C H A P T E R T H I R T E E N

Robust Desi n

Courtesy of Ford Motor Co.

EXHTBIT 13-1Rear seat belt experiment. This experiment was nm on a simulation model t0 explore many design

parameters and noise conditions.

265

266 Chaprat iS

What ls

',/

F'ord Motor Cornpany sal'ety errgineers were working with a supplier to better understandthe performance of rear seat belts. In any conventional seat belt system with lap andshoulder belts, if the lap portion of the belt rides upward, the passenger may slide beneathi t . potenttal ly result ing in abdorninal in jury. This phenomenon, cal lecl "submarining," isrelated to a large number of factors, including the nature of the col l is ion, the design of thevehicle, the properties of the seats and seat belts, and other conditions. Based on experi-mentat ion. s imulat ion, and analysis, Ford engineers hoped to determine which of thetrany factors were ntost cr i t ical to passenger safety and to avoicl ing submarining. Thettnage shou'n in Exhibi t 13- l depicts the modelused in Ford's simulat ion analysis.

This chapter presents a method for designing and conducting experiments to improvethe performance of products even in the presence of uncontrol iable var iat ions. Thisnlethod is known as robust desisn.

Robust Design?We define a robttst product (or process) as one that performs as intended even under non-ldeal conditions such as manufacturing process variations or a range of operating situa-t ions Wc use the term noise to descr ibe uncontro l led var iat ions that may af fect per for-rnance, and we say that a quality product should be robust to noise factors.

Robust design is the product development activity of improving the desired perfor_mance of the product while minimizing the effects of noise. In robust design *" ,r. .^-

, perlments and data analysis to identify robust setpoints for the design parameters we cancotr t ro l . A r t tb t ts t setpoint is a combinat ion of design parameter values for which theproduct perfotmance is as desired under a range of operating conditions and manufactur-i ng va r i a t rons ,

Conceptually, robust design is simple to understand. For a given performance target(sat'ely restraining rear-seat passengers, for example), there rnay be many combinatignsoiparumcter ' " 'a lues that r i , i l l l , ie lc i the desi r . 'd resul t . I {o \ \ , r . \er . sr )me of ihesr : cornbi r ra-t ions are more sensi t ive to uncontro l lab le var iat ion than others. Since the product wi l ll ikely oper:ate in the presence of various rroise factors. we would l ikp to choose the com-bination of parameter values that is least sensitive to uncontrollable variation. The robusr

- design process uses an experimental approach to finding these robust setpoints.To understand the concept of robust setpoints, consider two hypothetical factors af-

fecting some measure of seat belt performance, as shown in Exhibit l3-2. Assume thatfactor A has a l inear effect, fA, on performance and factor B has a nonlinear effect, fs.F'urther consider that we can choose setpoints for each factor: Al or A.2 for factor A, andB1 or 82 for factor B. Assuming that the effects of fa and fs are additive, a combinationof Al and B2 wi l l prov ide approximately the same level of overal l per formance as acombination of 42 and Bl. Manufacturing variations wil l be present at any chosen set-polnt, so that the actual value may not be exactly as specified. By choosing the value ofB l f o r f a c t o r B , w h e r e t h e s e n s i t i v i t y o f t h e r e s p o n s e t o f a c t o r B i s r e l a t i v e l y s m a l l , u n i n -tcnded r ,ar ia t ion in factor B has a re lat ive ly smal l in f luence on overai l product per for-mance. Therefore, the choice of Bl and A2 is a more robust combination of seipointsthan the combination of 82 and Al.

The robttst design process can be used at several stages of the product developmentprocess. As with most product development issues, the earlier that robustness can be con-sidered in the product development process, the better the robustness results can be. Ro-

Robust Design 267

A1 A2 81 82

1 .

EXHIBIT 13-2 noUuii design exploits nonlinear relationships to identify setpoints where the product performance

is less sensitive to variations. Inlhis e>iarnple, the chosen value for the factor A setpoint does not affect robustness'

whereas that of factor B does. Choosing Bl minimizes the effect of variation in factor B on overall performance'

bust design experiments can be used within the concept development phase as a way to

refine the spe;ifications and set realistic performance targets. While it is benefic.ial to

consider product robustness as early as the concept stage, experiments for robust design

are used nrost frequently during the detail design phase as a way to ensure the desired

product perfonnance und.l. u variety of conditions. In detail design, the robust design ac-

iiuity iS also known as parameter design, as this is a matter of choosing the right set-

points for the design parameters under our control. These include the product's materials'

dimensiOns, tolerances, manufacturing processes, and Operating instructtotrs

For many engineering design problems, equations based on fuudamental physicai prin-

ciples can be so-lved foi robuit parameter choices. However, engineers generally cannot

fuily model the kinds of uncertainties, variations, and noise factors that arise under real

conditions. Furthermore, the ability to develop accLlrate mathematical models is limited

for many engineering problems. For example, consider the difficulty of accurately model-

ing the seat Uett submarining problem under a wide variety of conditions' In such situa-

tio=ns, empirical investigationlhrough clesignerl experillents is necessary' SLrch expert-

