which are p1 ¼ 0:95 (normal) and p2 ¼ 0:05 (weak), respectively. The degradation of both subpopulations can be mod- eled by the Wiener process with linear drift, with the drift rates μ1 ¼ 8:4832 � 10�9 and μ2 ¼ 2:6875 � 10�8 for normal and weak units, respectively. The diffusion coefficients for the two sub- populations are identical, which is σ ¼ 2:2808 � 10�8. The product is designed to operate for τ ¼ 1200 units of time, and it fails if the degradation exceeds the failure threshold Xf ¼ 1:3093 � 10�5.

To eliminate the weak units in the population, a burn-in test is carried out. Assume that degradation of each unit after burn-in is measured by an equipment with the measurement error following a zero-mean Gaussian distribution, where the standard deviation is σϵ ¼ 3 � 10�7. The related costs for the burn-in implementation

Q. Zhai et al. / Reliability Engineering and System Safety 150 (2016) 126–135 131

are assumed to be

cs ¼ 1; cmea ¼ 0:5 and c0 ¼ 0:08:

6.1. Optimal burn-in under Model 1

First we study the optimal burn-in considering the mis- classification effect. We assume that the type-I and the type-II error costs are cα ¼ 69:95 and cβ ¼ 320, respectively. According to Eqs. (5) and (7), the optimal burn-in strategy can be found by simple search method, which is b1�;ξ1�b1�

� ¼ 75:98; 1:475 � 10� 6 �

and the minimal total cost is C�1 ¼ 9:20. Note that if we do not implement burn-in, all the units should be delivered to customers because p1cα 4p2cβ, and the total cost is 16. Thus, the burn-in test effectively removes the defected units and reduces the total cost.

For different measurement error σϵ, the optimal burn-in strategy and the total cost would be different, as shown in Fig. 1. Clearly, the optimal burn-in duration b1�, the optimal cutoff level ξ1�b1� and the optimal cost C

� 1 increase monotonically with the

standard deviation of measurement errors, which is consistent with the analysis in Section 3.2. Compared with the case where the measurement error is zero, the optimal burn-in strategy are much different for σϵ ¼ 3 � 10�7. If we mistakenly believe that the measurement error is zero, then we would derive the wrong burn- in duration b1�wrong ¼ 29:92 and cutoff level ξ

1� b1�;wrong ¼ 5:69 � 10

�7. Consequently, the actual total cost becomes 18.67, which is neither the optimal cost 4.75 we believe it to be, nor the true optimal cost 9.20. Moreover, it is even larger than the cost without burn-in (which is 16, as mentioned). Therefore, the analyst has to be quite cautious about measurement errors when designing the burn- in test.

Fig. 1. The optimal burn-in parameters and the total cost under different measurement error σϵ: the left panel is for the optimal b 1*, the middle is for the optimal ξ1*b1* , and

the right is for the optimal total cost. The cases where the measurement error is zero and the measurement error equals to 3 � 10�7 are highlighted.

6.2. Optimal burn-in under Model 2 and Model 3

As argued, if it is not easy to determine the misclassification cost, we can resort to Model 2 in Eq. (13). Here, a scrapped unit is disposed with a cost cd ¼ �30, where the negative value means the company can actually earn some money by selling these low- quality units. An accepted unit is sold with a price K ¼ 100 and a cost cf ¼ 390 is incurred if it fails in the field usage. These

parameters are chosen so that ~cα 0; 1ð Þ ¼ 69:95 and ~cβ 0; 1ð Þ ¼ 320, referring to Eqs. (15) and (16). Then, according to the discussion in Section 5.1, if the variation of the conditional probability

Pr max 0rsrτ

Xi bþsð ÞZXf jYi bð Þoξ2�b �

w.r.t. b is negligible, Model 1 will

approximate Model 2 well and we can derive a similar optimal burn-in strategy under Model 2. However, as shown in Fig. 2, the optimal burn-in strategy in this case is b2�; ξ2�b2�

� ¼ 68:49; 1:35 � 10�6 �

, and the minimal lifecycle cost is C�2 ¼ �86:38. Apparently, the optimal burn-in strategy under Model 2 is quite distinct from that under Model 1.

On the other hand, if we apply the approximation in Eq. (17), the optimal burn-in strategy under Model 3 can be easily obtained from Eq. (20) as b3�; ξ3�b3�

� ¼ 68:44; 1:35 � 10�6 �

and the mini- mum cost is C�3 ¼ �86:37. Clearly, Model 3 provides an almost identical burn-in strategy and the corresponding cost as that under Model 2, indicating that Model 3 is a good approximation for Model 2. Hence, one can study the sensitivity of the burn-in strategy and cost w.r.t. the measurement error under Model 3 in this case, since it is analytically easier to deal with.

