Reliability sensitivity analysis with random and interval variables

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2009; 78:1585–1617Published online 16 January 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2543

Reliability sensitivity analysis with random and interval variables

Jia Guo‡ and Xiaoping Du∗,†,§

Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology,400 West 13th Street, Rolla, MO 65409-0050, U.S.A.

SUMMARY

In reliability analysis and reliability-based design, sensitivity analysis identifies the relationship betweenthe change in reliability and the change in the characteristics of uncertain variables. Sensitivity analysisis also used to identify the most significant uncertain variables that have the highest contributions toreliability. Most of the current sensitivity analysis methods are applicable for only random variables.In many engineering applications, however, some of uncertain variables are intervals. In this work, asensitivity analysis method is proposed for the mixture of random and interval variables. Six sensitivityindices are defined for the sensitivity of the average reliability and reliability bounds with respect to theaverages and widths of intervals, as well as with respect to the distribution parameters of random variables.The equations of these sensitivity indices are derived based on the first-order reliability method (FORM).The proposed reliability sensitivity analysis is a byproduct of FORM without any extra function callsafter reliability is found. Once FORM is performed, the sensitivity information is obtained automatically.Two examples are used for demonstration. Copyright q 2009 John Wiley & Sons, Ltd.

Received 10 November 2007; Revised 20 November 2008; Accepted 28 November 2008

KEY WORDS: reliability; sensitivity analysis; intervals

1. INTRODUCTION

In reliability analysis [1–3] and reliability-based design [4–7], sensitivity analysis provides infor-mation about the relationship between reliability and the distribution parameters of a randomvariable. Sensitivity analysis can therefore identify the most significant uncertain variables thathave the highest contribution to reliability. When only random variables are involved, sensitivity

∗Correspondence to: Xiaoping Du, Department of Mechanical and Aerospace Engineering, Missouri University ofScience and Technology, 400 West 13th Street, Rolla, MO 65409-0050, U.S.A.

†E-mail: [email protected]‡Graduate Assistant.§Associate Professor.

Contract/grant sponsor: U.S. National Science Foundation (NSF); contract/grant number: CMMI-0400081Contract/grant sponsor: University of Missouri Research Board; contract/grant number: 7116Contract/grant sponsor: Intelligent Systems Center

Copyright q 2009 John Wiley & Sons, Ltd.

1586 J. GUO AND X. DU

analysis is usually performed for the probabilistic characteristics of a limit-state function, such asits moment, probability density function (PDF), and reliability. Such sensitivity analysis is usuallynamed probabilistic sensitivity analysis (PSA). Various PSA approaches have been reported in awide range of literature, including differential analysis [8, 9], variance-based methods [10], andsampling-based methods [10]. These types of PSA are briefly reviewed below.

1.1. Differential analysis (probability sensitivity coefficient)

The probability-based sensitivity measure is defined as the rate of change in a probability (P)

(reliability or the probability of failure) due to the change in a distribution parameter (qi ) of arandom input, namely �P/�qi . �P/�qi can be calculated by the finite difference method given by [2]

Sqi =P(qi +�qi )−P(qi )

�qi(1)

where qi is a distribution parameter, such as the mean or the variance of a random variable; �qiis a small step size of qi .

Various probability sensitivity measures have been proposed in literature [11–14]. Wu [11] andWu and Mohanty [12] propose a normalized cumulative density function (CDF)-based sensitivitycoefficient for the probability of failure with respect to the distribution parameters of randomvariables. The sensitivity is defined by

Sqi =�pf/pf�qi/qi

=∫

···∫

�

qifX(X)

� fX(X)

�qi

[fX(X)

pf

]dx=E

[qi� fX(X)

fX(X)�qi

]�

(2)

where fX is the joint PDF of all random variables, pf is the probability of failure, X is a vectorof random variables, and � denotes the failure region. The calculation of this sensitivity measureinvolves evaluating a multidimensional integral. A sampling method is usually used to estimatethis integral, which makes this method computationally expensive. Mavris et al. [13] extend Wu’smethod to evaluate the sensitivity of any probabilistic characteristics, such as the variance andmean of a limit-state function.

Another sensitivity measure related to reliability is the most probable point (MPP)-based sensi-tivity coefficients [14], defined as the gradient of a limit-state function at the MPP in the standardnormal space, normalized by the reliability index. Let G be a response calculated by a limit-state function G=g(X). After X is transformed into standard normal random variables U, theMPP u∗ =(u∗

1,u∗2, . . . ,u

∗nx ) is identified. The MPP is the shortest distance point from the origin

to the limit state g(U)=c, where c is a constant. The equation for the MPP search will begiven in Equation (7). The sensitivity of the MPP with respect to the i th random variable is thencalculated by

Si = (u∗i )

2

�2(3)

where � is the magnitude of u∗ or the reliability index. For the MPP-based reliability analysis, theprobability sensitivity coefficient does not require any additional computational efforts after theMPP is found. The sensitivity coefficient Si is just a byproduct of reliability analysis.

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2009; 78:1585–1617DOI: 10.1002/nme

RELIABILITY SENSITIVITY ANALYSIS 1587

1.2. Variance-based methods

Variance-based sensitivity analysis methods rely on the decomposition of the variance of a responseinto terms contributed by various sources of input variations. These sources can be classified intotwo types: main effects and total effects. A main effect refers to the effect of only one randomvariable, while a total effect is used to include both the individual effect of a random variableand the interaction of the random variable with other random variables. Although the methodsprovide a global sensitivity measure, their major limitation is that a variance is assumed to besufficient to describe the uncertainties encountered. This type of methods may lose accuracy whenthe variance is not a good measure of the distribution dispersion, such as in the case where aresponse distribution has high skewness and kurtosis [15].

1.3. Sampling methods

Sampling approaches, such as Monte Carlo sampling for sensitivity analysis, usually involve threesteps: (1) generating samples for uncertain input variables, (2) numerically evaluating a limit-state function and then obtaining samples of response variables, and (3) statistically analyzingresponses and quantifying their uncertainties, and then exploring the effects of the uncertainty ofinput variables on responses. Sampling methods are easy to use, but computationally expensivewhen reliability is high. Because the probability of failure is low in this case, a large number ofsamples are required to capture a failure event.

The current PSA methods handle only random variables that are assumed to follow certainprobability distributions. However, in many engineering applications, the information or knowledgemight be too insufficient to build probability distributions. As discussed in [16, 17], uncertaintyis sometimes represented by intervals due to the lack of knowledge. One example is that thetrue contact resistance in the vehicle crashworthiness design is hard to know; an interval is thenused based on the engineers’ best judgment [18]. Another example is a new design, where it isdifficult to determine the precise distribution of design variables, such as dimensions. Engineersoften define their design variables in the form of nominal values plus and minus certain tolerances,like 10±0.01mm. More examples of intervals can be found in [4, 16]. Sometimes even though avariable is random and follows a non-uniform distribution, only one interval estimate is availabledue to limited information or sparse samples. In this case, assigning an assumed distribution to thevariable may lead to erroneous results [19]. When intervals are involved, the current PSA methodsare no longer applicable.

Several methods of dealing with only interval variables have been reported for reliability analysisand reliability-based design [17, 20–34]. A few sensitivity analysis methods [35–38] for epistemicuncertainty (uncertainty due to the lack of knowledge) are potentially capable of dealing withinterval variables. These methods use intervals to represent epistemic uncertainty. For example,a sensitivity analysis approach on the basis of belief and plausibility measures is proposed byBae et al. [35, 36]. The results of this approach can help to guide the data collection to improvethe accuracy of reliability analysis and distinguish the dominant contributors of uncertainty.A sampling-based sensitivity analysis method is developed by Helton et al. [37]. It consists ofthree steps: an initial analysis to explore the model behavior, a stepwise analysis to indicate theeffects of uncertain variables on belief and plausibility functions, and a summary analysis to showa series of variance-based sensitivity analysis results. Considering the complexity of the mixture ofaleatory and epistemic uncertainties, Guo and Du [38] propose an approach to conduct sensitivity



analysis with this mixture. In their method, the most important epistemic variables are capturedunder the framework of the unified uncertainty analysis.

