Xin Q. Diesel Engine System Design

266 Diesel engine system design

engine power. This work is one of the earliest attempts that demonstrated

that RSM techniques can be successfully applied to internal combustion

engines.

RSM was also used by Rutter et al. (1996) in engine aftertreatment system

optimization for a catalytic converter of a gasoline engine. They used a

three-level CCF DoE design to study the optimum precious metal loading

of platinum (Pt), palladium (Pd) and rhodium (Rh).

3.4 Optimization of robust design for variability

and reliability

3.4.1 Overview of optimization for variability, reliability,

and robustness

Many related terms in reliability engineering and robust engineering are

used in the existing literature. Sometimes they are misused and this causes

confusion. This section claries the differences among the key concepts

related to optimization, uncertainty, variability, reliability, and robustness

for advanced nondeterministic probabilistic optimization.

Nondeterministic probabilistic optimization

The earlier sections mainly discuss the deterministic optimization techniques

to produce a single deterministic design solution for ‘design for target’ (either

‘design for nominal’ or ‘design for limit’). Under-design or over-design can

occur if the safety margin is not determined properly. The product may still

not perform well in use when all sources of variation present even with an

acceptable nominal design. Deterministic optimal design without considering

variations is generally not reliable, not robust or not economic. The variation

may cause the engine to operate at the conditions that violate the performance

or durability constraints. The deterministic design generated by stacking up

the worst-case tolerance is usually a non-economic over-design. Reliability-

based design optimization (RBDO) and robustness-based optimization are

required to handle variability. They are powerful optimization tools based

on nondeterministic approaches for diesel engine system design.

There are three types of input factors in nondeterministic optimization:

∑ deterministic control factor (also called deterministic design variables

in some other literatures)

∑ nondeterministic or random control factor (also called nondeterministic

or random design variables)

∑ random noise factor (also called random design parameters).

Both the mean and the standard deviation of the control factors can be

Diesel-Xin-03.indd 266 5/5/11 11:46:08 AM

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267Optimization techniques in diesel engine system design

varied, while the settings of the noise factor cannot be controlled or changed.

The output response is a function of the deterministic control factors,

nondeterministic control factors and random noise factors. For example,

fuel injection timing can be selected as a deterministic control factor; engine

compression ratio can be selected as a nondeterministic control factor; and

ambient temperature is a noise factor with random variations and it cannot

be controlled. The objective function of RBDO and robust design can be any

attribute measure such as performance, durability, packaging or cost. The

constraints in the optimization usually include an evaluation of probabilistic

reliability or failure rate. RBDO is used to optimize the system under the

constraint of maximum allowable probability of failure (or conversely

minimum acceptable reliability).

The need for nondeterministic probabilistic evaluation

The most efcient and economical way to handle the variations in design is to

use a probabilistic analysis rather than stacking up the worst-case tolerances.

The Monte Carlo simulation is the most widely used method in this area

and plays a central role in RBDO and robust optimization. The Monte Carlo

simulation uses random sampling to simulate the effect of the statistically

distributed factors on the statistical distribution of the responses. The necessity

of using a probabilistic analysis can be comprehended by using an example

from an investigation conducted by Savage et al. (2007). They presented a

methodology to analyze the effects of the variations from different vehicle

cooling system design variables on the radiator inlet coolant temperature. A

comprehensive list of all the inuential variables and the range of variation

were presented. The statistical distribution of the radiator inlet temperature

was predicted with the Monte Carlo simulation as a result of the variations in

nine factors (i.e., powertrain heat rejection, radiator heat rejection, condenser

heat rejection, coolant ow rate, ram airow rate, condenser air-side pressure

drop, radiator air-side pressure drop, motor speed, and motor voltage).