6.3. Sensitivity of the optimal burn-in strategy w.r.t. cost parameters

For Model 1, according to Appendix, the optimal burn-in duration b1� solves the following equation:

c0 ¼ ffiffiffiffi 2 π

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cαp1 Ucβp2

p exp �1

2 s2 þr

2

s2

� � � U μ2 �μ1 � �

σ2bþ2σ2ϵ � �

4 σ2bþσ2ϵ � �3

2

3k ¼ 1 2 ln 2 π �1 2

s�r s

� 2 þln μ2 �μ1

� � σ2bþ2σ2ϵ � �

4 σ2bþσ2ϵ � �3

2

;

where s ¼ μ2 �μ1 � �

b= 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p� ; r ¼ 12ln cαp1=ðcβp2Þ

� and

k ¼ ln c0=ðcβp2Þ �

. Apparently, the optimal burn-in strategy

depends on the log-ratio r and k. Note that for the cost parameters used in Section 6.1, we have r ¼ 0:712 and k ¼ �5:3. For the fol- lowing function:

l b;rð Þ ¼ 1 2 ln 2 π �1 2

s�r s

� 2 þln μ2 �μ1

� � σ2bþ2σ2ϵ � �

4 σ2bþσ2ϵ � �3

2

;

Q. Zhai et al. / Reliability Engineering and System Safety 150 (2016) 126–135132

we plot it w.r.t. b for different r (with other parameters unchan- ged). As shown in Fig. 3, l b; rð Þ moves towards right as r increases, indicating that the optimal burn-in duration b1� (i.e., the second intersection of l b; rð Þ with k) increases as r increases. On the other hand, for fixed r, increasing k would reduce the optimal b1�. This is reasonable as burn-in should be prolonged if the severity of type-I error increases, and it should be shortened when the burn-in operation cost c0 increases.

The variations of the optimal burn-in duration b1� and the ratio ξ1�b1� =b

1� w.r.t. the measurement error σϵ for different r are illu- strated in Fig. 4. It is shown that the ratio ξ1�b1� =b

1� also increases as

Fig. 2. Contour plot for the burn-in cost w.r.t. the burn-in duration and the cutoff level under Model 2.

Fig. 3. Variation of l b;rð Þ w.r.t. the burn-in duration b for different r ¼ 12 ln ðcαp1Þ=ðcβp2Þ

� � .

Fig. 4. Variations of optimal burn-in duration b1* and ratio ξ1*b1* =b 1* w.r.t. th

r increases, indicating that more units are classified as normal, i.e., burn-in tends to reduce the type-I error as its severity increases. For different r, measurement errors always increase both b1� and ξ1�b1� =b

1�, but the impact of measurement errors is smaller for small r, which is consistent with the analysis in Section 3.2.

As we have shown in Section 6.2, Model 3 approximates Model 2 well. Therefore, we consider the impact of different cost para- meters on the optimal burn-in strategy under Model 3 only. For Model 3, we have

C3 b; ξb � �

¼ cs þcmea þcdð Þþcf c0 cf b�p1

ccα bð Þ cf

Φ sþr bð Þ s

� �� þp2

ccβ bð Þ cf

Φ �sþr bð Þ s

� �! ;

where r bð Þ ¼ 12ln ccα bð Þp1=ðccβ bð Þp2Þ� . Let k1 ¼ c0=cf and k2 ¼ cd þKð Þ=cf ¼ 1�Λ. Then we have

ccα bð Þ=cf ¼ k2 �FIG τþb;Xfμ1; Xf σ2

� � ;ccβ bð Þ=cf ¼ FIG τþb;Xfμ2;

Xf σ2

� � �k2;

r bð Þ ¼ 1 2 ln

p1 k2 �FIG τþb; Xf μ1 ; Xf σ2

� � p2 FIG τþb;

Xf μ2 ; Xf σ2

� �k2

� 0 @

1 A;

and the optimal b3� can be obtained by minimizing

k1b�p1 ccα bð Þ cf

Φ sþr bð Þ s

� � þp2

ccβ bð Þ cf

Φ �sþr bð Þ s

� � ≝k1b�L3 b;k2ð Þ:

e measurement error under Model 1 for different r ¼ 12ln ðcαp1Þ=ðcβp2Þ � �

.

