Reliability-based robust design optimization: A multi-objective framework using hybrid quality loss function

Research Article

Published online 17 June 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/qre.1027

Reliability-based Robust DesignOptimization: A Multi-objectiveFramework Using Hybrid Quality Loss FunctionOm Prakash Yadav,a∗† Sunil S. Bhamareb and Ajay Rathoreb

In this globally competitive business environment, design engineers are constantly striving to establish new and effectivetools and techniques to ensure a robust and reliable product design. Robust design (RD) and reliability-based designapproaches have shown the potential to deal with variability in the life cycle of a product. This paper explores thepossibilities of combining both approaches into a single model and proposes a hybrid quality loss function-based multi-objective optimization model. The model is unique because it uses a hybrid form of quality loss-based objective functionthat is defined in terms of desirable as well as undesirable deviations to obtain efficient design points with minimumquality loss. The proposed approach attempts to optimize the product design by addressing quality loss, variability, andlife-cycle issues simultaneously by combining both reliability-based and RD approaches into a single model with variouscustomer aspirations. The application of the approach is demonstrated using a leaf spring design example. Copyright ©2009 John Wiley & Sons, Ltd.

Keywords: product design; multi-objective optimization; robust design; quality loss function; reliability-based design

1. Introduction

At the product design and development (PDD) stage, design engineers often encounter several design issues such as reliability,quality, and customer aspirations. Addressing these issues comprehensively at an early design stage is necessary to produce arobust, customer-focused, and economically competitive product that functions consistently during its intended service life.

Therefore, it is wise to address the variability issue in design parameters at the early design stage. Traditional deterministic methodsencapsulate variability by the use of a safety factor; however, such approaches can lead to a design, with both inconsistent and poorreliabilities, or over design. Reliability-based design1 and robust design (RD)2 approaches are effective tools for dealing with thevariability issue. Both approaches show the potential for dealing with variability in the life cycle of a product. However, neither an RDnor a reliability-based design approach, if used individually, will ensure quality and reliability simultaneously in a product3. Therefore,the merits of both approaches need to be integrated into a single model in order to ensure that a product is robust against the noisefactors and reliable over a specified time period. The idea of integrating them into a single model is not a new concept (see e.g.References4--7); however, a systematic approach to integrate them in a multi-objective environment has not been successful.

During the PDD stage, customer aspirations need to be incorporated along with predominant attributes such as robustness andreliability. However, any real-life product design involves several important product attributes, which often conflict with each other.For example, in RD it is essential to have a trade-off between the target and variance of each quality characteristic as well as a trade-offamong quality characteristics. Moreover, it is also essential to accommodate these responses against safety and reliability targets of aproduct design. This, however, raises issues such as how to achieve these accommodations and how robust and reliable the design isunder the given circumstances. Although existing methods such as RD and reliability-based design have proven to be very effective,the design community is still looking for a better approach to address these issues simultaneously at the early stage of PDD. What isreally needed is an integrated multi-objective optimization methodology to capture both reliability-based and RD characteristics andto resolve essential trade-offs so that a balanced optimization can be carried out to determine optimum levels of design variableswith the minimum quality loss.

In order to propose a comprehensive modeling approach, this study investigates all the existing tools and methods that attemptto optimize product design by considering all the issues highlighted earlier. Among these tools, Taguchi’s quality loss function (QLF)

aDepartment of Industrial and Manufacturing Engineering, North Dakota State University, Fargo, ND 58105, U.S.A.bDepartment of Mechanical Engineering, Malaviya National Institute of Technology, Jaipur-302017, India∗Correspondence to: Om Prakash Yadav, Department of Industrial and Manufacturing Engineering, North Dakota State University, Fargo, ND 58105, U.S.A.†E-mail: [email protected]

Copyright © 2009 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2010, 26 27--41

27

O. P. YADAV, S. S. BHAMARE AND A. RATHORE

and RD7, response surface methods (RSMs)8, reliability-based design1, and goal programming (GP)9, 10 seem to have potential toassimilate current issues. The GP has the capability to model real-life design situations in which designers are provided with certainperformance targets to achieve. The GP further allows consideration of three types of goal criteria (smaller-the-better, larger-the-better, and nominal-the-best) to meaningfully address real-life design situations. Taguchi’s7 QLFs also help to realistically approximatethe quality loss for the three types of goal criteria. Kim and Cho11use QLF as the objective function to optimize the RD of a productusing the GP method. However, they treat all deviational variables as undesirable while formulating the QLF-based objective function.Their approach aims to explore more innovative ways to integrate the existing tools such as reliability-based design, RD approach,and GP.

This paper presents a hybrid quality loss function (HQLF)-based multi-objective optimization approach to bring both quality andreliability issues simultaneously in a multi-objective environment. The concepts of variability optimization, RD, reliability-based design,multi-objective optimization, and quality loss are brought together to build the proposed model. The proposed approach ensuresreliable, robust, and concurrently cost-effective product design by satisfying all the desired quality characteristics. The technique isunique because it allows setting and achieving the target values for the quality characteristics, obtaining the expected values byminimizing their variances, and at the same time ensuring that the design meets the reliability target.

2. Variability optimization

Design engineers consider variability in control parameters as one of the root causes for poor product performance. Variations incontrol parameters result from variations in material properties, in the manufacturing processes and from degradation or wear, which,in turn, lead to variations in the actual expected values of the quality characteristic (�). The common practice has been to set upper andlower specification limits (USL and LSL) to represent acceptable performance12. Nevertheless, the variability in quality characteristicsmust be considered while obtaining optimum settings for control parameters. The effect of variation on the performance characteristicsis estimated or captured by approximation methods such as Taylor expansion method13, the perturbation method14, and Neumannexpansion method15. The Taylor series utilizes derivative information that is localized, while a polynomial fitting uses information fromdifferent points16. The existing literature indicates that designers use several approaches to deal with the variability optimization. RD,response surface-based RD, reliability-based design, and reliability-based RD approaches are widely used for variability optimization.The following sections provide a brief overview of these commonly used variability optimization approaches.

2.1. RD optimization

The RD approach provides a cost effective and efficient way to reduce the functional variation of a system without eliminating thecauses of variation7. RD makes a product’s performance insensitive to noise factors by exploiting the inherent nonlinearity of therelationship among the product parameters, noise factors, and quality characteristics. The existing RD approaches can be classified intothree broad categories: (i) Taguchi’s experimental design technique2, (ii) RSM-based RD, and (iii) general optimization techniques17, 18

based on the Taylor series expansion. In the general optimization approach, the Taylor series expansion method is used to developthe polynomial relationship for the expected mean value and the variance for a quality characteristic. The basic objective of all RDapproaches is to optimize the mean and minimize the performance variance in a way that satisfies critical performance targets and,consequently, suffers smaller quality loss19 as shown in Figure 1.

Vining and Myers20 suggested an optimization model aimed at minimizing the variance while keeping the process mean at thetarget value. Their generic form of the RD optimization model is given below.