The radiator inlet temperature had a normal distribution from 239.8∞F to

249.6∞F with a range of 9.8∞F by the Monte Carlo simulation. The worst-

case stacking-up method gave a distribution of 235.0∞F to 255.8∞F with a

range of 20.8∞F. They found that the Monte Carlo simulation provided a

more accurate variation range of the radiator coolant temperature.

Denition of variability-based optimization

Variability-based optimization applies optimization techniques in design for

variability to optimize the design in the presence of a range of variability

in order to make the design reliable or robust. Variability and uncertainty

are different. There are two types of uncertainties: (1) random uncertainty,

Diesel-Xin-03.indd 267 5/5/11 11:46:08 AM



268 Diesel engine system design

which is due to variations related to any attribute in the population and/or

with respect to time; and (2) epistemic uncertainty (Farizal and Nikolaidis,

2007; Donders et al., 2007), which is due to the deciencies in deterministic

simulation models caused by lack of knowledge.

Random uncertainty is also called variability. Strictly speaking, it further

includes two types of variations: (1) variations within the population at a xed

moment in time during the product life, including the variations inherent to

the physical system such as material characteristics, geometrical properties,

and manufacturing tolerances, or the variations caused by changes in usage,

test condition, and environment; and (2) variations with respect to time, i.e.,

a time-dependent degradation (for example, wear-out). For the rst type,

a design parameter may be used as the random variable in the statistical

distribution to characterize the probability, such as the stress used in the

stress–strength interference model. In the second type, time is the random

variable.

Epistemic uncertainty usually produces systematic modeling errors

and affects the entire population of a product in the same way. Random

uncertainty can be handled by a probabilistic approach, while epistemic

uncertainty cannot. RBDO and robust design are usually concerned with

random uncertainty or variability.

Less variability does not mean robustness. A design is called robust if its

response is insensitive to the variations in noise factors. Reliability refers to

the chance of failure of a product. Variability is the cause of failure rate or

reliability problems. Design for reliability (or RBDO) includes two distinct

categories of analysis, namely (1) design for variability (or variability-based

design optimization), which focuses on the variations at a given moment in

time in the product life; and (2) design for time-dependent gradation. Most of

the RBDO analysis in engine applications to date falls in the rst category.

It should be noted that the concepts of variability and reliability can refer

to any performance or durability issues, not just durability alone.

A variability analysis may help dene the engineering goal for the nominal

target value and identify the major contributors to the response so that the

variability can be reduced by changing the tolerances of the control factors.

If the statistical distribution is too wide, design tolerances must be tightened

to bring the variability within an acceptable range. On the other hand, if the

distribution is narrower than necessary, design tolerances can be relaxed in

order to reduce the design and manufacturing costs. In the optimization of

robust design, the mean and its functional limit tolerance range (from LSL or

lower specication limit to USL or upper specication limit) are optimized

simultaneously. A target failure rate (P

failure

) or reliability (R) is used in the

variability-based optimization as a constraint. Note that the reliability is

dened as R = 1 – P

failure

.

Diesel-Xin-03.indd 268 5/5/11 11:46:08 AM

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269Optimization techniques in diesel engine system design

Denition of reliability-based optimization

A design is called reliable if it satises all the requirements of performance

and durability attributes within a specied period of mission time in the

presence of variability. An example of unreliable design is as follows. The

same unit of product may function unreliably at different times on the same

day, or its performance or durability may deteriorate chronically over the

next two years. Another example of an unreliable product is that 5% of 100

units of the same product do not function reliably after the product has been

used for two years. Recall that if 5% of 100 units of the same product do

not function well before the product is put in use (service), the problem is

usually called a design quality or manufacturing quality problem.

A failure in terms of reliability is usually caused by one of the following:

(1) an improper mean value in the control factor; (2) an improper tolerance

in the variability control for the population in the control factor; or (3) the

degradation with respect to time in the mean (nominal) or the tolerance. The

failure can be either a performance attribute (e.g., emissions compliance, fuel

economy) or durability attribute (e.g., crack due to fatigue, engine failures

due to excessively hot radiator outlet coolant temperature).