Fig. 5. Contour plot for dL3=db under Model 3; the point representing the case in Section 6.2 is highlighted.

Fig. 6. Variations of optimal burn-in duration b3* and ratio ξ3*b3* =b 3* w.r.t. the measurement error under Model 3 for different k2 ¼ ðK þcdÞ=cf .

Burn-in duration b

C ut

of f l

ev el

ξ b

50 100 150

0.5

1

1.5

2

2.5

3

3.5

x 10 -6

b1* =76 ξb1*

1* =1.4774×10-6

C1 *=9.186

Burn-in duration b

C ut

of f l

ev el

ξ b

50 100 150

0.5

1

1.5

2

2.5

3

3.5

4 x 10

-6

b2* =68 ξb2*

2* =1.31×10-6

C2 *=-85.9366

Fig. 7. Variations of burn-in cost w.r.t. b and ξb under Model 1 (left) and Model 2 (right) obtained by Monte Carlo simulations.

Q. Zhai et al. / Reliability Engineering and System Safety 150 (2016) 126–135 133

Fig. 5 shows the contour plot of dL3=db w.r.t. b and k2. The point (k2 ¼ 7=39, b ¼ 68:44Þ with k1 ¼ 0:00021, which corresponds to the cost parameters in Section 6.2, is also marked. Note that the optimal b3� is determined by the second intersection of k1 and dL3=db, i.e., the second zero of dL3=db�k1. Then, for a fixed k1 (considering the contour line where dL3=db ¼ k1), b3� first increa- ses and then decreases as k2 increases. This can be interpreted in the following way. When K þcdð Þ is relatively small (compared with cf ), delivering weak units to the customer is costly, and the cutoff criterion is conservative. Thus, a considerable portion of normal units is misclassified as weak. In this case, when K þcdð Þ increases, burn-in is prolonged to mitigate the type-I error. In contrast, when K þcdð Þ is relatively large, selling a unit is more profitable. In this case, the company suffers less when selling a weak unit and it tends to shorten the burn-in duration to reduce the overall cost. On the other hand, for a fixed k2, the optimal b

3�

always decreases as k1 increases, i.e., an increased burn-in operation cost always incurs a shorter burn-in duration.

The variations of optimal burn-in duration b3� and the ratio ξ3�b3� =b3� w.r.t. the measurement error σϵ for different k2 with k1 ¼ 0:00021 are shown in Fig. 6. As explained, b3� is not monotonic w.r. t. k2 for fixed σϵ, but ξ

3� b3� =b

3� is always increasing w.r.t. k2 in this case. For fixed k2, measurement errors always increase b

3� and ξ3�b3� =b

3�, indicating that measurement errors always deteriorate the burn-in efficiency and the optimal burn-in strategy can be quite different when taking measurement errors into account.

6.4. Simulation studies

In this section, Monte Carlo simulations are carried out to validate our analysis. Specifically, in each simulation, we first conduct n times Bernoulli experiment to generate n units from the population, where each unit can be normal with probability p1 ¼ 0:95 and weak with probability p2 ¼ 0:05. Let Ij; j ¼ 1; …; n denote the class of the unit, where Ij ¼ 0 is the normal unit and Ij ¼ 1 is the weak unit. Then for each unit, we generate a degradation path

Q. Zhai et al. / Reliability Engineering and System Safety 150 (2016) 126–135134

according to the corresponding Wiener process

Xj tð Þ ¼ μ1 þ μ2 �μ1 � �

Ij � �

tþσB tð Þ; j ¼ 1; …; n: For a burn-in duration b, the corresponding measured degrada-

tion Yj;b at b can be generated. Then, given a specified cutoff level ξb, we can obtain the accepted units, disposed units as well as mis- classified units. For Model 2, we further generate the path for these accepted units until bþτ. The cost under Model 1 and Model 2 can be obtained for each specified burn-in duration and cutoff level. This simulation is replicated for m times to get an averaged cost for each b; ξb � �

. We set n ¼ 2000 and m ¼ 1000 in the simulation, and use other parameters as given in Sections 6.1 and 6.2.

Fig. 7 presents the contour plot of the per-unit burn-in cost for different b and ξb under Model 1 (left) and Model 2 (right). As shown in the figure, the optimal burn-in strategy under Model 1 is b1� ¼ 76 and ξ1�b1� ¼ 1:4774 � 10

�6 with the burn-in cost C�1 ¼ 9:186, while the optimal burn-in strategy under Model 2 is b2� ¼ 68 and ξ2�b2� ¼ 1:31 � 10

�6 with C�2 ¼ �85:94. Apparently, the simulation results are consistent with the analytical results in Sections 6.1 and 6.2, which validates our analysis.