All of the above methods are capable of identifying the most significant interval variables, butthey have some limitations. For example, it is difficult to use them to obtain information about howindividual intervals impact reliability, especially how reliability bounds will change after narrowinginterval bounds. In this work, we propose a sensitivity analysis method to handle the situationwhere both interval variables and random variables are involved. With this method, we attempt toanswer the following questions:

(1) How will the width of the reliability bounds change if the width of an interval is reducedor if the average of the interval is changed?

(2) How will the average reliability change if the width of an interval is reduced or if the averageof the interval is changed?

(3) How will the width of the reliability bounds change if a distribution parameter of a randomvariable is changed?

(4) How will the average reliability change if a distribution parameter of a random variable ischanged?

The answers to the above questions will provide useful information about improving reliabilityand reducing the impact of intervals and random variables on reliability. Hence, six sensitivityindices are proposed for answering these questions. Equations for the sensitivity indices are thenderived, and corresponding computational procedures are developed. The calculation of sensitivityindices requires searching the minimum and maximum reliability, or the probabilities of failure,over the intervals. To alleviate the computational burden, we use an efficient first-order reliabilitymethod (FORM)-based unified reliability analysis (URA) framework [39].

This paper is organized as follows: Section 2 provides a brief review of the unified reliabilityanalysis. In Section 3, the six sensitivity indices are defined, and the equations for calculatingthese sensitivity indices are derived. In Section 4, two engineering examples are used to illustratethe proposed method. Conclusions and future work are summarized in Section 5.

2. UNIFIED RELIABILITY ANALYSIS

Reliability analysis is one of the main steps of reliability sensitivity analysis. The proposedsensitivity analysis is based on the FORM [40, 41], which is applicable for random variables, andthe URA [39], which is applicable for the mixture of random and interval variables. Both methodsare briefly reviewed in this section.

2.1. Reliability analysis

In the reliability analysis where only random variables X are involved, reliability is defined by

R=Pr{G=g(X)�c}=1−Pr{G=g(X)<c}=1− pf (4)

where Pr{·} denotes a probability,G is a response, c is a specific limit state,X=(X1, X2, . . . , Xi , . . . ,

Xnx ) is a vector of random variables, g is a performance function, also called a limit-statefunction [42], and pf is the probability of failure.



If the joint PDF of X is fX, the probability of failure pf is calculated by

pf=Pr{G=g(X)<c}=∫g(X)<c

fX(x)dx (5)

The limit-state function g(X) is usually a non-linear function of X; therefore, the integrationboundary is non-linear. Since the number of random variables is usually high, multidimensionalintegration is involved. There is rarely a closed-form expression to Equation (5). The FORM iswidely used to easily evaluate the integral in Equation (5).

FORM involves three steps to approximate the probability integral: (1) transforming originalrandom variables X into standard normal random variables U, (2) searching for the MPP, and (3)calculating pf.

Step 1: Transformation, which is given by

ui =�−1{FXi (xi )} (6)

where FXi is the CDF of Xi and �−1 is the inverse CDF of a standard normal distribution.Step 2: MPP search, where the MPP u∗ is identified by

minU

‖U‖s.t. g(U)=c

(7)

in which ‖·‖ stands for the magnitude of a vector. Geometrically, the MPP is the shortest distancepoint from the limit state g(U)=c to the origin in U-space. The minimum distance �=‖u∗‖ iscalled the reliability index.

Step 3: Estimation of pf, which is given by

pf=�(−�) (8)

where � is the CDF of a standard normal distribution.The most computationally intensive work of FORM is the MPP search. The following recursive

algorithm [43] is used for the MPP search,

�(k) = �(k−1)+ g(u(k−1))

‖∇g(u(k−1))‖

u(k) = −�(k) ∇g(u(k−1))

‖∇g(u(k−1))‖

(9)

where ∇g(u(k)) is the gradient of g at u(k), ‖∇g(u(k))‖ is its magnitude, and k is the iterationcounter.

2.2. Unified reliability analysis

When both random and interval variables are present, random variables X are characterized byprobability distributions while interval variables Y reside on [yl ,yu]. The URA framework andcomputational method proposed in [39] are applicable to handle this situation. As shown in [39],the cumulative distribution function of the response G=g(X,Y) has its upper and lower bounds,and so does reliability Pr{G�c}. The URA [39] is used to find the reliability bounds.



Figure 1. The unified reliability analysis framework.

Figure 2. Flowchart of sequential single-loop procedure for pLf calculation.

The URA framework is illustrated in Figure 1. The inputs to the framework are random variablesX defined by a joint PDF and interval variables Y. The outputs are CDF bounds and reliabilitybounds.

The set of intervals Y is denoted by �Y and the event of failure is defined by g(X,Y)<c.According to [39], the lower and upper bounds of the probability of failure, pLf and pUf , arecalculated by

pLf =Pr{Gmax(X,Y)<c|Y∈�Y} (10)

and

pUf =Pr{Gmin(X,Y)<c|Y∈�Y} (11)

respectively. Gmin and Gmax are respectively the global minimum and maximum values of Gover �Y.

The evaluation of the upper and lower bounds of the probability of failure is essentially theevaluation of the minimum and maximum CDF of the limit-state function. Therefore, traditionalprobabilistic analysis (PA) methods can be used for the URA. The FORM is employed for the URA.

Figure 2 depicts the numerical procedure of the URA method. The procedure involves two typesof analysis. The first one is PA , which is responsible for the MPP search and the calculation of theprobability of failure. The second one is interval analysis (IA), which is responsible for the searchof the maximum and minimum values of G. The direct combination of PA and IA will involve adouble loop procedure, where PA is an outer loop and IA is an inner loop. For example, to find thelower bound of pf, at every iteration of the MPP search in the outer loop, the IA inner loops willbe called to find the maximum G in terms of Y. This method is inefficient due to the double-loopprocedure. The efficient computational method is then developed in [44]. The method involves an



efficient sequential single-loop procedure, where PA is decoupled from IA. The flowchart of thisefficient procedure is shown in Figure 2 for the pLf calculation. The solution is the MPP where Gis maximized. The MPP for pLf is then denoted by u∗,L in this paper. In addition the MPP for themaximum probability of failure pUf is denoted by u∗,U.

The probability Pr{Gmax(X,Y)<c|Y∈�Y} in Equation (10) is then computed by

Pr{Gmax(X,Y)<c|Y∈�Y}=�(−�)=�(−‖u∗,L‖) (12)

For the pUf calculation, the model of the MPP search is the same as in Figure 2 except that IAbecomes a minimization problem.

3. RELIABILITY SENSITIVITY ANALYSIS

When only random variables are involved, reliability sensitivity analysis is used to find the rate ofchange in the probability of failure (or reliability) due to the changes in distribution parameters(usually means and standard deviations). When both random variables and interval variables areinvolved, reliability analysis will generate two bounds of reliability or of the probability of failurepf. The gap between the maximum probability of failure pUf and the minimum probability offailure pLf represents the effect of interval variables on the probability of failure. In addition tothe traditional sensitivity analysis in terms of random variables, sensitivity analysis in terms ofinterval variables is also needed. In this work, six types of sensitivity are proposed with respectto both random variables and interval variables. The proposed sensitivity indexes are summarizedin Table I.