In RBDO, the constraints are dened for the required reliability, and the

design is optimized by changing the variable settings such that all the criteria

(e.g., performance, durability, packaging, cost, and reliability) are met. The

reliability estimation is usually expressed as a reliability value or failure

rate, and sometimes is expressed in terms of tolerance as multiples of the

standard deviation (s). For example, a 6s design has a higher reliability than

a 3s design, with a very small probability of failure in the order of 10

–10

.

However, the 6s design is also more expensive. The standard deviation may

apply to any variation such as the piece-to-piece variation among multiple

units or time-to-time variation for a given unit. The mean value is designed

at several multiples of the standard deviation (e.g., 3s) away from the

control limit with a prescribed failure rate. In fact, either a higher or lower

failure rate is undesirable because neither under-design nor over-design is

acceptable. More information on the RBDO theory can be found in Gu and

Yang (2003), Mourelatos and Liang (2005, 2006), Rahman et al. (2007),

and Donders et al. (2007).

Denition of robust optimization

As dened earlier, a design is called robust if its response is insensitive to

the variations in noise factors. Robust design is to improve the quality of

the product or process by minimizing the variation in the response without

eliminating the sources of variation. The focus in robust design is usually

to reduce the variation due to noise factors by changing the settings of the

Diesel-Xin-03.indd 269 5/5/11 11:46:08 AM



270 Diesel engine system design

control factors. In design for robustness and robust optimization, the objective

is to evaluate the ‘variance’ of a response parameter. Sometimes a separate

objective or constraint is also formulated for the mean of the response. A

reliability constraint does not necessarily present in this formulation. One

example of robust design is to minimize the variation range of the radiator

coolant temperature caused by the uctuation of heat rejection and ambient

temperature. Another example of robust design is to maintain a stable power

output of a diesel generator to make it insensitive to various noise factors.

More information on the robust design theory can be found in Gu and Yang

(2003) and Mourelatos and Liang (2005, 2006).

Robustness and reliability need to be integrated in the early stage

of the engine development process. Robustness should be included in

engine optimization as an objective, along with the nominal values of fuel

consumption, emissions, and NVH. A robust system exhibits low variation

of its functions in the presence of noise factors. Robust design is to achieve

the robust system by selecting the control factor levels that produce the least

variation in functions. Equally important is tolerance control.

The relationship among attribute optimality, reliability, and robustness

Reliability (low failure rate at time t) and robustness (insensitivity to noise

factors) are two distinct characteristics of a product. However, they share

some common analysis and design methods. They are both closely related

to variability. Therefore, they share a common probabilistic nature and

require a statistical approach to address. High reliability or high robustness

requires varying the mean and/or standard deviation of the control factors.

Both reliability and robustness can be considered in an optimization topic.

In summary, there are trade-offs among the three fundamental design

principles of a product: attribute optimality (e.g., lowest BSFC or cost),

reliability (e.g., long durability B10 life), and high robustness (e.g., small

variation in radiator coolant temperature in response to large changes in

ambient temperatures).

Adding a constraint in an optimization problem generally makes the

attribute objective less optimal. For example, when reliability is added as a

probabilistic constraint, the optimized engine BSFC in a nondeterministic

nominal design will become worse compared to the deterministic case without

any reliability constraint. If more margin is reserved for variability (e.g.,

demanding a higher reliability), the design for BSFC will become even less

optimal. RBDO is the process to move from a deterministic optimum to the

reliability optimum. Similarly, if reliability is formulated as a constraint in an

optimization topic, it is impossible to satisfy robustness without sacricing

the attribute optimality because a reliability optimum may not be robust (i.e.,

insensitive to variation).