7. Conclusions

Burn-in is an effective tool to eliminate defective units and to improve the reliability of delivered units. Degradation-based burn- in can exploit the correlation between the degradation of quality characteristics (QC) and the failure time, which can greatly shorten the burn-in test and thus is quite suitable for highly reliable pro- ducts. In the implementation of the degradation-based burn-in, the measurement of the QC is a critical step and measurement errors are inevitable. Existing degradation-based burn-in models seldom considered measurement errors and could lead to inferior burn-in decisions. This paper investigated the optimal burn-in strategies under two different cost models when measurement errors were taken into account. The first model involved the misclassification cost during burn-in, while the second considered the burn-in implementation cost and field failure cost. Through analytical analysis, we showed the intimate relationship between the two models and proposed an accurate approximation method to approximate the second model, which greatly simplifies the analysis. The results of the paper could help engineers to design proper burn-in tests when facing measurement errors in the measurement process. The illustrative example showed that properly accounting for the measurement errors could sig- nificantly reduce the lifecycle cost of the product.

In practice, some stochastic processes, e.g., the Gamma process [32,33] or the inverse Gaussian process [34,35], may be proper for degradation modeling in cases where the underlying degradation is monotone. As a direction for future works, it is interesting to investigate the impact of measurement errors on the optimal burn-in policies when the degradation is modeled by other pro- cesses. Comparing the optimal burn-in strategies under different cost models for these degradation processes is another direction for future study.

Acknowledgments

The authors would like to thank the Associate Editor and three anonymous referees for their constructive comments which con- siderably help to improve the original manuscript. This research was supported by the Fundamental Research Funds for the Central Universities under Grant No. YWF-14-KKX-008 and the Innovation Foundation of BUAA for PhD Graduates under Grant No. YWF-14- YJSY-035.

Appendix A

Properties related to cMC bð Þ in Model 1

Let

s ¼ μ2 �μ1 � �

b

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p ; r ¼ 1 2 ln

cαp1 cβp2

! ;

then cMC bð Þ in Eq. (7) can be rewritten as

cMC sð Þ ¼ cαp1Φ �s� r s

� þcβp2Φ �sþ

r s

� :

Taking derivative w.r.t. s, we have

dcMC ds

¼ cαp1 �1þ r s2

� ϕ �s�r

s

� þcβp2 �1�

r s2

� ϕ �sþr

s

� ¼ cαp1 �1þ

r s2

� 1ffiffiffiffiffiffi 2π

p exp �1 2

s2 þ2rþr 2

s2

� � � þcβp2 �1�

r s2

� 1ffiffiffiffiffiffi 2π

p exp �1 2

s2 �2rþr 2

s2

� � � ¼ �

ffiffiffiffi 2 π

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cαp1 Ucβp2

p exp �1

2 s2 þr

2

s2

� � � o0 for s40;

where cαp1 Uexp �rf g ¼ cβp2 Uexp rf g ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cαp1 Ucβp2

p . Note that

∂s ∂b

¼ μ2 �μ1 � �

σ2bþ2σ2ϵ � �

4 σ2bþσ2ϵ � �3

2

40;

therefore cMC bð Þ is monotonically decreasing w.r.t. b (∂cMC=∂bo0). On the other hand, we have

∂s ∂σ2ϵ

¼ � μ2 �μ1 � �

b

4 σ2bþσ2ϵ � �3

2

o0 for b40;

indicating that for the same burn-in duration, the misclassification cost with a larger measurement error is always higher; in turns, longer burn-in is required to achieve the same cMC.

Properties of optimal burn-in duration in Model 1

Referring to the preceding proof, we have

dC1 db

¼ c0 � ffiffiffiffi 2 π

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cαp1 Ucβp2

p exp �1

2 s2 þr

2

s2

� � � U ∂s ∂b

:

It can be proved that the function

L bð Þ ¼ exp �1 2

s2 þr 2

s2

� � � U ∂s ∂b

first increases and then decreases w.r.t. b. Hence, depending on the value c0=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cαp1 Ucβp2

p , there may be two zeros, one zero or no zero

for dC1=db. If there is no zero or only one zero for dC1=db, indi- cating that dC1=dbZ0 and C1 is always increasing w.r.t. b, then the optimal burn-in duration is b1� ¼ 0, i.e., no burn-in is required. If there are two zeros for dC1=db, then the second zero corresponds to a local minimum for C1. If this local minimum is smaller than the total cost without burn-in C1ð0Þ, then a nontrivial global optimal burn-in duration exist. Otherwise, b1� ¼ 0 (to be clear, we clarify that the optimal b1 is zero even when this local minimum equal to C1ð0Þ). In any case, dC1=db b ¼ 0 ¼ c0 40

�� , indicating that a short burn-in is no better than none.