Minimize �2fi(xj)

Subject to fi(xj)=Ti, i=1,. . . , n (1)

xlj ≤xj ≥xu

j , j =1, 2. . . , J (2)

xj ≥0

Performance

f (x) RobustDesign

OriginalDesign

TargetPerformance

CriticalPerformance

FailureRegion

Figure 1. Robust design phenomenon4

28



where fi(xj) is the ith quality characteristic function, �2fi(xj) represents its variance, Ti is the target for ith quality characteristic,

xj is jth design variable, xlj and xu

j are the lower limit and upper limit on the jth design variable, respectively, and fi(xj) represents

a constraint equation for the expected value of the ith quality characteristic. The literature provides further details on general RDoptimization methods18, 21.

The original RD approach suggested by Taguchi using Signal-to-noise ratios has some pitfalls and is shown to be inefficient22--24.As a result, an RSM-based RD approach (combined array design) gained momentum and appeared to be a better alternative8, 25.Jeang26 adopted the RSM approach for data analysis for an experimental model in order to determine the optimal tolerances inan assembly for quality improvement and cost reduction at the early design stage. Gao et al.27 showed that the RSM is an efficientapproach to parametric model building and design optimization. Kim et al.28 proposed a response surface-based RD approach tominimize the total sensitivity of the design variables of the field coil shape for the High Temperature Superconductor Motor. Theoptimization technique based on the RSM was applied to the RD of the HTS magnet. Recently, Giovagnoli and Romano29 introduceda modification of the dual response surface modeling that incorporates the option of stochastically simulating some of the noisefactors when their behavior is known. Chen30 adopted the RSM and nonlinear programming methods to integrate both parameterand tolerance design methods and demonstrate that RSM is superior to the Taguchi approach and is a natural fit for RD problems.Results show that dual response is compatible with the ordinary least-square method or generalized linear model to solve the robustparameter design problem. It is important to note that RSM is a powerful optimization approach using designed experiments and isvery useful when the mathematical model for any quality characteristic under consideration is unknown.

The issue of multiple objectives in RD was adequately addressed by many authors18, 31, 32. They considered RD as the bi-objectivedesign optimization problem. In their efforts, these authors have made attempts to optimize expected mean values and to minimizethe variance of the quality characteristic function. An ever-increasing need to optimize the mean and variance of performancecharacteristics along with the quality loss issue further forced design engineers to explore a better alternative modeling approach. TheQLF-based optimization enables an easy generation of various non-inferior alternatives from a customer’s viewpoint6. Ramakrishnanand Rao33, Reddy et al.34, Anthony35, and Kim and Cho11 have attempted to use QLF as an objective function to formulate the RDoptimization problem to minimize the quality loss.

2.2. Reliability-based design optimization

In reliability-based design, the consideration of design parameters as random variables provides an optimum design in the presenceof variability among design parameters36--38. The available literature on reliability-based design optimization (RBDO) methods canbe classified into two broad categories: probability level (P-level) methods39 and performance level (G-level) methods40. The P-levelmethods allow the incorporation of the available statistical information for accurate measures of safety by probabilistic calculations.Optimization using the probabilistic method is called RBDO. A typical RBDO problem is formulated as follows41:

Minimize f (d,�X ,�P)

Subject to P(Gi(d, X, P)≥0)≥Ri , i=1,. . ., n

dL ≤d≤dU, �LX ≤�X ≤�U

X

(3)

where d∈Rk is the vector of deterministic design variables, X∈Rm is the vector of random design variables, P∈Rq is the vector ofrandom parameters, f () is the objective function, n is the number of constraints, k is the number of deterministic design variables,m is the number of random design variables, q is the number of random parameters, Ri is the desired reliability, dL and �L

x are thelower limits on deterministic and random variables, respectively, and dU and �U

x are the upper limits on deterministic and randomvariables, respectively.

The G-level methods are widely used in structural design problems. The key to these G-level methods is the concept of limitstate function (g=0), which is mainly used to divide the design space into a safe region (g>0) and a failure region (g<0)42. In theRBDO approach, the limit state function (g=0) is approximated at the most probable point (MPP) using any of the approximateprobability integration methods such as the first-order reliability method (FORM) or the second-order reliability method (SORM)42, 43.Geometrically, the distance between the MPP and the nominal point is a direct measure of the reliability or safety index �, which isoften used to represent the reliability level40. In the design space, different failure surfaces are represented at different levels of safetyby varying the distances to the origin as shown in Figure 2.

In general, a larger � corresponds to a higher reliability level. In structural and mechanical design problems, constraints are definedon the target reliability level, and design is optimized such that all design criteria are met. For more discussion on RBDO, readers areadvised to refer References44--52.

2.3. Reliability-based robust design optimization

The reliability-based design approach accepts variability and uses the limit state function to separate out the stress and strengthprobability density functions (pdf) to achieve the desired reliability level. This indicates that the reliability-based design methods donot attempt to minimize the variability53, and, hence, concentrate on the rare events at the tails of the probability distribution54. Onthe other hand, the RD optimization approach attempts to minimize the variance for a given quality characteristic. RD improves thequality of a product by minimizing the effects of the causes of variation without eliminating the causes, which explains that thesetwo approaches can complement each other, whereas their independent applications in design optimization may not ensure the


29


β

Z2

g < 0

g > 0

0

Failure Surfaces

Z1

Figure 2. Designs corresponding to different failure surfaces

functional stability throughout product’s operational life3. This motivated the design community to integrate the RD element intoRBDO in order to provide better and reliable design in the fiercely competitive global market55.

The reliability-based robust design optimization (RBRDO) deals with two objectives of design methodologies subject to uncertain-ties, i.e. reliability and robustness. The reliability constraints deal with the probability of failure, while the robustness minimizes theproduct quality loss. In general parameter design, the RBRDO is given as follows6:

Minimize C(X; d)

Subject to P(Gi(X; d)≤0)≥�(�i), i=1,. . . , np

dL ≤d≤dU, d ∈Rndv and X ∈Rnrv

(4)

where C(X; d) is the objective function, d =�(X) is the design vector, X is the random vector, and the probabilistic constraint is describedby the performance function Gi(X; d), its probability distribution, and the prescribed reliability target �i , and np, ndv, and nrv are thenumber of probabilistic constraints, design variables, and random variables, respectively. The design robustness that is associatedwith the product quality is captured in the form of quality loss in the objective function equation.

The idea of integrating robust and reliability-based approaches in a single model is not a new concept. Wang and Wu5 used thereliability-based robust approach in cases with input uncertainties that were not well defined due to lack of data. Youn et al. 6 suggestedthe performance moment integration approach for RBRDO. Xi et al. 56 attempted to integrate a derivative-free probability analysismethod to RBRDO. This paper proposes to combine both approaches in an HQLF-based multi-objective optimization approach. Thefollowing section provides a detailed discussion on the proposed approach.