Diesel-Xin-03.indd 270 5/5/11 11:46:08 AM

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271Optimization techniques in diesel engine system design

Although a RBDO topic and a robust optimization topic can be formulated

separately to address different issues, in many scenarios they need to be

formulated together as a multi-objective optimization topic in order to

optimize a system mean attribute and simultaneously minimize its standard

deviation (sensitivity to noise) subject to the same probabilistic reliability

constraint. Such a formulation is called combined robust-RBDO. The two

objectives (the mean attribute and the robustness) usually have a trade-off,

i.e., improving one must sacri ce the other. The trade-off can be analyzed

by a Parero optimality frontier calculation.

Mathematical formulation of deterministic optimization

For comparison purpose, the formulation of deterministic design optimization

is given by:

Min

imize

(

)

s

ubject to

()

≤ 0

fX

(fX (

XX

)XX )

gX

()gX()

dr

,

dr

,

fX

dr

fX

XX

dr

XX

p

)

p

)

)XX )

p

)XX )

id

()

id

()

gX

id

gX

()gX()

id

()gX()

,

XX,XX

,,

= 1,

,

≤

im

= 1,im= 1,

im

, im,

XX

≤XX ≤

X

d

XX

d

XX

L

XX

L

XX

d

XX

d

XX

im◊◊im

im◊im

ddd

X

d

X

d

X

d

X

U

Ï

Ì

Ô

Ï

Ô

Ï

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ó

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ó

Ô

3.28

where X

d

is a vector of k

d

-dimensional deterministic control factors

(deterministic design variables), g

i

(X

d

) is a non-probabilistic constraint, X

r

is

a vector of k

r

-dimensional nondeterministic random control factors (random

design variables),

X

r

is the mean of X

r

, X

p

is a vector of k

p

-dimensional

nondeterministic random noise factors (random design parameters),

X

p

X

p

X

is

the mean of X

p

, and

XX

d

XX

d

XX

L

XX

L

XX

d

XX

d

XX

U

and

XX and XX

are the lower and upper bounds of X

d

.

Mathematical formulation of RBDO

A typical formulation of nondeterministic RBDO is expressed as follows:

Min

imize

(

)

s

ubject to

((

fX

(fX (

XX

)XX )

Pg

((Pg((

XX

((XX((

X

dr

,

dr

,

fX

dr

fX

XX

dr

XX

p

)

p

)

)XX )

p

)XX )

id

((

id

((

XX

id

XX

((XX((

id

((XX((

r

,

XX,XX

,,

XX,,XX

r

,,

r

pi

X

pi

X

ppip

X

p

X

pi

X

p

X

d

Ri

pi

Ri

pi

m

X

d

X

d

)

pi

)

pi

≤ 0)

pi

≤ 0)

pi

≥

pi

≥

pi

Ri Ri

pi

Ri Ri

pi

Ri

pi

Ri

pi

= 1,

,

Ri,Ri

,

Ri,Ri

Ri Ri,Ri Ri

◊◊

◊

LLL

d

U

r

L

rr

U

XX

d

XX

d

XX

d

XX

r

XX

r

L

XX

L

X

rr

X

rr

≤

XX XX

d

XX

d

XX

d

XX≤ XX

XX ≤XX

rr

XX XX

≤

rr

≤

rr

Ï

Ì

Ô

Ï

Ô

Ï

ÔÔÔ

Ô

Ì

Ô

Ì

ÔÔÔ

Ô

ÔÔÔ

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ó

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ó

Ô

Ó

Ô

Ó

Ô

Ó

Ô

Ó

3.29

where

X

r

is a vector of random control factors (random design variables)

whose mean and standard deviation can be varied, and

X

p

X

p

X

is a vector of

random noise factors (random design parameters) whose mean and standard

deviation cannot be changed. R

i

is the desired reliability constraint, and

g

i

(X

d

, X

r

, X

p

) > 0 indicates failure. Note that, as an example, this particular

formulation only varies the deterministic control factors and the means of the

Diesel-Xin-03.indd 271 5/5/11 11:46:09 AM



272 Diesel engine system design

random control factors (not their standard deviations) to reach a reliability-

constrained optimum.