Proof for Proposition 3.2

Consider two different measurement errors σϵ;1 and σϵ;2, and the corresponding minimal burn-in cost C�1 σϵ;i

� � ¼ C1 b1�i ;σϵ;i

� ,

where σϵ;1 oσϵ;2 and b1�i is the optimal burn-in duration. According to the impact of the measurement error on the

Q. Zhai et al. / Reliability Engineering and System Safety 150 (2016) 126–135 135

misclassification cost, we know that

cMC b 1� 2 ;σϵ;1

� rcMC b1�2 ;σϵ;2

� :

Thus the following relation always holds:

C�1 σϵ;1 � �

¼ C1 b1�1 ;σϵ;1 �

rC1 b1�2 ;σϵ;1 �

rC1 b1�2 ;σϵ;2 �

¼ C�1 σϵ;2 � �

:

If we further suppose that at least one of b1�i is strictly larger than 0, then at least one out of the above two inequalities holds strictly, i.e.,

C1 b 1� 1 ;σϵ;1

� oC1 b1�2 ;σϵ;1

� orC1 b

1� 2 ;σϵ;1

� oC1 b1�2 ;σϵ;2

� ;

and then C�1 σϵ;1 � �

oC�1 σϵ;2 � �

holds.

Proof for Proposition 4.1

For C2 b; ξb � �

in Eq. (13), after differentiation w.r.t. ξb and some simplification, we have

∂C2 ∂ξb

¼ cfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p X2 i ¼ 1 piϕ

ξb �μibffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p !

Λ� Z þ1 0

SIG τ; y μi ; y2

σ2

� � 1 σc ϕ

y�Xf þμξb;i σc

� � dy

� � ;

where σc ¼ σϵσ ffiffi b

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p , μξb;i ¼ μibσ

2 ϵ þξbσ2b

σ2bþσ2ϵ and

SIG τ; y μi ; y

2

σ2

� ¼ 1�FIG τ; yμi;

y2

σ2

� :

Let ∂C2∂ξb ¼ 0 we have

Λ ¼

P2 i ¼ 1

piϕ ξb �μibffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p � � Rþ1

0 SIG τ;

y μi ; y

2

σ2

� 1 σc ϕ

y�Xf þμξb ;i σc

� dy

P2 i ¼ 1

piϕ ξb �μibffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p � �

¼ 1�w ξb � �� �

h1 ξb � �

þw ξb � �

h2 ξb � �

¼ h1 ξb � �

þw ξb � �

h2 ξb � �

�h1 ξb � �� �

;

where

ξb � �

¼ p2ϕ

ξb �μ2bffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2b þσ2ϵ

p � �

p1ϕ ξb �μ1bffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p � �

þp2ϕ ξb �μ2bffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2bþσ2ϵ

p � �

¼ p2 p1exp � 1σ2bþσ2ϵ μ2 �μ1

� � b ξb �

μ1 þμ2ð Þb 2

� n o þp2

is a monotonically increasing weight function w.r.t. ξb. On the other hand,

hiðξbÞ ¼ Zþ1 0

SIG τ; y μi ; y2

σ2

� � 1 σc ϕ

y�Xf þμξb;i σc

� � dy

is monotonically decreasing w.r.t. ξb for i ¼ 1; 2, and it is easy to prove that h1 ξb

� � 4h2 ξb

� � for given ξb. Hence

H ξb � �

¼ h1 ξb � �

þw ξb � �

h2 ξb � �

�h1 ξb � �� �

is a monotonically decreasing function w.r.t. ξb. Note that H ξb � �

-

1 as ξb-�1, and H ξb � �

-0 as ξb-þ1. Then, for a given Λ, if ΛA 0; 1ð Þ, one can find a unique solution for Λ ¼ h ξb

� � . If Λr0,

then ∂C2=∂ξb o0 for all ξb and the optimal cutoff level is �1. Proposition 4.1 is thus proved.

Appendix A. Supplementary material

Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ress.2016.01.015.

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