..nt, .un be used to directly support decision making and can also be used to improve

the accuracy of mathematical models'

In the case of the seat belt design problem, Ford's engineers wished to test a range ot

seat belt design parameters and collision conditions. However, crash testing is vety ex-

pensive, so Fird^worked with its seat belt supplier to develop a simulation model which

was caliUraled using experimental crash data. Considering the hundreds of possible de-

sign parameter combinaiions, collision conditions' and other factors of interest' the engi-

n"-..s chor. to explore the simuiation model using a carefully planned exlreriment' Al-

though simulation requires a great deal of computational effort, the simulation model stil l

alloied Ford engineers to rui dozens of experiments under a wide variety of conditions'

which would nolhaye been possible using physical crash testing'

Response toFactor B

268 Cirapter 13

For the Fo'r'ci seat belt riesign teani, the goals of this designeo experiment wei:e io learn:

" What cotnbination of seat, seat belt,'an<l attaehnrent parameters minimizeri rear-seatpassenger submnrlning during & $r6$n,

' How submarinlng ls afflcted by uncontrollable conditions, What oonrbination of de-stgtl parantctefr is rDost robust to such noise lactors,?

Design of ExperimentsTlre approach to robust design presented in this chapter is based on a method called de-

,, sign Qf experiztt,zls (DOE). In this method, the team identifies the parameters that can'./' be controlled and the noise factors it wishes to investigate. The team then designs, con-

ducts, and analyzes experiments to help determine the parameter setpoints to achieve ro-bust performance.

In Japan during the 1950s and I960s, Dr. Genichi Taguchi developecl techniques toapply DOE to improve the quality of products and manufacturing processes. Beginningwith the qual i ty movement of the 1980s, Taguchi 's approach to experimental designstarted to have an impact on engineering practice in the United States; particularly at FordMotor Cotnpany, Xerox Corporation, AT&T Bell Laboratories, and through the Ameri-can Supplier Institute (which was created by Ford).

Taguchi receives credit for promoting several key ideas of experimental design for thedevelopment of robust products and processes. These contributions include introducingnoise factors into experiments to observe these effects and the use of a signal-to-noiseratio metric including both the desired performance (signal) and the undesired effects(noise). While statisticians had been showing engineeis how to run experiments fordecades, it rvas not until Taguchi's methods were widely explained to the U.S. manufac-turing industry during the 1990s that experiments became commonly utilized to achieverobust design.

DOE is not a substitute for technical knowledge of the system under investigation.In fact, the team should use its understanding of the product and how it operates tochoose the right parameters to investigate by experiment. The experimental risults canbe used in conjunction with technical knowledge of the system'in order to make thebest choices of parameter setpoints, Furthermore, the experimental results can be usedto build better mathematical models of the product's function. In this rvay, experimen-tat ion complements technical knowledge. For example, Ford engineers have basicmathematical models of seat belt performance as a function of passenger sizes and col-lision types. These models allow Ford to size the mechanical elements and to determinethe belt attachment geometry. Based on empirical and simulation data, Ford's analyti-cal models and seat belt design guidel ines gain precision over t ime, reclucing the needfor time'consurning empirical and simulation studies, Eventually, this technical knowl-edge may improve to the point where only confirming tests of new seat belt configura-tions are required.

Basic experimental design and analysis for product development can be successfullyplanned and executed by the development team. However, the held of DOE has many ad.vanced methods to address a number of complicating factors and yield more useful ex-perimental results. Development teams thus can benefit from consulting with a statisti-cian or DOE expert who can assist in designing the experiment and choosing the bestanalytical approach.

-------iFt-

Robust DesiPn 269

The Robust Design ProcessTo develop a robust product through DOE, we suggest this seven-step process:

l . Ident i fycontrol factors,noisefactors,andperformancemetr ics.2. Formulate an objective function'

3. DeveloP the exPerimental Plan'4. Run the exPeriment.5. Conduct the analYsis.

6. Select and confirm factor setpoints'

. Reflect and rePeat.