3. Proposed approach

As discussed earlier, the reliability-based design improves product reliability but does not minimize variability in qualitycharacteristics54, whereas the RD approach minimizes the variability but ignores the probabilistic nature of the random variable.Therefore, the proposed approach attempts to capture the merits of both approaches by providing a comprehensive reliability-basedRD methodology. Further, the proposed methodology addresses the quality loss issue in product design by developing a QLF-basedobjective function for a multi-objective optimization problem. In the development of the objective function, the proposed modelconsiders both undesirable and desirable deviations57. Here, deviation is defined as the difference between the targeted value ofany given constraint (or quality characteristics) and the actual value obtained by the model solution. The following sections providedetailed discussion on the development of the reliability-based RD methodology using the HQLF as an objective function.

3.1. Understand and classify the nature of quality characteristics

Since a product is built to satisfy the customer needs and expectations, it is important to translate the identified customer needs intoappropriate quality characteristics. The quality function deployment (QFD)58 matrix provides an effective platform to map customerneeds into primary quality characteristics. The proposed approach advocates using the QFD matrix to translate identified customerrequirements into appropriate quality characteristics that will drive customer-focused design efforts. After identifying the primaryquality characteristics, the next, and the most important, step is to understand and classify the nature of these product qualitycharacteristics that need to be optimized. The types of classification include nominal-the-best, larger-the-better, and smaller-the-betterquality characteristics. The classification further helps to understand and capture the desirability of deviational variables unlike thetraditional GP model and other suggested approaches that seek to minimize the total deviation. In order to capture the desirabilityof deviational variables, we classify all deviational variables as desirable and undesirable variables. For example, for a smaller-the-better-type characteristic (f (x)≤t), d+ represents the overachievement of the target value and hence is classified as undesirable. Theconventional GP method minimizes the undesirable deviations in order to ensure that the final value is closer to the specified upperlevel of target. However, these existing methods do not make any attempt to further minimize the quality characteristic value onthe lower side of the target value that is always desirable and even expected in a less-than-equal-to type of constraint. Therefore, to

30



tt /0y

L(y)

Figure 3. Quality loss function for a smaller-the-better-type characteristic

t0

L(y)

t / y

Figure 4. Quality loss function for a larger-the-better-type characteristic

achieve the function value less than the specified target, this study suggests maximizing the desired deviational variable, also knownas underachievement, d−, in the same objective function. The inclusion of the desirable deviation compels the model to explore thedesign space further below the set target limit, which is always desirable for efficient product design. Figure 3 illustrates that thequality loss for a smaller-the-better characteristic will be less if the actual value, t′, is reduced further below the assigned target t. Theabove reasoning motivates us to incorporate desirable deviational variables in the objective function definition.

On the contrary, for a larger-the-better quality characteristic (f (x)>=t), d− denotes the underachievement of the target value and,hence, is considered to be an undesirable deviation, whereas d+represents the overachievement and that is treated as desirable.Figure 4 shows the decrease in quality loss with the increase in actual value t′ beyond the assigned target value t. This indicates thatoverachievement is always desirable for larger-the-better type of quality characteristics and, hence, should be incorporated in theoptimization model.

For nominal-the-best (f (x)= t) quality characteristics, the objective is to achieve the actual value, t′, which is the same as assignedtarget value t. This requirement clarifies that both positive and negative deviations (d+, d−) for nominal-the-best type of qualitycharacteristic are considered to be undesirable and, hence, should be minimized. Now, the classification of deviational variables asdesirable and undesirable for each category of quality characteristic facilitates the process of defining the proposed hybrid objectivefunction.

3.2. Develop the quality loss function

The proposed approach applies Taguchi’s QLF7 concept to develop an appropriate QLF-based objective function. The quadratic lossfunction can meaningfully approximate the quality loss in most situations7, 59, 60. Equation (5) gives the quality loss for a nominal-the-best quality characteristic where the deviation on either side of the target value is undesirable2.

L(y)=k(y−m)2 (5)

where k is a constant called quality loss coefficient and (y−m) represents deviation from the target value. The quality loss for asmaller-the-better type characteristic is obtained from Equation (5) by substituting m=0:

L(y)=ky2 (6)

The larger-the-better-type quality characteristics do not take negative values, but zero is their worst value. As their values becomelarger, quality losses become progressively smaller. This shows that the reciprocal of such a characteristic has the same qualitativebehavior as a smaller-the-better-type quality characteristic. Therefore, the quality loss for a larger-the-better characteristic can begiven as follows:

L(y)=k

[1

y2

](7)

Equations (6) and (7) represent well-tested functions to calculate quality loss for smaller-the-better- and larger-the-better-type vari-ables, respectively. However, in an effort to calculate total quality loss by adding Equations (6) and (7), the objective function continuity


31


problem appears when both variables have zero values. Moreover, Equation (7) becomes an indeterminate function when the desirabledeviational variable value is zero. Therefore, in order to address these two concerns and make the quality loss-based objective functioncontinuous at zero, this paper proposes to modify these equations using the data transformation approach. The exponential trans-formation y∗+ =exp(y) is used for the undesirable deviational variables and the negative exponential transformation y∗− =exp(−y)is used for the desirable deviational variables. Thus, Equation (6) is modified to calculate the quality loss for a smaller-the-better(undesirable)-type variable:

L(y)=k(y∗+)2 =k[exp(y)]2 (8)

Equation (7) is modified to calculate the quality loss for a larger-the-better (desirable)-type variable:

L(y)=k(y∗−)2 =k[exp(−y)]2 (9)

These modified equations are still in quadratic form and satisfy both the major concerns. Equation (8) is monotonically increasing,whereas Equation (9) is monotonically decreasing, and both equations have same function value at zero. The modified equationsensure the continuity of the total QLF at zero value of deviational variables when these two equations are added together. Equation(9) now avoids the problem of the indeterminate function when the variable value is zero.

We now develop the QLF for our model utilizing deviational variables for each category of quality characteristic. For a nominal-the-best characteristic, both overachievement, d+

1 , and underachievement, d−1 , deviations are undesirable and, therefore, can be treated

as smaller-the-better variables. The appropriate QLF can be written using Equation (8):

L(d1)=k�(exp(d+1 ))2 +(exp(d−

1 ))2� (10)

For a smaller-the-better type characteristic, the overachievement, d+2 , is always undesirable, whereas underachievement, d−

2 , is desir-able since it further lowers the actual value of the smaller-the-better characteristic from the assigned target value. This understandingallows us to treat the undesirable deviational variable as a smaller-the-better and the desirable deviational variable as a larger-the-better type. The overall quality loss for a smaller-the-better-type constraint can be formulated using Equations (8) and (9):

L(d2)=k�{exp(d+2 )}2 +{exp(−d−

2 )}2� (11)

Similarly, the underachievement, d−3 , is always undesirable for a larger-the-better type characteristic, whereas the overachievement,

d+3 , is desirable because overachievement takes the actual value beyond the assigned target value. Based on the above rationale,

we can treat underachievement as a smaller-the-better-type variable and overachievement as a larger-the-better-type variable. Theoverall quality loss for a larger-the-better-type constraint can be written by combining Equations (8) and (9):

L(d3)=k�{exp(d−3 )}2 +{exp(−d+

3 )}2� (12)

The classification of deviational variables as desirable and undesirable variables and the subsequent definitions of quality loss for eachcategory of quality characteristic provide a strong basis for developing a new objective function for the multi-objective optimizationproblem. The proposed objective function represents the total quality loss due to deviational variables and is achieved by adding upEquations (10)–(12):

L(d)=L(d1)+L(d2)+L(d3) (13)

The proposed objective function, L(d), is termed the HQLF-based objective function. It makes use of both desirable and undesirabledeviations and simultaneously minimizes the undesirable deviations and maximizes the desirable deviations to explore the designspace further below/above the set target limit(s).