Mathematical formulation of robust optimization

A typical reliability-based robust design optimization formulation (i.e.,

including reliability as a constraint) is expressed as follows:

Min

imize

(

,

)

s

ubject to

((

,

fX

(fX (

XX

Pg

((Pg((

XX

((XX((

, XX,

vd

fX

vd

fX

(fX (

vd

(fX (

rp

,

rp

,

XX

rp

XX

, XX,

rp

, XX,

id

((

id

((

XX

id

XX

((XX((

id

((XX((

r

,,

≤ 0)

,

= 1,

,

XR

≤ 0)XR≤ 0)

≥XR ≥

im

= 1,im= 1,

im

, im,

pi

≤ 0)

pi

≤ 0)

,

pi

,

XR

pi

XR

≤ 0)XR≤ 0)

pi

≤ 0)XR≤ 0)

≥XR ≥

pi

≥XR ≥

,XR ,

pi

,XR ,

)

XR)XR

pi

)

pi

)

pi

XR

pi

XR)XR

pi

XR

im◊◊im

im◊im

≤

XX

≤XX ≤

X

XX

≤XX ≤

X

d

XX

d

XX

L

XX

L

XX

d

XX

d

XX

d

X

d

X

U

r

XX

r

XX

L

XX

L

XX

r

rrr

U

dr

pt

ar

ptarpt

ge

argear

t

fX

dr

fX

dr

XX

dr

XX

dr

pt

XX

pt

C

pt

C

pt

(,

dr

(,

dr

fX(,fX

dr

fX

dr

(,

dr

fX

dr

,

dr

,

dr

XX ,XX

dr

XX

dr

,

dr

XX

dr

pt

XX XX

pt

XX

pt

XX

pt

≤

pt

≤

pt

)

pt

)

pt

)

pt

(

This is an optional constr

aint

)

ÏÏÏ

Ì

Ô

ÏÏÏ

Ô

ÏÏÏ

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ó

Ô

3.30

where f

v

(X

d

, X

r

, X

p

) is the variance of objective function f, and C

target

is a

constant of the mean performance target. f

v

= 0 indicates extremely robust,

and it indicates the variations in the input factors do not create any change

in the output response. This particular formulation is to reduce the attribute

variation at the target mean for the attribute. The variation measure f

v

in

robust design is given by

fX

XX

f

x

vd

fX

vd

fX

rp

XX

rp

XX

i

n

i

x

i

(,

fX(,fX

vd

(,

vd

fX

vd

fX(,fX

vd

fX

,

XX ,XX

rp

,

rp

XX

rp

XX ,XX

rp

XX

)

XX )XX

rp

)

rp

XX

rp

XX )XX

rp

XX

=

=1

2

S

∂

f∂f

∂

x∂x

Ê

Ë

Ê

Ë

Ê

Á

Ê

Ë

Á

Ë

Ê

Ë

Ê

Á

Ê

Ë

Ê

ˆ

¯

ˆ

¯

ˆ

˜

ˆ

¯

˜

¯

ˆ

¯

ˆ

˜

ˆ

¯

ˆ

s

2

3.31

The variation can also be expressed by the spread of the probability density

function of the objective function, which is simply the percentile difference

(Mourelatos and Liang, 2006):

D

Rf

DRfD

f

Rf

f

Rf

RR

f

RR

f

Rf = Rf

–

21

f

21

f

RR

21

RR

f

RR

f

21

f

RR

f

–

21

–

3.32

where

ff

RR

ff

RR

ff

1

ff

RR

ff

1

ff

RR

ff

ff and ff

RR

and

RR

ff

RR

ff and ff

RR

ff

2

are a low and high percentile of f, respectively. For

example, the 5th and 95th percentiles can be used.