3.1. Type I sensitivity ��p/��i

��p/��i is the sensitivity of the width of the pf bounds, �p, with respect to the interval width ofthe i th interval variable Yi , �i . �p is defined by

�p = pUf − pLf (13)

Table I. Six sensitivity indices.

Sensitivity type Description Input

Type I ��p/��i Sensitivity of the width of the pf bounds, �p , with respect to thewidth of interval variable Yi , �i

Interval

Type II � pf/��i Sensitivity of the average pf, pf, with respect to the width ofinterval variable Yi , �i

Interval

Type III ��p/�yi Sensitivity of the width of the pf bounds, �p , with respect to theaverage of interval variable Yi , yi

Interval

Type IV � pf/�yi Sensitivity of the average pf, pf, with respect to the average ofinterval variable Yi , yi

Interval

Type V ��p/�qi Sensitivity of the width of the pf bounds, �p , with respect to adistribution parameter, qi , of random variable Xi

Random

Type VI � pf/�qi Sensitivity of the average pf, pf, with respect to a distributionparameter, qi , of random variable Xi

Random



The width of Yi is calculated by

�i = yUi − yLi (14)

where yLi and yUi are the lower and upper bounds of Yi , respectively.To obtain a unique sensitivity index, we define the change of �i , �(�i ) in such a way that Yi

expands in both directions equally; namely, yLi is decreased by �(�i )2 and yUi is increased by �(�i )

2 .There are infinite ways that Yi can change by �(�i ), for example, [yLi , yUi ] can change to[

yLi − 3�(�i )

4, yUi + �(�i )

4

]or

[yLi − �(�i )

4, yUi + 3�(�i )

4

]

Our definition makes the change unique.This type of sensitivity can identify interval variables that have the largest impact on the width

of the pf bounds. If the gap of the pf bounds is too wide, decisions will be difficult to make. Tonarrow the width of pf bounds efficiently, more information about the important interval variablesshould be collected, and then their widths can be reduced. Sensitivity analysis will provide a usefulguidance to the collection of more information.

To derive the equations for ��p/��i , we consider all the situations where the maximum orminimum pf occurs on the lower bound, upper bound, or at an interior point of Yi . Next wedemonstrate how to derive ��p/��i , when the maximum pf occurs on the upper bound of Yi andthe minimum pf occurs on the lower bound of Yi . The derivations of other cases are given inAppendix B, and the common equations used in derivations are given in Appendix A.

The problem can be stated asGiven: G=g(X,Y), Y∼i =(Y1,Y2, . . . ,Yi−1,Yi+1, . . . ,Yny), yi =(yLi + yUi )/2, pLf occurs at yLi ,

and pUf occurs at yUi .Find: ��p/��i .

��p

��i= �(pUf − pLf )

��i= �[pUf (yi + 1

2�i ,Y∼i )− pLf (yi − 12�i ,Y∼i )]

��i

= �[pUf (yi + 12�i ,Y∼i )]

�(yi + 12�i )

�(yi + 12�i )

��i− �[pLf (yi − 1

2�i ,Y∼i )]�(yi − 1

2�i )

�(yi − 12�i )

��i

=(1

2

)�[pUf (yi + 1

2�i ,Y∼i )]�(yi + 1

2�i )−(

−1

2

)�[pLf (yi − 1

2�i ,Y∼i )]�(yi − 1

2�i )

= 1

2

(�pUf�yUi

+ �pLf�yLi

)(15)

�pUf /�yUi and �pLf /�yLi then need to be calculated. In this case, the MPPs of pLf and pUf are onone bound of Yi . Let h be the bound and pf be pUf or pLf . Then

�pf�h

= �[�(−�)]�h

=−�(−�)��

�h(16)



where �(·) is the PDF of a standard normal distribution. Next, we will show how to calculate��/�h.

Let the MPP be u∗ =(u∗1,u

∗2, . . . ,u

∗nx ) and the corresponding intervals Y at the MPP be y. In

the U-space after X are transformed into U, the limit-state function becomes g(U,Y), and at theMPP the limit-state function is g(u∗,y). Let ∇g(u∗) be the gradient of g(U,Y) in terms of U atthe MPP; namely,

∇g(u∗)=(

�g�U1

∣∣∣∣u∗,y

,�g�U2

∣∣∣∣u∗,y

, . . . ,�g�Un

∣∣∣∣u∗,y

,

)

For brevity, without losing generality, we will drop Y or y in the limit-state function expressionin the following derivations. At the MPP, the following equation holds [40, 41]:

u∗i =−�

∇g(u∗)‖∇g(u∗)‖ (17)

where ∇g(u∗)/‖∇g(u∗)‖ is the unit vector of the gradient, and the gradient is calculated at theMPP, therefore a constant. Then

�Ui

�h=−��

�h

�g�Ui

∣∣∣∣u∗i

‖∇g(u∗)‖ (18)

Recall that yi is on one bound h of the interval variable Yi at the MPP, where G=g(u∗,h)

reaches the limit state and hereby becomes a constant. Then

�G�h

=nx∑i=1

�g�Ui

�Ui

�h+ �g

�h=0 (19)

Therefore, Equation (19) becomes

nx∑i=1

�g�Ui

�Ui

�h+ �g

�h=

nx∑i=1

�g�Ui

⎛⎜⎜⎜⎜⎝−��

�h

�g�Ui

∣∣∣∣u∗i

‖∇g(u∗)‖

⎞⎟⎟⎟⎟⎠+ �g

�h=−��

�h

∑nxi=1

(�g�Ui

∣∣∣∣u∗i

)2

‖∇g(u∗)‖ + �g�h

= −��

�h‖∇g(u∗)‖+ �g

�h=0 (20)

We then obtain

��

�h=

�g�h

‖∇g(u∗)‖ (21)



Substituting ��/�h in Equation (16) into Equation (21) yields

�pf�h

=−�(−�)��

�h=−�(−�)

�g�h

‖∇g(u∗)‖ = −�(−�)

‖∇g(u∗)‖�g�h

(22)

Using the results from Equations (22) and (15), we get the equation of Type I sensitivity whenpmaxf occurs on the upper bound of Yi and pmin

f occurs on the lower bound of Yi as follows:

��p

��i= 1

2

(�pUf�yUi

+ �pLf�yLi

)=−1

2

⎛⎝ �(−�U)

‖∇g(u∗,U)‖�g�Yi

∣∣∣∣∣yUi

+ �(−�L)

‖∇g(u∗,L)‖�g�Yi

∣∣∣∣∣yLi

⎞⎠ (23)

where �U is the reliability index at the maximum pf, �L is the reliability index at the minimumpf, u∗,U is the MPP for the maximum pf, and u∗,L is the MPP for the minimum pf. The equationsof Type I sensitivity for other situations are given in Appendix B.

3.2. Type II sensitivity � pf/��i

� pf/��i is the sensitivity of the average pf, pf, with respect to �i . pf is defined by

pf= pUf + pLf2

(24)

The relationship among pUf , pLf , �p, and pf is illustrated in Figure 3. This type of sensitivity

quantifies the rate of change of the mean value of pf due to the change of the interval width ofYi . The equations of this type of sensitivity are given in Appendix C.

3.3. Type III sensitivity ��p/�yi

��p/�yi is the sensitivity of the width of the probability of failure �p with respect to the averageof the i th interval variable, yi . yi is defined by

yi = yUi + yLi2

(25)

The relationship among yLi , yUi , �i , and yi is illustrated in Figure 4. This type of sensitivity

is useful when we can control the averages of the interval variables during reliability-basedoptimization. We can efficiently decrease the reliability gap by shifting averages of interval variablesto which the probability of failure is highly sensitive. The equations of this type of sensitivity aregiven in Appendix D.