3.3. Proposed HQLF-based multi-objective optimization model

The proposed HQLF-based multi-objective optimization model uses Equation (13) as its objective function, which needs to be mini-mized. Each quality characteristic of the product defines an objective that needs to be achieved or optimized. At the same time, ourgoal is to make these quality characteristics insensitive to noise factors; therefore, we must minimize the variation in each quality char-acteristic under the given operating conditions. Another critical input to the model is the reliability goal or target. The optimal productdesign should be able to achieve the given reliability target. These three requirements define four major categories of constraints forthe proposed model as discussed below:

(a) The first category of constraints includes the desired targets for each identified quality characteristic. Depending on the natureof the quality characteristics, these constraints could be defined as nominal-the-best, larger-the-better, or smaller-the bettertype. The total number of constraints under this category is equal to the total number of quality characteristics that designersdesire to optimize.

(b) The second category of constraints contains variability minimization targets for each quality characteristic. In order to makethese quality characteristics insensitive to input factor variability, designers can specify the variability targets for each givenquality characteristic. Since the objective of these constraints is to minimize the variability in quality characteristics, the modeltreats them as smaller-the-better-type constraints. The total number of constraints under this category is equal to the totalnumber of quality characteristics being considered in the design problem.

32



(c) The reliability requirement defines the third category of constraints for the proposed model. Because continuous design changesor modifications are directed toward reliability improvement, the reliability target constraint is considered as a larger-the-better-type objective.

(d) The final or fourth category of model constraints comprises model parameter range values. Although these parameters aredecision variables in the proposed model, the designers have to provide the possible range of values for each design parameterbased on other engineering considerations such as manufacturing capability, material properties, and specification requirementsfor other components in the assembly.

After formulating the HQLF-based objective function and defining the different categories of constraints, we now present thecomplete multi-objective design optimization model. The proposed model captures the merits of both reliability-based design andRD approaches, and uses the objective function that is developed based on quality loss concept. We term the proposed model theHQLF-based RBRDO model. The generic form of the HQLF-based multi-objective optimization model is given below:

Min Z =n∑

i=1[w1i(k{exp(d+

1i )}2 +k{exp(d−1i )}2)]+

n∑i=1

[w2i(k{exp(d+2i )}2 +k{exp(−d−

2i )}2)]

+n∑

i=1[w3i(k{exp(d−

3i )}2 +k{exp(−d+3i )}2)] (14)

S.t fi(xj)+d−1i −d+

1i =T1i (15)

Vi(xj)+d−2i −d+

2i =T2i , i=1,. . . , n (16)

P

(�gl

�gl

)+d−

3l −d+3l =�tl

, l =1,. . . , m (17)

xLj ≤xj ≤xU

j , j =1,. . . , J

xj, d±i ≥0 (18)

where

�g = Y − S

�g =√

�2Y +�2

S

fi(xj) = fi(xj)+1

2

[∑ �2fi(xj)

�2xj

×�2xj

]Taylor series Equation

Vi(xj) = ∑(�fi(xj)

�xj

)2

×�2xj

(19)

where w1i , w2i , and w3l are weights assigned to mean, variance, and reliability goals for each quality characteristic, respectively; kis a constant and defined as coefficient of quality loss; d+

1i , d+2i , and d−

3l are undesired deviational variables; d−1i , d−

2i , and d+3l are

desired deviational variables for each category of constraints; d+1i is overachievement from the target value of the ith constraint; d−

1iis underachievement from the target value of the ith constraint; d+

2i is overachievement from the variance value of the ith constraint;

d−2i is underachievement from the variance value of the ith constraint; d+

3l is overachievement from the reliability index value of the

reliability constraint; d−3l is underachievement from the reliability index value of the reliability constraint; fi(x) represents a function;

Vi(x) represents a variance function of ith quality characteristic constraint; T1i is specified as target value for expected mean; T2i istarget for variance of the ith quality characteristic; �tl represents target reliability index for reliability requirement constraint; xL

j and

xUj are lower and upper limits on the jth control parameter; Y is yield strength; S is maximum allowable stress for the given material

type; and �g and �g are the mean and standard deviation of the limit state functions, respectively. In the model, different prioritiescan be assigned to each quality criteria as well as to each constraint category, which essentially allows designers to align productdevelopment efforts with corporate strategy and manufacturing capabilities.

The underlying principle of combining both reliability-based design and RD approaches using HQLF-based multi-objective opti-mization model and its functional mechanics can figuratively be explained using the stress–strength interference model as shown inFigure 5, where f (S) and f (R) are pdf of stress and strength, respectively.

Figure 5 shows the interference between the initial distributions for both stress and strength of the given material. However, theapplication of both RD and reliability-based design significantly reduces variability and shifts the mean to avoid interference, therefore


33


Stress & strength (N/m2)

Reduction in variance is achieved by robust design approach

Shift in pdf is being guided by reliability constraint

f (S)

f (S)

,f (R)

f (R)

Interference

Figure 5. Reliability-based robust design approach. This figure is available in colour online at www.interscience.wiley.com/journal/qre

improving product reliability. The incorporation of the reliability constraint further ensures a sufficient shift in the mean of strengthdistribution as shown in Figure 5. The applicability of the proposed approach is illustrated in detail using a leaf spring in the nextsection.

4. Application

In this section, the mechanics of the proposed methodology are demonstrated by considering a leaf spring design example. Theleaf spring is a flexible element in an automotive suspension system. The basic function of a leaf spring is to absorb the bumps andirregularities of a road surface and to support the vehicle load. Several researchers61--64 have used leaf spring as a test example, butthey have not comprehensively dealt with the design problem in view of customer needs. By means of a leaf spring example, weintend to demonstrate the proposed approach as a customer-focused methodology driven by customer needs and requirements.From the existing literature and through our interaction with design engineers from the auto industry, we understand that there arefour important customer needs to be addressed while designing a suspension system: balanced drive, comfortable ride, safety, anddirectional stability. The leaf spring is a key element in the suspension system, so it should be carefully designed to satisfy all theidentified customer needs. These identified customer needs are then translated into appropriate quality characteristics by using theQFD matrix. The house of quality matrix (QFD) mapping process helped us to identify four quality characteristics: weight, free height,spring rate, and stress57. In order to design a very effective suspension system, it is important to optimize the leaf spring design thatsatisfies these four quality criteria and achieves a specified reliability target for a given life cycle.