It should be noted that in many formulations of robust optimization the

mean is optimized (maximize or minimize) and the variance is minimized

simultaneously with a trade-off in such a multi-objective formulation.

Mathematical formulation of combined robust-RBDO

The uni ed methodology of combining robust design with RBDO has been

proposed by several authors (Gu et al., 2004; Mourelatos and Liang, 2005,

Diesel-Xin-03.indd 272 5/5/11 11:46:10 AM



273Optimization techniques in diesel engine system design

2006). Essentially, it is the combination of equations 3.29 and 3.30. The

topic then becomes a multi-objective optimization for two or more objectives

(e.g., at least one for the mean of an attribute and the other for the variance

of an attribute). For example, a combined robust-RBDO formulation can

be as follows:

Min

imize

and

s

ubj

,

)

(,

,

)

ms

and

ms

and

(,

ms

(,

ms

,

ms

,

)

ms

)

fX

(,fX(,

ms

fX

ms

fX

ms

(,

ms

(,fX(,

ms

(,

ms

(,

fX

(,

ms

(,

ms

XX

ms

XX

ms

,

ms

,XX ,

ms

,

)

ms

)XX )

ms

)

ms

)

XX

)

ms

)

fX

(,fX(,

XX

,XX ,

)XX )

dr

(,

dr

(,

,

dr

,

ms

dr

ms

dr

ms

(,

ms

(,

dr

(,

ms

(,

ms

(,

dr

(,

ms

(,

,

ms

,

dr

,

ms

,

fX

dr

fX

(,fX(,

dr

(,fX(,

(,

ms

(,fX(,

ms

(,

dr

(,

ms

(,fX(,

ms

(,

ms

(,

fX

(,

ms

(,

dr

(,

ms

(,

fX

(,

ms

(,

ms

XX

ms

dr

ms

XX

ms

XX

ms

dr

ms

XX

ms

,

ms

,XX ,

ms

,

dr

,

ms

,XX ,

ms

,

pd

)

pd

)

(,

pd

(,

ms

pd

ms

pd

ms

and

ms

and

pd

and

ms

and

)

ms

)

pd

)

ms

)

ms

)

pd

)

ms

)

ms

)XX )

ms

)

pd

)

ms

)XX )

ms

)

ms

)

XX

)

ms

)

pd

)

ms

)

XX

)

ms

)

fX

pd

fX

(,fX(,

pd

(,fX(,

rp

,

rp

,

)

rp

)

XX

rp

XX

,XX ,

rp

,XX ,

)XX )

rp

)XX )

ect to

eect to e

((

,

)

≤ 0)

,

= 1,

Pg

((Pg((

XX

((XX((

, XX,

XR

) XR)

≤ 0)XR≤ 0)

≥XR ≥

,XR ,

i

id

((

id

((

XX

id

XX

((XX((

id

((XX((

rp

,

rp

,

XR

rp

XR

i

,

i

,

◊◊

,

≤

◊

m

XX

≤XX ≤

X

d

XX

d

XX

L

XX

L

XX

d

XX

d

XX

d

X

d

X

U

≤

XX

≤XX ≤

X

r

XX

r

XX

L

XX

L

XX

rr

≤

rr

≤

X

rr

X

U

Ï

Ì

Ô

Ï

Ô

Ï

Ô

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ô

Ì

Ó

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ó

Ô

Ó

Ô

Ó

Ô

Ó

Ô

Ó

3.33

where

m

fX

m

fX

m

XX

dr

fX

dr

fX

XX

dr

XX

p

(,

fX(,fX

dr

(,

dr

fX

dr

fX(,fX

dr

fX

,

XX ,XX

dr

,

dr

XX

dr

XX ,XX

dr

XX

)