Figure 3. pUf , pLf , �i , and pf.



Figure 4. yLi , yUi , �i , and yi .

3.4. Type IV sensitivity � pf/�yi

� pf/�yi is the sensitivity of the average probability of failure pf with respect to yi . It tells us howmuch the average probability of failure will change given the change in the midpoint of an intervalvariable. The equations of this type of sensitivity are given in Appendix E.

3.5. Type V sensitivity ��p/�qi

��p/�qi is the sensitivity of the width of the probability of failure �p with respect to a distributionparameter, qi , of random variable Xi . For example, for a normal distribution, qi would be themean �i or standard deviation �i , while for uniform distribution, qi could be one of the intervalbounds. As shown previously, the pf gap �p is mainly caused by interval variables [38]. On theother hand, the value of pf primarily depends on random variables. The equations of this type ofsensitivity are given in Appendix F.

3.6. Type VI sensitivity � pf/�qi

� pf/�qi is the sensitivity of the average probability of failure pf with respect to a distributionparameter, qi , of random variable Xi . The equations of this type of sensitivity are given inAppendix G. The equations of Type V and VI sensitivities for a normal distribution are also givenin Appendices F and G, respectively.

3.7. Equations of all the sensitivity indices

The equations for all the above sensitivity indices are summarized in Tables II–IV.In the above table, w is given in Equation (A11) in Appendix A.The procedure to calculate the sensitivity indices is illustrated in Figure 5. First, unified reliability

analysis is conducted to obtain MPPs and interval variables at pUf and pLf . Then depending onthe location of the interval variables, either interior or on a bound, at the MPP, the correspondingequations from Tables II–IV are used to calculate the sensitivity indices.

4. NUMERICAL EXAMPLES

Two examples are used to demonstrate our proposed sensitivity measures with random and intervalvariables. The first example deals with normally distributed variables, while the second examplehandles random variables with both normal and non-normal distributions.



TableII.Ty

pesIandII

sensitivitiesforintervals.

Case

Type

I��

p/��

i(A

ppendixB)

Type

II�p

f/��

i(A

ppendixC)

1pU f

occurs

atyU i

;pL f

occurs

atyL i

−1 2

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yU i

+�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yL i

]−1 4

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yU i

−�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yL i

]

2pU f

occurs

atyL i

;pL f

occurs

atyU i

1 2

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yL i

+�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yU i

]1 4

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yL i

−�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yU i

]

3pU f

occurs

atyU i

;pL f

occurs

atan

interior

point

−1 2

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yU i

]−1 4

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yU i

]

4pU f

occurs

atyL i

;pL f

occurs

atan

interior

point

−1 2

[�(�

U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yL i

]1 4

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yL i

]

5pL f

occurs

atyU i

;pU f

occurs

atan

interior

point

1 2

[�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yU i

]−1 4

[�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yU i

]

6pL f

occurs

atyL i

;pU f

occurs

atan

interior

point

−1 2

[�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yL i

]1 4

[�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yL i

]7

pU fandpL f

both

occurat

interior

points

00

TableIII.Ty

peIIIandIV

sensitivitiesforintervals.

Case

Type

III��

p/�y

i(A

ppendixD)

Type

IV�p

f/�y

i(A

ppendixE)

1pU f

occurs

atyU i

;pL f

occurs

atyL i

−�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yU i

+�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yL i

−1 2

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yU i

+�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yL i

]

2pU f

occurs

atyL i

;pL f

occurs

atyU i

− �(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yL i

+�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yU i

−1 2

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yL i

+�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yU i

]

3pU f

occurs

atyU i

;pL f

occurs

atan

interior

point

−�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yU i

−1 2

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yU i

]

4pU f

occurs

atyL i

;pL f

occurs

atan

interior

point

−�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yL i

−1 2

[�(−

�U)

‖∇g(u∗

,U)‖

�g �Yi

∣ ∣ ∣ yL i

]

5pL f

occurs

atyU i

;pU f

occurs

atan

interior

point

�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yU i

−1 2

[�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yU i

]

6pL f

occurs

atyL i

;pU f

occurs

atan

interior

point

�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yL i

−1 2

[�(−

�L)

‖∇g(u∗

,L)‖

�g �Yi

∣ ∣ ∣ yL i

]7

pU fandpL f

both

occurat

interior

points

00



Table IV. Type V and VI sensitivities for random variables.

Case Type V ��p/�q (Appendix F) Type VI � pf/�q (Appendix G)

General −�(−�U)u∗,Ui�U

�w�qi

+�(−�L)u∗,Li�L

�w�qi

− 12

[�(−�U)

u∗,Li�U

�w�qi

+�(−�L)u∗,Li�L

�w�qi

]

Figure 5. The procedure to calculate sensitivity indices.

Figure 6. A double-lap joint design of adhesive.

4.1. Example 1—adhesive bonding example

A double-lap joint design of a rubber-modified epoxy-based adhesive [45] is illustrated in Figure 6.The design consists of aluminum outer adherends and an inner steel adherend. The assembly iscured at 250◦F and is stress-free at temperature T1. The completed bond is subjected to an axialload P at a service temperature T2. The coefficients of thermal expansion for the outer and inneradherend �o and �i are 6×10−6 and 13×10−6 in/(in◦F), respectively. The modulus Eo and thethickness to, of the outer adherend, and the modulus Ei and the thickness ti, of the inner adherend,are random variables. The shear modulus G, width b, length L , of the adhesive, and the lap-shearstrength of adhesive Sa are also random variables. Their distributions are given in Table V.

Because it is difficult to spread the adhesive uniformly over the surface, the thickness of theadhesive is estimated to be in an interval shown in Table VI. The temperature change, �T =T2−T1,is difficult to fit into some probability distribution since the temperature field is unknown. Aninterval is therefore assigned for �T as listed in Table VI.



Table V. Random variables X .

Variable Mean Standard deviation Distribution

X1(Eo) 10×106 psi 0.1×106 psi NormalX2(Ei) 30×106 psi 0.3×106 psi NormalX3(to) 0.15 in 0.0015 in NormalX4(ti) 0.10 in 0.001 in NormalX5(G) 0.2×106 psi 0.002×106 psi NormalX6(b) 1 in 0.01 in NormalX7(L) 1.1 in 0.011 in NormalX8(P) 2000 psi 20 psi NormalX9(Sa) 4100 psi 41 psi Normal

Table VI. Interval variables.

Variable Lower bound Upper bound

Y1(h) 0.0195 in 0.0205 inY2(�T ) −131.0◦F −129.0◦F

The limit-state function is the safety margin for strength requirement of the joint, which isdefined by the difference between the lap-shear strength of adhesive and the maximum shear stress�max. The equation is obtained at x=0.5 where the maximum shear stress occurs. The function isgiven by

G=g(X,Y)= Sa−�max

where �max=�(0.5).

�(x) = P

4b sinh(L/2)cosh(x)+

[P

4bcosh(L/2)

(2Eoto−Eiti2Eoto+Eiti

)

+ (�i−�o)�T

(1/Eoto+2/Eiti)cosh(L/2)

]sinh(x)

and

=√G

h

(1

Eoto+ 2

Eiti

)

The failure event is defined by F={X,Y|g(X,Y)<0}.The analysis results are listed in Tables VII–X. To verify the proposed method, additional relia-

bility analyses are also conducted. The results are shown as ‘Numerical verification’ in Table VIII(for interval variables) and Table X (for random variables). Each parameter (the average or widthof an interval variable, or a distribution parameter of a random variable), with respect to whicha sensitivity index is to be calculated, is increased by a small step size. An additional reliabilityanalysis for that parameter is then performed. The rate of change in the reliability analysis resultswith respect to the parameter is computed. The rate should be very close to the sensitivity index



Table VII. Bounds of the probability of failure.