Step 1: ldentify Control Factors, Noise Factors,and Fedormance Metrics

The.,rpbust design procedure begins with identification of three lists: control factors,

noise fdctors, and performance metrics for the experiment:

, Control factors: These are the design variables to be varied in a controlled manner

during the experiment, in order to explore the product's performance uuder the many

combinations of pararneter setpoints. Experiments are generally run at two or three

discrete levels (setpoint values) of each factor. These parameters are called control

factors because they are among the variables that can be specified tbr productiorl

and/or operation of ihe product.lor example, the webbing stiffness and coefficient of

fr ict ionarecontrol factorsof interestfort l reexperlment.. Noise factors.. Noise fa'ctors are variables that cannot be explicitly controlled during

the minufacturing and operation of the product. Noise factors may include manufac-

turing variances, changes in materials pioperlies, multiple user scenarios or operating

condltio,,s, and even Jeterioration or misuse of the product. if through special tech-

niques the team can control the noise factors during the experiment (but not in produc-

tion or operation), then variance cair deliberately be induced duping the experiment to

m,n'fi :ill;:,ff Tli,i;,:li[:T,:TL::h.iilffi ,'i[":';;:L*, j'l''Jmize the effects of,this variation. For seat belts to be used with a range of seats, the

shape of the seat and the seat fabric must be considered noise factors' The goal is to

design a seat belt system that works well regardless of the values of these factors'

Performance metrics: These are the product specifications of interest in the experi-

ment. Usually the experiment is analyzed with one or two key product specifications

as the perforrnance metrics in order to find control factor setpolnts to optimize this

performance. These mefics may be derived directly from key spee ifications wherc rtl '

bo.tn.r, is of critical concern. iSee Chapter 5, Product Specifications') i"or example,

how far the passenger's back or buttocks move fonvard during the collision would be

possible performance metrics for the seat belt experiment'

For the seat belt design problem, the team held a meeting to list the cotltrol factors, noise

factors, and performance metrics, As Taguchi teaches, they placed these lists into a single

graphic, cal|ed a paranleter diagram \ar p.diagrai?''':, as shown in Exhibit 13-3"

&

$70 Chapter 13

EXHIBIT,I3.3Parameterdiagram used todesign the seatbelt experirnent.Bold textindicates thepcrfonnance

metric used andthe controlfactors and noiselactors chosen

for exploration.

Step 2:

Control Factors

Belt webbing st i f fnessBelt webbing fr ict ionLap belt force l imiterUpper anchorage st i f fnessBuckle cable st i f fnessFront seatback bolsterTongue fr ict ionAt tachment geomet ry

Performance Metrics

Back ang leS l i p o f bu t t ocksH ip r o ta t i onFo rwa rd knee mo t i on

Noise Factors

Shape of rear seatType of seat fabricSeverity of coll isionWear of componentsPosi t ioq ing of passengerPosi t ion ing of bel ts on bodvS ize o f passenge rType of c loth ing fabr icWeb manu fac tu r i ng va r i a t i onsLatch manufactur ing var iat ions

After listing the various factors, the team nrust decide which ones will be explored byexperlment' when a large number of parameters are suspected of potentiallylffectingperformance, the selection of critical uuriubl., can be subsiantially narrowed uy J\ing un-alytical models and/or by running a sctreening ,xp"ri^ent with two levels fbr each ofmany factors' Then a finer experiment is run with two or more levels of the few parame-ters believed to affect performance.

Ford engineers consit lered the l ists shown in Exhibi t J3-3. They chose to fbcus the ex-periment on exploration of seven seat belt parameters, holdingconstant the geometric lo-cations of the three aftachment points. They decided to use ;back angle at peak,, as theoutput metric, the angle that the passenger's back makes with respect to vertlcal at themoment of maximum restraint ' Back angle is a smal ler- is-better performance metrc,measured in radians.A primary concem in this experiment was the effect of three particular noise factor.s:seat shape, fabric type, and severity of collision. Through prelimrnary analysis, the teanrfound the best and worst combinations of these nois. cJnditions with respect to the sub-marining effect. These three noise factors were thereblr combined,i"r" i*J ._,r;n;;-;,r.conditions fbr the

]lurposes of the experiment. This approach, known as contpoundednoise' can be helpful when many noise factors must be considered. (See Testing NoiseFactors in Step 3.)

Formulate an Objective Function

rust be transformed into an objective /itnctrcnnce. Several objective functions are useful inrance concems. They can be forniulated eitner, and they include;

' Maximizing; This type of function is used for performance dimensions where largervalues are better, such as maximum deceleration before belt slippage, common formsof this objective function q are n = p or n : Liz, where pr i, tn. ,n.un of the experimen-tal observations under a given test condition.