Once the quality characteristics are identified, the next step is to classify them according to their desirability. The actual valuesof weight, spring rate, and induced stress are always desirable to be smaller than the target values, and so they are classified assmaller-the-better type, while free height is considered as a nominal-the-best type of characteristic. Table I presents a list of theselected quality characteristics for a leaf spring design, their mathematical models, the nature of goal criteria, and the correspondingtarget values. One of the motivations for selecting the leaf spring example is the availability of mathematical models for identifiedquality characteristics. The standard mathematical equations for these quality characteristics are easily available in text books andspring design manuals65, 66. However, in the absence of standard mathematical models, we must develop response surface modelsfor each quality characteristic using the experimental data. In order to assign optimum targets for mean and variance, a bi-objectiveoptimization model11 is formulated separately for each quality characteristic. Kim and Cho11 used the Taylor series approximation toevaluate an approximated mean value and standard deviation in the bi-objective optimization model. The solution of the bi-objectiveoptimization model for each quality characteristic is treated as targets and given in Table I.

In the mathematical equations given in Table I, n represents the number of leaves, l denotes length, b is breadth, t denotes the thick-ness of each leaf, and Ro is the radius of curvature of leaf spring. Other nomenclatures include � as material density =78×103 N / m3, Pas load on spring =4500 N, and E as elastic modulus =2.10×103 MPa. The design equations are clearly influenced mainly by the controlparameters such as the number of leaves (n), length (l), width (b), thickness (t), radius of curvature (Ro), and stiffness factor (St). For aspecific automotive vehicle, the ranges of these control parameters are selected from the manufacturer’s catalogue and are presentedin Table II.

4.1. HQLF-based multi-objective optimization model

4.1.1. Constraints development. For the leaf spring design, the HQLF-based multi-objective optimization model is formulated toachieve a better trade-off among quality characteristics and to obtain the optimum values of the design parameters. The proposedmodel contains nine main constraints and parameter ranges. The first four constraints represent the assigned target values for the fouridentified quality characteristics: weight, free height, spring rate, and stress. The following equations define these four constraints,respectively:

n×�× l× b× t ≤ 83.63 (20)

l2

8×Ro+ �2

l8×Ro

+ l2

8×Ro3×�2

Ro = 0.10795 (21)

34



Table I. Quality characteristics with target values

Quality characteristics Type of goal criteria Mean target Variance target

Weight (N)= Less-than-equal-to goal�×n× l×b×t f (x)≤m 83.63 41.71

Free height (mm)= Equal-to type goall2

8×R0f (x)=m 107.95 0.0889

Spring rate (N/m)= Less-than-equal-to goal8×E×n×b×t3×Stiff. Factor

3×l2f (x)≤t 251.72 77.3

Stress (MPa) Less-than-equal-to goal

= 3×P×l2×n×b×t2 f (x)≤m 895.56 67.8

Table II. Range of control parameters

Control parameters Range

No. of leaves (n) 3≤n≤6Length (l) 600≤ l ≤3000 mmBreadth (b) 40≤b≤120 mmThickness (t) 6≤ t ≤25 mmRadius of curvature (Ro) 350≤Ro≤700 mmStiffness factor (St) 1≤St≤1.5

8×n×St×E× b× t3

3× l2+ 8×n×St×E× b× t

l2×�2

t + 16×n×St×E× b× t3

l4×�2

l ≤ 251.72 (22)

3×P× l

2×n× b× t2+ 3×P× l

2×n× b3 × t2×�2

b + 9×P× l

2×n× b× t4×�2

t ≤ 895.56×106 (23)

The next set of constraints includes four constraints that represent the variance targets for each of the four quality characteristics asshown below:

(n×�× l× t)2 ×�2b +(n×�× l× b)2 ×�2

t +(n×�× b× t)2 ×�2l ≤41.71 (24)

(l

4×Ro

)2

×�2l +

(l2

8×Ro2

)2

×�2Ro ≤0.0000889 (25)

(8×n×St×E× t3

3× l2

)2

×�2b+

(8×n×St×E× b× t2

l2

)2

×�2t +(

16×n×St×E× b×t3

3× l3

)2

×�2l ≤77.3 (26)

(3×P× l

2×n× b2 × t2

)2

×�2b +

(3×P× l

n× b× t3

)2

×�2t +

(3×P

2×n× b× t2

)2×�2

l ≤67.8×106 (27)

The ninth constraint represents the reliability constraint and specifies the reliability target for the product. The following equationdefines the reliability constraint for the leaf spring design:

P

(�g

�g

)≥�t (28)

Other constraints are defined as design parameter ranges and non-negativity constraints. The goals of all the four quality characteristicsand their variability reduction targets are sought to be achieved within the given range of each design parameter values as listed inthe model. In multi-objective optimization, original constraints are converted into equality constraints by introducing two deviationalvariables known as overachievement and underachievement. The desirability of these two types of deviational variables is used toarticulate the hybrid form of objective function as discussed in Sections 3.1 and 3.2.

4.1.2. HQLF-based objective function. The proposed HQLF-based objective function is defined in terms of desirable and undesirabledeviations. The four quality characteristic target constraints are considered first to define their corresponding deviational variables(desirable and undesirable). The QLF expression is derived for the expected mean function of each quality characteristic. Among the


35


four quality characteristics, the weight is treated as a smaller-the-better-type characteristic, and its quality loss expression isderived by using Equation (11) as given below:

Min Z11 = [{exp(d+11)}2 +{exp(−d−

11)}2] (29)

Another quality characteristic free height is considered as a nominal-the-best type and the corresponding quality loss due to deviationfrom the assigned target is developed by using Equation (10):

Min Z12 = [{exp(d+12)}2 +{exp(d−

12)}2] (30)

Similarly, the other two quality characteristics, spring rate and stress, are treated as smaller-the-better types and their qualityloss expressions are derived by using Equation (11):

Min Z13 = [{exp(d+13)}2 +{exp(−d−

13)}2] (31)

Min Z14 = [{exp(d+14)}2 +{exp(−d−

14)}2] (32)

In addition, the proposed model contains four variance constraints that are defined as smaller-the-better type. The quality lossexpressions for all four variance constraints are formulated using Equation (11):

Min Z21 = [{exp(d+21)}2 +{exp(−d−

21)}2] (33)

Min Z22 = [{exp(d+22)}2 +{exp(−d−

22)}2] (34)

Min Z23 = [{exp(d+23)}2 +{exp(−d−

23)}2] (35)

Min Z24 = [{exp(d+24)}2 +{exp(−d−

24)}2] (36)

Further, the reliability target constraint is defined as a larger-the-better type, and the corresponding QLF is developed usingEquation (12):

Min Z3 = [{exp(d−3 )}2 +{exp(−d+

3 )}2] (37)

The total quality loss expression can be derived by taking weighted sum of Equations (29)–(37):

Z =w1i(Z11 +Z12 +Z13 +Z14)+w2i(Z21 +Z22 +Z23 +Z24)+w3iZ3 (38)

4.1.3. The final model. The final HQLF-based objective function is obtained by taking a weighted sum of Equations (29)–(37) asshown in Equation (38). The weights assigned to each of the quality characteristics and/or constraint categories are calculatedusing a pair wise comparison approach and analytic hierarchy process67. For the sake of simplicity, the value of the quality loss coefficientis considered as unity. The complete HQLF-based multi-objective optimization model is formulated below:

MinZ = wi

⎡⎣{exp(d+

11)}2 +{exp(−d−11)}2 +{exp(d+

12)}2 +{exp(d−12)}2

+{exp(d+13)}2 +{exp(−d−

13)}2 +{exp(d+14)}2 +{exp(−d−

14)}2

⎤⎦

+wi

⎡⎣{exp(d+

21)}2 +{exp(−d−21)}2 +{exp(d+

22)}2 +{exp(−d−22)}2

+{exp(d+23)}2 +{exp(−d−

23)}2 +{exp(d+24)}2 +{exp(−d−

24)}2

⎤⎦

+wi[{exp(d−3 )}2 +{exp(−d+

3 )}2] (39)

S.t n×�× l× b× t+d−11 −d+

11 =83.63

l2

8×Ro+ �2

l8×Ro

+ l2

8×Ro3×�2

Ro +d−12 −d+

12 =0.10795

8×n×St×E× b× t3

3× l2+ 8×n×St×E× b× t

l2×�2

t + 16×n×St×E× b× t3

l4×�2

l +d−13 −d+

13 =251.72

36



3×P× l

2×n× b× t2+ 3×P× l

2×n× b3 × t2×�2

b + 9×P× l

2×n× b× t4×�2

t +d−14 −d+

14 =895.56×106

(n×�× l× t)2 ×�2b +(n×�× l× b)2 ×�2

t +(n×�× b× t)2 ×�2l +d−

21 −d+21 =41.71

(l

4×Ro

)2

×�2l +

(l2

8×Ro2

)2

×�2Ro +d−

22 −d+22 =0.0000889

(8×n×St×E× t3

3× l2

)2

×�2b +

(8×n×St×E× b× t2

l2

)2

×�2t +

(16×n×St×E× b× t3

3× l3

)2

×�2l

+d−23 −d+

23 =77.3

(3×P× l

2×n× b2 × t2

)2

×�2b +

(3×P× l

n× b× t3

)2

×�2t +

(3×P

2×n× b× t2

)2×�2

l +d−24 −d+

24 =67.8×106

P

(�g

�g

)+d−

3 −d+3 = �t

d±11, d±

12, d±13, d±

14, d±21, d±

22, d±23, d±

24, d±3 ≥ 0∑

wi = 1

3< = n<=6

0.6< = l <=3.0 m, �l =0.4398 m

0.04< = b<=0.12 m, �b =0.00575 m

0.006< = t <=0.025 m, �t =0.00102 m

0.35< = Ro<=0.75 m, �Ro =0.08106 m

1< = St <=1.5

where

�g = Y − 3×P× l

2×n× b2 × t2

�g =

√√√√�2Y +

(3×P× l

2×n× b2 × t2

)2

×�2b +

(3×P× l

n× b× t3

)2

×�2t +

(3×P

2×n× b× t2

)2×�2

l

� = 78×103 N / m3

P = 4500 N

E = 210×109 N / m2

�t = 0.999

Y = 1700×106 N / m2, �Y =120×106 N / m2

The proposed HQLF-based objective function is of quadratic nature. Therefore, the sequential quadratic programming (SQP)technique68--70 provides a handy tool to solve such a quadratic programming problem. To solve our model, we took advantage of theSQP algorithm on the MATLAB 7.1 platform. The results of the proposed model are given in Tables III and IV and are discussed next.

4.2. Discussion

In the leaf spring example, the HQLF-based procedure forces the model to minimize the undesirable deviation variables and maximizethe desirable deviation variables. The inclusion of the desirable deviational variables in the objective function provides a bettertrade-off among quality characteristics and a more efficient design solution. Tables III, IV, and V provide the comparisons for controlparameters, quality characteristics, and quality loss.


37


Table III. Comparison for control parameters

HQLF-based

Design variables Deterministic Robust Reliability-based robustNo. of leaves 5 4 4Length 0.86006 0.8512 0.81973Breadth 0.0405 0.041899 0.042867Thickness 0.006004 0.0061872 0.006002Radius of curvature 0.60000 0.69901 0.69943

Table IV. Comparison for quality characteristics

HQLF-based

Constraints Target Deterministic (actual) Robust (actual) Reliability-based robust (actual)

Spring weight (N) 83.63 81.54864014 68.84684 65.79734Free height (m) 0.10795 0.154104834 0.129566 0.12009Spring rate (N/m) 251.72 331.635 306.811 308.894Stress (MPa) 8.95E+08 7.956E+08 8.96E+08 8.96E+08

Table V. Comparison based on three performance indicators

Deterministic Robust Reliability-based robust

Quality loss 255 131 121Directional derivative −1.07e−11 −1.69e−12 −1.51e−13

value of final iteration

The results and the performance of the proposed approach are compared with the deterministic and robust design approachesin Tables III and IV. Table III provides the comparison of the design parameter values, and Table IV gives the quality characteristic(constraint) values for three different models: deterministic, robust, and proposed reliability-based RD methods. As seen in TableIV, the proposed model achieves a significant reduction in weight compared with the other two methods, which results in furthermaterial cost savings. The comparison indicates that the proposed model further improves the free height parameter by bringingit closer to the specified target. The other two quality characteristic values are almost unchanged and are the same as given bythe RD methodology. This comparison clearly demonstrates that the integration of these two approaches and the consideration ofHQLF-based objective function achieve a better trade-off among conflicting characteristics and thus provides a better design solution.

Further, the inclusion of the reliability target constraint ensures that the model satisfies the specified reliability target while achievingappropriate trade-off among other quality characteristics. The model selects design parameters in such a way that it satisfies the givenreliability constraint, and further trade-off among the quality characteristics results in the reduction of spring weight. Since springweight is a smaller-the-better type, the reduction in weight below the target value is considered to be an improvement in the modelsolution. We believe that the consideration of desirable deviations in the objective function has forced the model to explore broaderdesign space and find out better design solutions. This has been demonstrated by achieving smaller objective function value (qualityloss) for the proposed model as seen in Table V.

The results and performance of the proposed model are further compared with each other on the basis of two performance criteria:the objective function value (total quality loss) and the directional derivative of the final iteration required to converge to the optimalsolution. Since the objective function is the minimization of quality loss, the smaller the quality loss value the better the optimizationmethod. The total quality loss criterion helps to judge the robustness as well as how efficiently these models achieve trade-off amongthe assigned requirements without sacrificing much on any one of them. The smaller quality loss means more robust product design6.Table V indicates that the proposed HQLF-based model minimizes the total quality loss to 121, which is smaller than the other twomethods.

The performance of the proposed model is further analyzed on the basis of model accuracy. The accuracy of any given model ismeasured by verifying the magnitude of the directional derivative of the final iteration. The search direction should essentially be inthe hatched region16, 71 as shown in Figure 6. It is generally indicated by the negative function value. When the magnitude of thedirectional derivative value becomes smaller, the decent direction becomes steeper, which helps to take the search more closelyto the efficient point as shown in Figure 6.