Probability of failure pLf pUf pf �p

pf 7.797×10−5 1.067×10−2 5.374×10−3 1.059×10−2

Table VIII. Sensitivity with respect to interval variables.

Proposed method Numerical verification

Type of sensitivity Y1 Y2 Y1 Y2

Type I ��p/��i 5.009×10−2 9.309×10−6 5.071×10−2 9.379×10−6

Type II � pf/��i −2.494×10−2 −4.655×10−6 −2.525×10−2 −4.669×10−6

Type III ��p/�yi −9.978×10−2 −1.862×10−5 −1.001×10−1 −1.868×10−5

Type IV � pf/�yi 5.009×10−2 9.309×10−6 5.071×10−2 9.379×10−6

Table IX. The change of �p and pf with 1% increase in �i and yi .

Type of sensitivity �+1%1 �+1%

2

Type I ��p/��i 5.009×10−7 1.862×10−7

Type II � pf/��i −2.494×10−7 −9.310×10−8

Type III ��p/�yi −1.996×10−3 2.241×10−5

Type IV � pf/�yi 1.002×10−3 −1.210×10−5

calculated from the proposed method. Both Tables VIII and X show good consistency and verifythe accuracy of the proposed method.

The sign of a sensitivity index gives a possible direction for improvement. For example, inTable VIII ��p/��1 and ��p/��2 are both positive, while ��p/�y1 and ��p/�y2 are both negative.Therefore, if we wish to reduce the bounds of pf, we could narrow the intervals of the thicknessof adhesive (�1) and the temperature change (�2) or increase their averages (y1 and y2). A similarconclusion can be drawn for ��p/�yi and � pf/�yi .

To better interpret the sensitivity analysis results, the percentage change in Table IX is alsoincluded. �+1%

yi indicates the change in �p or pf corresponding to the 1% increase in �i or

yi , respectively. For instance, if �1 is increased by 1%, or �1 is increased by (yU1 − yL1 )×1%=(0.0205−0.0195)×1%=1.0×10−5 in, the width of the probability of failure bounds �p wouldincrease by (5.009×10−2)×(1.0×10−5)=5.009×10−7, where the multiplier is the change in �1while the multiplicand is the Type I sensitivity index. Similarly, the average probability of failurepf would change by (−2.494×10−2)×(1.0×10−5)=−2.494×10−7. Since the sign is negative,pf would actually decrease. This example indicates how the change in input uncertainty impactsreliability or the probability of failure. A sensitivity index also tells us the relative importance ofuncertain variables. For example, Y1 has higher �+1% of Type I–IV sensitivity indices than thoseof Y2; Y1 is therefore more significant than Y2 in terms of its impact on �p and pf.



Table X. Sensitivity with respect to random variables.

Proposed method Numerical validation

Type V ��p/�q Type VI � pf/�q Type V ��p/�q Type VI � pf/�q

X1(�1) 1.091×10−10 5.475×10−11 1.097×10−10 5.509×10−11

X1(�1) 4.566×10−11 2.295×10−11 4.631×10−11 2.328×10−11

X2(�2) −2.097×10−11 −1.054×10−11 −2.083×10−11 −1.046×10−11

X2(�2) 5.066×10−12 2.550×10−12 4.997×10−12 2.515×10−12

X3(�3) 7.270×10−3 3.650×10−3 7.315×10−3 3.673×10−3

X3(�3) 3.044×10−3 1.530×10−3 3.087×10−3 1.552×10−3

X4(�4) −6.292×10−3 −3.161×10−3 −6.250×10−3 −3.139×10−3

X4(�4) 1.520×10−3 7.650×10−4 1.499×10−3 7.546×10−4

X5(�5) 9.818×10−9 4.931×10−9 9.780×10−9 4.911×10−9

X5(�5) 7.402×10−9 3.723×10−9 7.330×10−9 3.687×10−9

X6(�6) −1.913×10−3 −9.608×10−4 −1.912×10−3 −9.601×10−4

X6(�6) 1.405×10−3 7.068×10−4 1.404×10−3 7.060×10−4

X7(�7) −8.018×10−3 −4.026×10−3 −8.011×10−3 −4.022×10−3

X7(�7) 2.715×10−2 1.365×10−2 2.731×10−2 1.373×10−2

X8(�8) 9.421×10−7 4.731×10−7 9.428×10−7 4.735×10−7

X8(�8) 6.815×10−7 3.428×10−7 6.817×10−7 3.429×10−7

X9(�9) −1.079×10−6 −5.417×10−7 −1.078×10−6 −5.411×10−7

X9(�9) 1.832×10−6 9.213×10−7 1.831×10−6 9.210×10−7

Table X shows sensitivities in terms of the mean and standard deviation of random variables.The positive signs of ��p/�q and � pf/�q imply that the distribution parameters need to be loweredto reduce �p and pf. In addition, the negative ones suggest that distribution parameters need to beincreased to reduce �p and pf. From this table, it can be concluded that X7 has the highest impacton �p and pf because it has the highest sensitivity index value. Given the positive signs of Type Vand VI sensitivity indices of X7, reducing the mean and variance of X7 would be more efficientthan adjusting other random variables in order to lower �p and pf.

4.2. Example 2—cantilever tube

In Example 1, all random variables are normally distributed. In this example, some random variablesfollow non-normal distributions. The cantilever tube shown in Figure 7 is subject to external forcesF1, F2, and P , and torsion T [44]. The limit-state function is defined as the difference betweenthe yield strength S and the maximum stress �max, namely,

G=g(X,Y)= S−�max

where �max is the maximum von Mises stress on the top surface of the tube at the origin and isgiven by

�max=√

�2x +3�2zx



Figure 7. Cantilever tube.

The normal stress �x is calculated by

�x = P+F1 sin 1+F2 sin 2A

+ Mc

I

where the first term is the normal stress due to the axial forces, and the second term is the normalstress due to the bending moment M , which is given by

M=F1L1 cos 1+F2L2 cos 2

and

A = �

4[d2−(d−2t)2]

c = d/2

I = �

64[d4−(d−2t)4]

The torsional stress �zx at the same point is calculated by

�zx = Td

2J

where J =2I .The random and interval variables are given in Tables XI and XII, respectively.The results of reliability analysis and sensitivity are listed in Tables XIII–XV. It is noted that

sensitivity indices of ��p/��i and ��p/�yi are all positive, while sensitivity indices of � pf/��iand � pf/�yi are all negative. In this case, the direction of the change in �p will be opposite to thedirection of change in pf whenever we adjust �i and yi . For instance, decreasing �1 will result ina lower �p and a higher pf.

In this example, uniform distributions and a Gumbel distribution are involved. In Table XVI, thesensitivities in terms of the parameters of these two distributions are also calculated. It is indicatedthat Type V and VI sensitivities of uniformly distributed variables, X3 and X4, are all positive.Hence, if we raise or lower the bounds of X3 and X4, the change of �p and pf will follow thesame direction.



Table XI. Random variables.

Variable Parameter 1 Parameter 2 Distribution

X1(t) 5mm (mean) 0.1mm (std∗) NormalX2(d) 42mm (mean) 0.5mm (std) NormalX3(L1) 119.75mm (lb†) 120.25mm (ub‡) UniformX4(L2) 59.75mm (lb) 60.25mm (ub) UniformX5(F1) 3.0 kN (mean) 0.3 kN (std) NormalX6(F2) 3.0 kN (mean) 0.3 kN (std) NormalX7(P) 12.0 kN (mean) 1.2 kN (std) GumbelX8(T ) 90.0Nm (mean) 9.0Nm (std) NormalX9(Sy) 220.0MPa (mean) 22.0MPa (std) Normal

∗: std—standard deviation.†: lb—the lower bound of a uniform distribution.‡: ub—the upper bound of a uniform distribution.