XX )XX

p

)

p

XX

p

XX )XX

p

XX

is the mean of the objective function f(X

d

, X

r

, X

p

), and

s

fX

XX

dr

fX

dr

fX

XX

dr

XX

p

(,

fX(,fX

dr

(,

dr

fX

dr

fX(,fX

dr

fX

,

XX ,XX

dr

,

dr

XX

dr

XX ,XX

dr

XX

)

XX )XX

p

)

p

XX

p

XX )XX

p

XX

is the standard deviation of the objective function (however,

not necessarily the same objective function or attribute). There is a trade-off

between these two objectives. The response surface methodology (RSM) can

be used to link the input to the output to evaluate the objective functions

and the constraints. In fact, not only the mean can be changed, the standard

deviation of the random control factors can also be changed in order to

optimize the objective functions. There is a trend to regard robust design

optimization as a subset of the generalized RBDO when a reliability constraint

is required for the robust design problem.

Gu et al. (2004) introduced a formulation and its solution algorithm of

RBDO combined with robust design. They also addressed the advantages of

combining the Monte Carlo simulation with response surface models in RBDO.

Mourelatos and Liang (2005, 2006) illustrated the design trade-offs between

reliability and robustness in combined robust-RBDO optimization.

From two-step robust optimization to one-step optimization

In the conventional robust design theory, for example Taguchi’s two-step

optimization theory in quality and variability control (Fowlkes and Creveling,

1995), the mean and tolerance (deviation or variation) of the objective

performance attribute are varied in two steps. In Taguchi’s approach, the

concept of RSM is not used. Because RSM allows more than one parameter

to be modi ed at one time, using RSM to optimize both mean and tolerance

simultaneously becomes feasible in robust design. By introducing the

powerful RSM and Pareto optimality into robust design, the variability-

control optimization problem can be handled with the above formulations

in just one step, more effectively than the conventional two-step approach

without RSM.

Diesel-Xin-03.indd 273 5/5/11 11:46:11 AM

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274 Diesel engine system design

Solution algorithms of RBDO and robust optimization

There are many commercial software packages available for solving RBDO

problems. Usually, RBDO problems require a dual-loop solution approach

which includes an outer loop for optimization and an inner loop for calculating

the reliability to evaluate the probabilistic constraint. The dual-loop approach

is computationally intensive. In order to improve the computational efciency

of RBDO problems, many approximate solution methods have been developed

to convert the dual-loop to a single-loop. Detailed discussion on the solution

algorithms of RBDO and robust optimization are provided by Gu and Yang

(2003), Mourelatos and Liang (2005, 2006) and Donders et al. (2007).

3.4.2 Basics of statistics and probability distribution

selection

Statistical probability distributions

There is a statistical spread or variability in almost any value that can be

measured in a population. A single number is often inadequate to represent

an item. Instead, probability distributions are often more appropriate. There

are many statistical distributions. The probability distribution of any random

variable can be characterized by certain distribution parameters such as mean

value, standard deviation, minimum and maximum values. Their denitions

are introduced in Table A.1 in the Appendix. The standard deviation is a

measure of the variance or scatter around the mean value. The larger the

standard deviation, the wider the range of the variability.

The probability distribution of a random variable is described completely

by its cumulative distribution function (CDF), whose value at each real

number x is the probability that the variable is smaller than or equal to x. A

probability distribution is called discrete if its cumulative distribution function

only increases in discrete jumps. Discrete distributions are characterized by

a probability mass function.

On the other hand, a probability distribution is called continuous if its

cumulative distribution function is continuous. If the distribution of a real

number x is continuous then x is called a continuous random variable.