Thus, the accuracy of the obtained solution to the minimum point is verified by the magnitude of the directional derivative of finaliteration. It is found to be lower in the case of the proposed methodology compared with both the robust and deterministic design

38



Figure 6. An illustration of a decent direction

models (see Table V). The change in the gradient vector is smaller; therefore, the obtained design point can be considered as themost accurate.

5. Conclusion

The paper proposes a structured and systematic HQLF-based methodology for RBRDO. The proposed HQLF-based multi-objectiveoptimization approach integrates both robustness and reliability attributes into a single model. The proposed approach obtains bettertrade-offs among all the quality characteristics. The findings of the HQLF-based reliability-based robust model are more efficientthan the deterministic as well as robust approaches. The proposed approach is applicable to a general design situation for achievingoptimum design parameter values by minimizing the hybrid quadratic loss function and satisfying the performance characteristictargets of expected values and variability. The model provides a robust and reliable design solution at a competitive cost. The modelapplicability has been demonstrated by a leaf spring example, but further validation is necessary before claiming its overall success.For future research, we intend to capture life-cycle issues in the multi-objective optimization model by understanding and modelingfailure mechanisms as well as degradation behavior.

Acknowledgements

The authors would like to sincerely thank the anonymous referees and editor for their valuable comments.

References1. Kapur KC, Lamberson LR. Reliability in Engineering Design. Wiley: New York, 1977.2. Phadke MS. Quality Engineering using Robust Design. Prentice-Hall: Englewood Cliffs, NJ, 1989.3. Yang G. Life Cycle Reliability Engineering. Wiley: NJ, 2007.4. Tong T, Yang RJ. A reliability-based design optimization procedure for minimal variations. AIAA-94-1415-CP, 1994.5. Wang W, Wu J. Reliability-based robust design. Proceedings of 39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material

Conference, Long Beach, CA, 20–23 April 1994; 2956--2964.6. Youn BD, Choi KK, Yi K. Performance moment integration approach for reliability-based robust design optimization. Mechanics Based Design of

Structures and Machines 2005; 33:185--213.7. Taguchi G. System of Experimental Design. Unipub/Kraus International: New York, 1987.8. Myers RH, Montgomery DC. Response Surface Methodology: Process and Product Optimization Using Designed Experiments (2nd edn). Wiley:

New York, 2002.9. Charnes A, Cooper W, Ferfusan R. Optimalestimationofexecutivecompensationbylinearprogramming. Management Science 1955; 1(2):138--151.

10. Cohon JL. Multi-objective Programming and Planning. Academic Press: New York, 1978.11. Kim JY, Cho BR. Development of priority-based robust design. Quality Engineering 2002; 14(3):355--363.12. Zang TA, Hemsch MJ, Hilburger MW, Kenny SP, Luckring JM, Maghami P, Padula SL, Stroud WJ. Needs and opportunities for uncertainty-based

multidisciplinary design methods for aerospace vehicles. NASA/TM-2002-211462, 2002.13. Jung DH, Lee BC. Development of a simple and efficient method for robust optimization. International Journal for Numerical Methods in Engineering

2002; 53:2201--2215.14. Rahman S, Rao BN. A perturbation method for stochastic meshless analysis in elastostatics. International Journal for Numerical Methods in Engineering

2001; 50:1961--1991.15. Yamazaki F, Shinozuka M. Neumannexpansionforstochasticfinite elementanalysis. Journal of Engineering Mechanics (ASCE) 1988; 114:1335--1354.16. Papalambros PY, Wilde DJ. Principles of Optimal Design: Modeling and Computation. Cambridge University Press: New York, 1991.17. Eggert RJ, Mayne RW. Probabilistic optimal-design using successive surrogate probability density functions. Journal of Mechanical Design

(ASME) 1993; 115(3):385--391.18. Chen W, Wiecek MM, Zhang J. Quality utility—a compromise programming approach to robust design. Journal of Mechanical Design (ASME)

1999; 121(2):179--187.19. Zang C, Friswell MI, Mottreshead JE. A review of robust optimal design and its application in dynamics. Computer and Structures 2005; 83:315--328.20. Vining GG, Myers RH. Combining Taguchi and response surface philosophies: A dial response approach. Journal of Quality Technology 1990;

22:38--45.21. Su J, Renaud JE. Automatic differentiation in robust optimization. AIAA Journal 1997; 35(6):1072.22. Nair VN (ed.). Taguchi’s parametric design: A panel discussion. Technometrics 1992; 34:127--161.


39


23. Box GEP. Signal-to-noise ratios, performance, criteria, and transformation. Technometrics 1988; 30:1--40.24. Box GEP, Bisgaard S, Fung CA. An explanation and critique of Taguchi’s contribution to quality engineering. Quality and Reliability Engineering

International 1988; 4:123--131.25. Montgomery DC. Design and Analysis of Experiments (7th edn). Wiley: New York, 2009.26. Jeang A. Robust tolerance design by response surface methodology. International Journal of Advance Manufacturing Technology 1999; 15:399--403.27. Gao XK, Low TS, Liu ZJ, Chen SX. Robust design for torque optimization using response surface methodology. IEEE Transactions on Magnetic 2002;

38(2):1141--1144.28. Kim Y-K, Hong J-P, Lee G-H, Jo Y-S. Application of response surface methodology to robust design for racetrack type high temperature

superconducting magnet. IEEE Transactions on Applied Superconductivity 2002; 12(1):1434--1437.29. Giovagnoli A, Romano D. Robust design via simulation experiments: A modified dual response surface approach. Quality and Reliability Engineering

International 2008; 24:401--416.30. Chen J. Integrated robust design using response surface methodology and constrained optimization. Unpublished PhD Thesis, University of

Waterloo, Waterloo, ON, Canada, 2008.31. Chen W, Allen JK, Mistree F, Tulsi K. A procedure for robust design. ASME Journal of Mechanical Design (ASME) 1996; 18:478--485.32. Chen W, Allen JK, Mistree F. The robust concept exploration method for enhancing concurrent systems design. Journal of Concurrent Engineering:

Research and Applications 1997; 5(3):203--217.33. Ramakrishnan B, Rao SS. A robust optimization approach using Taguchi’s loss function for solving nonlinear optimization problems. Advances in

Design Automation (ASME) 1991; DE-32(1):241--248.34. Reddy PB, Nishina K, Babu AS. Unification of robust design and goal programming for multi-response optimization—a case study. Quality and

Reliability Engineering International 1997; 13:371--383.35. Antony J. Multi-response optimization in industrial experiments using Taguchi’s quality loss function and principal component analysis. Quality

and Reliability Engineering International 2000; 16:3--8.36. Tu J, Choi KK, Park YH. A new study on reliability-based design optimization. Journal of Mechanical Design (ASME) 1999; 121:557--564.37. Wu YT, Shin Y, Sues R, Cesare M. Safety-factorbasedapproachforprobabilistic-baseddesignoptimization. Forty-secondAIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dynamics and Material Conference, Seattle, WA, 2001.38. Youn BD, Choi KK, Park YH. Hybrid analysis method for reliability-based design optimization. Journal of Mechanical Design (ASME) 2003; 125:

221--232.39. Du X, Chen W. Sequential optimization and reliability assessment method for efficient probabilistic design. DETC-DAC34127, ASME Design

Engineering Technical Conferences, Montreal, Canada, September 2002.40. Hasofer AM, Lind NC. Exactand invariant second-moment code format. Journal of the Engineering Mechanics Division (ASCE) 1974; 100(EM1):

111--121.41. Liang J, Mourelatos ZP, Tu J. A single-loop method for reliability-based design optimization. International Journal of Product Development

2008; 5(1–2):76--92.42. Melchers RE. Structural Reliability Analysis and Prediction (2nd edn). Wiley: New York, 1999.43. Yang RJ, Gu L. Experience with approximate reliability-based optimization methods. AIAA (American Institute of Aeronautics and Astronautics)

2003–1781, 2003; 3474--3481.44. Brunner FJ. Reliability approach for vehicle safety components. Quality and Reliability Engineering International 1989; 5(4):291--297.45. Liaw LD, DeVries RI. Reliability-based optimization for robust design. International Journal of Vehicle Design 2001; 25(1/2):64--77.46. Wu Y-T, Millwater HR, Cruse TA. An advance probabilistic analysis method for implicit performance function. AIAA Journal 1990; 28(9):1663--1669.47. Wu Y-T. Computational method for efficient structural reliability and reliability sensitivity analysis. AIAA Journal 1994; 32(8):1717--1723.48. Zhang YM, Liu QL. Reliability-based design of automobile components, proceedings of the institution of mechanical engineers part D. Journal of

Automobile Engineering 2002; 216(D6):455--471.49. Zhang YM, Wen BC, Andrew YTL. Reliability analysis for rotor rubbing. Journal of Vibration and Acoustics (ASME) 2002; 124(1):58--62.50. Zhang YM, Liu QL, Wen BC. Practical reliability-based design of gear pairs. Mechanism and Machine Theory 2003; 38(12):1363--1370.51. Zhang YM, Wang Sh, Liu QL, Wen BC. Reliability analysis of multi-degree-of-freedom nonlinear random structure vibration systems with correlation

failure modes. Science in China (Series E) 2003; 46(5):498--508.52. Ramu P, Qu X, Youn BD, Haftka RT, Choi KK. Inverse reliability measures and reliability-based design optimization. International Journal of Reliability

and Safety 2006; 1(1/2):187--205.53. Siddall JN. A new approach to probability in engineering design and optimization. Journal of Mechanism, Transmissions, and Automation in Design

(ASME) 1984; 106:5--10.54. Doltsinis I, Kang Z. Robust design of structures using optimization methods. Computer Methods in Applied Mechanics and Engineering 2004; 193:

2221--2237.55. Ueno K. Company-wide Implementations of Robust-technology Development. ASME Press: New York, 1997.56. Xi Z, Youn BD, Gorsich DA. Reliability-based robust design optimization using the EDR method. SAE/SP-2119, Paper No. 2007-01-0550, 2007.57. Bhamare SS, Yadav OP, Rathore APS. A hybrid quality loss function based multi-objective design optimization. Quality Engineering 2009; DOI:

10.1080/08982110902762626.58. Hauser J, Clausing D. The house of quality. Harvard Business Review 1988; 66(3):63--73.59. Chandra MJ. Statistical Quality Control. CRC Press: Boca Raton, FL, 2001.60. Taguchi G, Chowdhary S, Wu Y. Taguchi’s Quality Engineering Handbook. Wiley: New York, 2004.61. Rajendran I, Vijayarangan S. Optimal design of a composite leaf spring using genetic algorithms. Computers and Structures 2001; 79:1121--1129.62. Al-Qureshi HA. Automobile leaf springs from composite materials. Journal of Materials Processing Technology 2001; 118(1):58--61(4).63. Erol S, Gratton M. Design, analysis and optimization of composite leaf springs for light vehicle applications. Computers and Structures 1999; 44:

195--204.64. Adali S, Walker M, Verijenko VE. Multi-objective optimization of laminated plates for maximum pre-buckling, buckling and post-buckling strength

using continuous and discrete ply angles. Computers and Structures 1996; 35:117--130.65. Wahl AM. Mechanical Springs (2nd edn). McGraw-Hill: New York, 1963.66. SAE. Manual on design and application of leaf springs. SAE HS 7888, 1990.67. Saaty TL. The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation. McGraw-Hill: New York, NY, 1980.68. Biggs MC. Constrained minimization using recursive quadratic programming. Towards Global Optimization, Dixon LCW, Szergo GP (eds.).

North-Holland: Amsterdam, 1975; 341--349.69. Han SP. A globally convergent method for nonlinear programming. Journal of Optimization Theory and Applications 1977; 22:297.70. Powell MJD. A Fast Algorithm for Nonlinearly Constrained Optimization Calculations Numerical Analysis (Lecture Notes in Mathematics, vol. 630),

Watson GA (ed.). Springer: Berlin, 1978.71. Arora JS. Introduction to Optimum Design. McGraw-Hill: New York, NY, 1989.

40



Authors’ biographies

Om Prakash Yadav is an assistant professor in the Industrial and Manufacturing Engineering Department at North Dakota StateUniversity, U.S.A. He completed his PhD in Industrial Engineering at Wayne State University, Detroit, U.S.A. He has over 20 years ofteaching, research, industry, and consulting in India and U.S.A. He has spent more than 3 years of his working experience in the autoindustry in North America. He received his BS in Mechanical Engineering from Malaviya National Institute of Technology (MNIT), Jaipur(India) and MS in Industrial Engineering from the National Institute of Industrial Engineering (NITIE), Bombay (India). He has publishedover 30 research papers in the area of quality, reliability, product development, and operations management. His research interestsare focused around product development, reliability and quality engineering, concurrent engineering, and manufacturing systemsengineering.

Sunil S. Bhamare is a Senior Lecturer in Government Polytechnic, Ratnagiri, India. He completed his PhD in Mechanical Engineeringat Malviya National Institute of Technology, Jaipur, India. He received his BE in Mechanical Engineering and ME in ManufacturingTechnology from India. He has been invited by the Department of Industrial and Manufacturing Engineering, North Dakota StateUniversity (NDSU), Fargo, U.S.A. to pursue part of PhD studies. He has presented papers on simulation and modelling, physics-of failuretheories and decision making framework for early stages of product design and development.

Ajay Rathore is a Professor in the Mechanical Engineering Department at Malaviya National Institute of Technology, Jaipur, India. Hecompleted his PhD from Liverpool John Moores University, Liverpool, U.K. He has over 25 years of teaching, research, and consultingexperience in India. He has received his BS in Mechanical Engineering from MREC Jaipur and MBA from the University of RajasthanJaipur (India). He has published over 15 research papers in the area of quality, productivity improvement, product development, andoperations management. His research interests are focused around optimization, supply chain management, productivity improve-ment, product development, and total quality management.


41

Documents

Reliability-based robust design optimization: A multi-objective framework using hybrid quality loss function