Table XII. Interval variables.

Variable Lower bound Upper bound

Y1(1) 0◦ 10◦Y2(2) 5◦ 15◦

Table XIII. Bounds of probability of failure.

Probability of failure pLf pUf pf �p

pf 1.437×10−4 1.631×10−4 1.530×10−4 1.940×10−5

Table XIV. Sensitivity of interval variables.


Type of sensitivity Y1 Y2 Y1 Y2

Type I ��p/��i 1.038×10−4 5.861×10−5 1.034×10−4 5.837×10−5

Type II � pf/��i −5.192×10−5 −2.930×10−5 −5.170×10−5 −2.919×10−5

Type III ��p/�yi 2.077×10−4 1.172×10−4 2.068×10−4 1.167×10−4

Type IV � pf/�yi −1.038×10−4 −5.861×10−5 −1.034×10−4 −5.837×10−5

5. CONCLUSIONS

When information or knowledge is not adequate to build probability distributions, interval variablesmay be used. In this case, PSA approaches are no longer applicable. An effective sensitivityanalysis method is proposed to handle the mixture of random variables and interval variables.

With the presence of both random and interval variables, reliability and the probability of failurereside between their lower and upper bounds. In this work, based on the unified reliability analysis



Table XV. The change of �p and pf with 1% increase in �i and yi .

Type of sensitivity �+1%y1 �+1%

y2

Type I ��p/��i 1.038×10−5 5.861×10−6

Type II � pf/��i −5.192×10−6 −2.930×10−6

Type III ��p/�yi 1.039×10−5 5.860×10−6

Type IV � pf/�yi −5.190×10−6 −2.931×10−6

Table XVI. Sensitivity of random variables.


Type V ��p/�q Type VI � pf/�q Type V ��p/�q Type VI � pf/�q

X1(�1) −5.822×10−2 −4.886×10−1 −5.820×10−2 −4.886×10−1

X1(�1) 1.614×10−2 1.457×10−1 1.615×10−2 1.458×10−1

X2(�2) −2.413×10−2 −1.888×10−1 −2.413×10−2 −1.888×10−1

X2(�2) 1.393×10−2 1.088×10−1 1.394×10−2 1.089×10−1

X3(a3) 1.093×10−3 8.412×10−3 1.093×10−3 8.413×10−3

X3(b3) 1.130×10−3 8.697×10−3 1.137×10−3 8.742×10−3

X4(a4) 1.123×10−3 7.893×10−3 1.124×10−3 7.894×10−3

X4(b4) 1.162×10−3 8.143×10−3 1.167×10−3 8.163×10−3

X5(�5) 7.630×10−8 6.197×10−7 7.631×10−8 6.197×10−7

X5(�5) 8.347×10−8 7.033×10−7 8.355×10−8 7.040×10−7

X6(�6) 3.908×10−8 3.117×10−7 3.908×10−8 3.117×10−7

X6(�6) 2.192×10−8 1.779×10−7 2.193×10−8 1.780×10−7

X7(�7) 5.002×10−9 4.256×10−8 5.002×10−9 4.256×10−8

X7(�7) 5.139×10−10 5.670×10−9 5.143×10−10 5.674×10−9

X8(�8) 5.678×10−8 5.050×10−7 5.688×10−8 5.049×10−7

X8(�8) 1.363×10−9 1.402×10−8 1.363×10−9 1.402×10−8

X9(�9) −2.887×10−12 −2.457×10−11 −2.886×10−12 −2.457×10−11

X9(�9) 8.708×10−12 8.108×10−11 8.740×10−12 8.146×10−11

(URA) framework [39], we have explored various sensitivity indices with respect to both randomand interval variables. Four new types of sensitivity for interval variables include the sensitivitiesof the width and average of the probability of failure bounds with respect to the interval widthand with respect to the mean of each interval variable. Two new types of sensitivity for randomvariables include the sensitivities of the width and average of the probability of failure with respectto the distribution parameters of each random variable. Equations for the six sensitivity indices arederived. Through the URA and the FORM, the sensitivity indices are calculated after reliabilityanalysis is completed without calling the limit-state function again. The sensitivity indices aretherefore a byproduct of reliability analysis.

The advantages of the proposed methods are as follows: (1) the method is easy to use because itemploys the FORM, which is widely used in industry, (2) sensitivity information is just a byproductof reliability analysis, (3) both random and interval variables can be handled by reliability analysis



at the same time, and (4) the computation is efficient without a double-loop procedure or MonteCarlo simulation involved.

The method has some limitations. Since it is based on only the FORM, the method cannot bedirectly extended to the second-order reliability method. The method assumes the existence ofglobal optimal solution if optimization is used for IA. The method may not provide an accuratesolution if a global optima is not reached. It is well known that FORM may not be accuratewhen multiple MPPs exist. The proposed method exhibits the same behavior of the multiple MPPssituation.

Future work would be the further improvement of efficiency and the inclusion of more sensitivityindices. For higher efficiency, efficient interval arithmetic could be used for IA. Other sensitivitymethods, such as those suggested in [46], could also be incorporated.

APPENDIX A: COMMON EQUATIONS

1. Derivative of pf with respect to one bound of an interval variable Yi .�pf/�h is given in Equation (22) and is rewritten below.

�pf�h

= −�(−�)

‖∇g(u∗)‖�g�h

(A1)

where pf could be pLf or pUf , and � could be �L and �U.2. Derivative of pf with respect to the width of an interval, �i .

If pf occurs at yUi ,

�pf��i

= �pf(yUi ,Y∼i )

��i= �pf(yi + 1

2�i ,Y∼i )

��i= �pf(yi + 1

2�i ,Y∼i )

�(yi + 12�i )

�(yi + 12�i )

��i(A2)

Equation (A2) can then be simplified to

�pf��i

= 1

2

�pf(yi + 12�i ,Y∼i )

�(yi + 12�i )

= 1

2

�pf�yUi

(A3)

Similarly, if pf occurs at yLi , the equation becomes

�pf��i

=−1

2

�pf(yi − 12�i ,Y∼i )

�(yi − 12�i )

=−1

2

�pf�yLi

(A4)

If pf occurs at an interior point �yi , which is not dependent of �i , then it can be shown that

�pf��i

= �pf(�yi ,Y∼i )

��i= �pf(

�yi ,Y∼i )

��yi

��yi

��i= �pf(

�yi ,Y∼i )

��yi

·0=0 (A5)

3. Derivative of pf with respect to the average of an interval, yi If pf occurs at yUi , one canobtain

�pf�yi

= �pf(yUi ,Y∼i )

�yi= �pf(yi + 1

2�i ,Y∼i )

�yi= �pf(yi + 1

2�i ,Y∼i )

�(yi + 12�i )

�(yi + 12�i )

�yi(A6)



and therefore

�pf�yi

= �pf(yi + 12�i ,Y∼i )

�(yi + 12�i )

= �pf�yUi

(A7)

Similarly, if pf occurs at yLi ,

�pf�yi

= �pf(yi − 12�i ,Y∼i )

�(yi − 12�i )

= �pf�yLi

(A8)

If pf occurs at an interior point �yi ,

�pf�yi

= �pf(�yi ,Y∼i )

�yi= �pf(

�yi ,Y∼i )

��yi

��yi

�yi= �pf(

�yi ,Y∼i )

��yi

·0=0 (A9)

4. Derivative of pf bound with respect to a distribution parameter qi

�pf�qi

= ��(−�)

�qi=−�(−�)

��

�u∗i

�u∗i

�qi(A10)

If the CDF of Xi is FXi (xi ), then

u∗i =�−1[FXi (x

∗i )]=w(q1,q2, . . . ,qm) (A11)

where m is the number of distribution parameters.Then from �=‖u∗

i ‖, one obtains

�pf�qi

=−�(−�)u∗i

�

�w

�qi(A12)

APPENDIX B: EQUATIONS FOR TYPE I SENSITIVITY ��p/��i

Case 1: pLf occurs at yLi and pUf occurs at yUi (see Section 3).Case 2: pLf occurs at yUi and pUf occurs at yLi .