Diesel engine system design often handles continuous random variables, for

example, the noise factor of piece-to-piece variation. The normal distribution,

continuous uniform distribution, Weibull distribution, Beta distribution, and

Gamma distribution are well-known continuous distributions (see Table

A.2 in the Appendix). The continuous distributions are characterized by a

probability density function (PDF). The relationship between CDF f

CDF

(x)

and PDF f

PDF

(x) is given by

Diesel-Xin-03.indd 274 5/5/11 11:46:11 AM



275Optimization techniques in diesel engine system design

fx

PX

xf

uu

fx

CD

fx

FP

fx

FP

fx

PX

FP

PX

xf

FP

xf

DF

FPDFFP

x

()

fx()fx

FP

()

FP

fx

FP

fx()fx

FP

fx

=

FP

=

FP

(

PX (PX

FP

(

FP

PX

FP

PX (PX

FP

PX

≤

FP

≤

FP

)

xf )xf

FP

)

FP

xf

FP

xf )xf

FP

xf

xf =xf

xf

FP

xf =xf

FP

xf

(

xf (xf

FP

(

FP

xf

FP

xf (xf

FP

xf

DF

(

DF

FPDFFP

(

FPDFFP

xf (xf

xf

FP

xf (xf

FP

xf

)d

uu)duu

–

•

Ú

xf

Ú

xf

Ú

xf

Ú

xf

FP

xf

Ú

xf

FP

xf

xf (xf

Ú

xf (xf

xf

FP

xf (xf

FP

xf

Ú

xf

FP

xf (xf

FP

xf

–

Ú

–

3.34

The probability that x is between the two points a and b is

Pa

xb

fx

x

fx

PD

fx

F

fx

a

b

(

Pa( Pa

≤

xb ≤xb

)

xb )xb

=

(

fx (fx

fx

PD

fx (fx

PD

fx

F

fx (fx

F

fx

)d

Ú

a

Ú

a

(

Ú

(

3.35

The integral of a continuous probability function is equal to one, given by

fx

xf

fx

PD

fx

FP

fx

FP

fx

xf

FP

xf

DF

FPDFFP

()

fx()fx

FP

()

FP

fx

FP

fx()fx

FP

fx

d

xfd xf

FP

d

FP

xf

FP

xfd xf

FP

xf

xf= 1 wxf

xf

FP

xf= 1 wxf

FP

xf

he

xfhexf

xf

FP

xfhexf

FP

xf

xfrexf

xf

FP

xfrexf

FP

xf

FP

xf xf

FP

xf

()

x()x

≥ 0

–

•

Ú

–

Ú

–

3.36

A histogram is a useful plot to show the frequency occurrence of the event

values vs. the values distributed in different bins. Narrow bins will collect

few data points and reduce the frequency occurrence or the population count.

Large bins may lump together different ranges which are really different,

thus distorting the real distribution of the data. Probability density can be

interpreted as the ratio of frequency occurrence of the data in the bin to the

bin size.

Reliability in statistics

The reliability function is the integral of the probability density function

from x to in nity, given by

Rx

PX

xf

uu

fx

PD

xf

PD

xf

F

x

fx

CD

fx

F

fx

()

Rx()Rx

=

(

PX (PX

≥

)

xf )xf

xf =xf

(

xf (xf

PD

(

PD

xf

PD

xf (xf

PD

xf

F

(

F

xf (xf

)d

uu)duu

=

1 –

()

fx()fx

∞

Ú

xf

Ú

xf

Ú

xf

Ú

xf

x

Ú

x

xf (xf

Ú

xf (xf

3.37

where X is the mean time to failure. The hazard function in reliability

engineering, an instantaneous failure rate representing the tendency to fail,

is de ned as

hx

fx

Rx

r

hx

r

hx

()

hx()hx

=

()

fx()fx

()

Rx()Rx

3.38

Statistical distribution parameters

The population mean is the  rst moment-generating function of the probability

density function about the origin (see Table A.1 in the Appendix). The

variance of a distribution is equal to the second moment-generating function

about the mean. The skewness of a distribution is equal to the third moment-

generating function about the mean. Skewness is a measure of asymmetry of

Diesel-Xin-03.indd 275 5/5/11 11:46:13 AM

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Xin Q. Diesel Engine System Design

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