��p

��i= �[pUf (yi − 1

2�i , y∼i )− pLf (yi + 12�i , y∼i )]

��i(B1)

Using Equations (A3) and (A4) gives

��p

��i=−1

2

(�pUf�yLi

+ �pLf�yUi

)(B2)

Then from Equation (A1),

��p

��i=−1

2

(�pUf�yLi

+ �pLf�yUi

)=−1

2

⎡⎣ −�(−�U)


∣∣∣∣∣yLi

+ −�(−�L)


∣∣∣∣∣yUi

⎤⎦ (B3)



Case 3: pLf occurs at an interior point �yi and pUf occurs at yUi .

��p

��i= �[pUf (yi + 1

2�i ,Y∼i )− pLf (�yi ,Y∼i )]

��i(B4)

Using Equations (A3) and (A5), one obtains

��p

��i= 1

2

�pUf�yUi

(B5)

Applying the results from Equation (A1) yields

��p

��i= 1

2

�pUf�yUi

= 1

2

⎡⎣ −�(−�U)


∣∣∣∣∣yUi

⎤⎦ (B6)

Case 4: pLf occurs at an interior point �yi and pUf occurs at yLi .

��p

��i= �[pUf (yi − 1

2�i ,Y∼i )− pLf (�yi ,Y∼i )]

��i(B7)

Using Equations (A4) and (A5) yields

��p

��i=−1

2

�pUf�yLi

(B8)

Applying Equation (A1) yields

��p

��i=−1

2

⎡⎣ −�(−�U)


∣∣∣∣∣yLi

⎤⎦ (B9)

Case 5: pLf occurs at yUi and pUf occurs at an interior point �yi .

��p

��i= �[pUf (

�yi ,Y∼i )− pLf (yi + 1

2�i ,Y∼i )]��i

(B10)


��p

��i=−1

2

�pLf�yUi

(B11)

Using Equation (A1) yields

��p

��i=−1

2

⎡⎣ −�(−�L)


∣∣∣∣∣yUi

⎤⎦ (B12)

Case 6: pLf occurs at yLi and pUf occurs at an interior point �yi .

��p

��i= �[pUf (

�yi ,Y∼i )− pLf (yi − 1

2�i ,Y∼i )]��i

(B13)




��p

��i= 1

2

�pLf�yLi

= 1

2

⎡⎣ −�(−�L)


∣∣∣∣∣yLi

⎤⎦ (B14)


��p

��i= 1

2

⎡⎣ −�(−�L)


∣∣∣∣∣yLi

⎤⎦ (B15)

Case 7: pLf and pUf occur at two interior points �yi1 and �

yi2, respectively.

��p

��i= �[pUf (

�yi1,Y∼i )− pLf (

�yi2,Y∼i )]

��i(B16)


��p

��i=0 (B17)

APPENDIX C: EQUATIONS FOR TYPE II SENSITIVITY � pf/��i

Case 1: pLf occurs at yLi and pUf occurs at yUi .

� pf��i

= �{ 12 [pUf (yi + 12�i ,Y∼i )+ pLf (yi − 1

2�i ,Y∼i )]}��i

(C1)


� pf��i

= 1

4

(�pUf�yUi

− �pLf�yLi

)(C2)

From Equation (A1)

� pf��i

= 1

4

⎛⎝ −�(−�U)


∣∣∣∣∣yUi

− −�(−�L)


∣∣∣∣∣yLi

⎞⎠ (C3)

Case 2: pLf occurs at yUi and pUf occurs at yLi .

� pf��i

= �{ 12 [pUf (yi − 12�i ,Y∼i )+ pLf (yi + 1

2�i ,Y∼i )]}��i

(C4)


� pf��i

= 1

4

(−�pUf

�yLi+ �pLf

�yUi

)(C5)



From Equation (A1)

� pf��i

= 1

4

⎡⎣− −�(−�U)


∣∣∣∣∣yLi

+ −�(−�L)


∣∣∣∣∣yUi

⎤⎦ (C6)


� pf��i

= �{ 12 [pUf (yi + 12�i ,Y∼i )+ pLf (

�yi ,Y∼i )]}

��i(C7)


� pf��i

= 1

4

�pUf�yUi

(C8)

From Equation (A1)

� pf��i

= 1

4

⎡⎣ −�(−�U)


∣∣∣∣∣yUi

⎤⎦ (C9)


� pf��i

= �{ 12 [pUf (yi − 12�i ,Y∼i )+ pLf (

�yi ,Y∼i )]}

��i(C10)


� pf��i

=−1

4

�pUf�yLi

(C11)

Applying Equation (A1), one obtains

� pf��i

=−1

4

⎡⎣ −�(−�U)


∣∣∣∣∣yLi

⎤⎦ (C12)

Case 5: pLf occurs at yUi and pUf occurs at an interior point �yi

� pf��i

= �{ 12 [pUf (�yi ,Y∼i )+ pLf (yi + 1

2�i ,Y∼i )]}�(�i )

(C13)


� pf��i

= 1

4

�pLf�yUi

(C14)




� pf��i

= 1

4

[−�(−�L)


∣∣∣∣yUi

](C15)


� pf��i

= �{ 12 [pUf (�yi ,Y∼i )+ pLf (yi − 1

2�i ,Y∼i )]}��i

(C16)


� pf��i

=−1

4

�pLf�yLi

(C17)

From Equation (A1)

� pf��i

=−1

4

⎡⎣ −�(−�L)


∣∣∣∣∣yLi

⎤⎦ (C18)


yi2, respectively.

��p

��i= �{ 12 [pUf (

�yi1,Y∼i )+ pLf (

�yi2,Y∼i )]}

��i(C19)


��p

��i=0 (C20)

APPENDIX D: EQUATIONS FOR TYPE III SENSITIVITY ��p/�yi


��p

�yi= �(pUf − pLf )

�yi= �[pUf (yi + 1

2�i ,Y∼i )− pLf (yi − 12�i ,Y∼i )]

�yi(D1)


��p

�yi= �pUf

�yi− �pLf

�yi= �pUf

�yUi− �pLf

�yLi(D2)

From Equation (A1)

��p

�yi= −�(−�U)


∣∣∣∣∣yUi

− −�(−�L)


∣∣∣∣∣yLi

(D3)




��p


�yi= �[pUf (yi − 1

2�i ,Y∼i )− pLf (yi + 12�i ,Y∼i )]

�yi(D4)


��p

�yi= �pUf

�yi− �pLf

�yi= �pUf

�yLi− �pLf

�yUi(D5)

Applying the results of Equation (A1) yields

��p

�yi= −�(−�U)


∣∣∣∣∣yLi

− −�(−�L)


∣∣∣∣∣yUi

(D6)


��p


�yi= �[pUf (yi + 1

2�i ,Y∼i )− pLf (�yi ,Y∼i )]

�yi(D7)


��p

�yi= �pUf

�yi= �pUf

�yUi(D8)

By Equation (A1)

��p

�yi= −�(−�U)


∣∣∣∣∣yUi

(D9)


��p


�yi= �[pUf (yi − 1

2�i ,Y∼i )− pLf (�yi ,Y∼i )]

�yi(D10)


��p

�yi= �pUf

�yi= �pUf

�yLi(D11)

By Equation (A1)

��p

�yi= −�(−�U)


∣∣∣∣∣yLi

(D12)


��p


�yi= �[pUf (

�yi ,Y∼i )− pLf (yi + 1

2�i ,Y∼i )]�yi

(D13)




��p

�yi=−�pLf

�yi=−�pLf

�yUi(D14)


��p

�yi= − −�(−�L)


∣∣∣∣∣yUi

(D15)


��p


�yi= �[pUf (

�yi ,Y∼i )− pLf (yi − 1

2�i ,Y∼i )]�yi

(D16)


��p

�yi=−�pLf

�yi=−�pLf

�yLi(D17)


��p

�yi= − −�(−�L)


∣∣∣∣∣yLi

(D18)


yi2, respectively.

��p

�yi= �[pUf (

�yi1,Y∼i )− pLf (

�yi2,Y∼i )]

�yi(D19)


��p

�yi=0 (D20)

APPENDIX E: EQUATIONS FOR TYPE IV SENSITIVITY � pf/�yi


� pf�yi

=�

(pUf + pLf

2

)

�yi=

�{1

2

[pUf

(yi + 1

2�i ,Y∼i

)+ pLf

(yi − 1

2�i ,Y∼i

)]}�yi

(E1)


� pf�yi

= 1

2

(�pUf�yi

+ �pLf�yi

)= 1

2

(�pUf�yUi

+ �pLf�yLi

)(E2)




� pf�yi

= 1

2

⎡⎣ −�(−�U)


∣∣∣∣∣yUi

+ −�(−�L)


∣∣∣∣∣yLi

⎤⎦ (E3)


� pf�yi

=�

(pUf + pLf

2

)

�yi=

�{1

2

[pUf

(yi − 1

2�i ,Y∼i

)+ pLf

(yi + 1

2�i ,Y∼i

)]}�yi

(E4)


� pf�yi

= 1

2

(�pUf�yi

+ �pLf�yi

)= 1

2

(�pUf�yLi

+ �pLf�yUi

)(E5)


� pf�yi

= 1

2

⎡⎣ −�(−�U)


∣∣∣∣∣yLi

+ −�(−�L)


∣∣∣∣∣yUi

⎤⎦ (E6)


� pf�yi

=�

(pUf + pLf

2

)

�yi=

�{1

2

[pUf

(yi + 1

2�i ,Y∼i

)+ pLf

(�yi ,Y∼i

)]}�yi

(E7)


� pf�yi

= 1

2

�pUf�yi

= 1

2

�pUf�yUi

(E8)


� pf�yi

= 1

2

−�(−�U)


∣∣∣∣∣yUi

(E9)


� pf�yi

=�

(pUf + pLf

2

)

�yi=

�{1

2

[pUf

(yi − 1

2�i ,Y∼i

)+ pLf (

�yi ,Y∼i )

]}�yi

(E10)


� pf�yi

= 1

2

�pUf�yi

= 1

2

�pUf�yLi

(E11)




� pf�yi

= 1

2

−�(−�U)


∣∣∣∣∣yLi

(E12)


� pf�yi

=�

(pUf + pLf

2

)

�yi=

�{1

2

[pUf (

�yi ,Y∼i )+ pLf

(yi + 1

2�i ,Y∼i

)]}�yi

(E13)


� pf�yi

= 1

2

�pLf�yi

= 1

2

�pLf�yUi

(E14)


� pf�yi

= 1

2

−�(−�L)


∣∣∣∣∣yUi

(E15)


� pf�yi

=�

(pUf + pLf

2

)

�yi=

�{1

2

[pUf (

�yi ,Y∼i )+ pLf

(yi − 1

2�i ,Y∼i

)]}�yi

(E16)


� pf�yi

= 1

2

�pLf�yi

= 1

2

�pLf�yLi

(E17)


� pf�yi

= 1

2

−�(−�L)


∣∣∣∣∣yLi

(E18)


yi2, respectively.

� pf�yi

= �{ 12 [pUf (�yi1,Y∼i )+ pLf (

�yi2,Y∼i )]}

�yi(E19)

Using Equation (A9) gives

� pf�yi

=0 (E20)



APPENDIX F: EQUATIONS FOR TYPE V SENSITIVITY ��p/�qi

��p

�qi= �(pUf − pLf )

�qi= �pUf

�qi− �pLf

�qi(F1)

Using Equation (A12) gives

��p

�qi=−�(−�U)

u∗,Ui

�U�w

�qi+�(−�L)

u∗,Li

�L�w

�qi(F2)

where u∗,Ui is the MPP at pUf and u∗,L

i is the MPP at pLf .Specifically, for a normal distributed random variable Xi ∼N (�i ,�i ),

w(�i ,�i )=�−1[FXi (x∗i )]=�−1

[�

(x∗i −�i�i

)]= x∗

i −�i�i

(F3)

and then

�w

��i=− 1

�i(F4)

and

�w

��i=− x∗

i −�i�2i

=−u∗i

�i(F5)

From Equation (F2), we can obtain the following sensitivities:(1) qi =�i

��p

��i=−�(−�U)

u∗,Ui

�U�w

��i+�(−�L)

u∗,Li

�L�w

��i=�(−�U)

u∗,Ui

�U�i−�(−�L)

u∗,Li

�L�i(F6)

(2) qi =�i

��p

��i= −�(−�U)

u∗,Ui

�U�w

��i+�(−�L)

u∗,Li

�L�w

��i

= �(−�U)u∗,Ui

�Uu∗,Ui

�i−�(−�L)

u∗,Li

�Lu∗,Li

�i=�(−�L)

(u∗,Ui )2

�L�i−�(−�L)

(u∗,Li )2

�L�i(F7)

APPENDIX G: EQUATIONS FOR TYPE VI SENSITIVITY � pf/�qi

�( pf)

�qi=

�

(pUf + pLf

2

)

�qi= 1

2

(�pUf�qi

+ �pLf�qi

)(G1)



Using Equation (A12), it can be easily shown that

� pf�qi

=−1

2

[�(−�U)

u∗,Li

�U�w

�qi+�(−�L)

u∗,Li

�L�w

�qi

](G2)

Applying the results from Equation (F4) for a normal distributed random variable Xi ∼N (�i ,�i ),the following sensitivities are obtained.

(1) qi =�i

� pf��i

= −1

2

[�(−�U)

u∗,Ui

�U�w

��i+�(−�L)

u∗,Li

�L�w

��i

]

= 1

2

[�(−�U)

u∗,Ui

�U�i+�(−�U)

u∗,Li

�U�i

](G3)

(2) qi =�i

� pf��i

= −1

2

[�(−�U)

u∗,Ui

�U�w

��i+�(−�L)

u∗,Li

�L�w

��i

]

= 1

2

[�(−�U)

u∗,Ui

�Uu∗,Ui

�i+�(−�L)

u∗,Li

�Lu∗,Li

�i

]

= 1

2

[�(−�U)

(u∗,Ui )2

�U�i+�(−�L)

(u∗,Li )2

�L�i

](G4)

ACKNOWLEDGEMENTS

This work is partly supported by the U.S. National Science Foundation (NSF) grant CMMI-0400081,University of Missouri Research Board Grant #7116, and the Intelligent Systems Center at MissouriUniversity of Science and Technology. The presented views are those of the authors and do not representthe position of the funding agencies.

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Reliability sensitivity analysis with random and interval variables