--q-

Robust Desisn 271

Minintixing: This type of function is used for perfomrance dimensions where smallervalues are better, such as back angle at peak deceleration, Common forms of this ob-jective function are I = p or 1'l = 02, where o2 is the variance of the experimental ob-servations under a given test condition. Altematively, such minimization objectivescan be formulated as luncl ions to be uraxint ized, such as 4 = l l p or | ,= I o: .

, Target vulue:'lhis type oi'flnction is uscd for perfbrrnlnce dirncnsious u'licre valuesclosest to a desired setpoint or target are best, such as amount of belt slackening beforerestraint. A commorr maximizing form of this obtect ive funct ion is q = l fp - 02"* hcrc t rs t l te t ; -rrg. ' t r ulr tc,

' Signal-to-rtoise ratio: This type of function is used particularly to measure robust-ness. Taguchi formulates this metr ic as a rat io with the desired response in the nu-merator and the var iance' in the response as the denominator. Generai ly the nieanvalue of the desired response, such as the mean back angle at peak, is not difficultto adjust by changing contl'ol factors. In the denominator, we place the variance ofthis response (the noise response), which is to be minirnized, such as the var iancein back angle result ing from noise condit ions. In pract ice, reducing var iauce isrnore di f f icul t than changing the mean. By computing this rat io, we can highl ightrobust factor sett ings for which the noise response is relat ively lbw as compared tothe signal response. A common maximizing tor 'ni of this ob. lect ive funct ion is t l :

l0 log (ltzlo').

The Ford statistician consulting with the team suggested two objective f'unctrotls: the av-erage back angle at peak and the range ofthe back angle at peak (the drf fereuce bctweenthe maximum and rninimum back angle at peak at the two noise conditions to be tested).Both of these are objectives to be minimized. Together these two metrics would providedeeper insight i r t to the behavior of the s5,51911r than ei ther one alone.

I

Step 3: Develop the Experimental PlanStatisticians have developed rnany types of'efficient experiurental plans. These plans layout how to vary thefaetor /evcrls (values of the control factors and possibly also sourc ofthe noise factors) in a series of experiments in order to explore the system's behavior,Sorhe DOE plans are more efficient for cliaracterizing certain types of systems, whileotliers provide more complete analysis.

Experimental DesignsA critical concern in designing experiments is the cost of setting up and running the ex-perimentaltrials.In situations where this cost is low, running a large number of trials andusing an experimental design with resolution high enough to explore more factors, factorcombinations, and interactions may be feasible. On the other hand, when the cost of ex-perimentation is high, efficient DOE plans can be used that simuitaneously change sev-eral factors at once. Some of tlte most popular experinlentd designs ate listed below and

depicted in Exhibit 13-4. Each one has important uses,

. Full factorial: This design involves the systematic exploration of every combittation oflevels of each factor. This allows the team to identify all of the multifactor interaction

A2B1 a2 B1 B2

c 1 l c 2 c 1 l c 2 c1 c2 c1 c2D1 ) ' o2 )1 DzlDl lD '1

,,,1-lF I1 n X X X X X x X

t t X X

FIG 1 x X v. X X

x x X

61 x x x X x xx x x x x x x x x x x

IrzG 1 X x x X

X X

272 Chuprer' 13

Full-Factorial Matr ix 1/2 Fractional Factorial Matrix

L8 Orthogonal Array(1/16 Fract ional Factor ia l

EXHIBIT 13 '4 Severa l a l temat ive expernnenta l p lans ib r seven fac to rs (A , ts , C , D, E , F , and G) a t rwo leve lscach. The f t r l l - lac to r ia l e rper i rnent conta ins 21 = 128 t r ia ls , whr le the L8 or thogona l a r ray des ign conta ins on ly 8 t r ia ls ,denoted by the x l l rarks i t t the nratr ices. The L8 ofthogonal array plan rs the one Lrsed for the seat belt cxperinrent and ;sshorvn in conventional ror.v, 'column format in Exhibit l3-5.

S o r r r c e : [ r r l c l i o n r r l t i l c l o ! i a l l o y o u t 5 n d t p r c r l t i o n r R o s s ( 1 9 9 6 )

A1 A2

B 1 B2 81 82

c 1 l c 2 c2 ? 1 c2 c1 c2D1 D2IDl )'1 >2 ) l D2IDl irlor ) 1 )2

t 131 X X X

x I X ' x x

ez3 1 x X x

x

1/4 Fractional Fastorial Matrix 1./8 Fractional Factorial Matrix

A1 A2B1 82 B 1 s2

c l l c z c l c2 c1 c2 c l c2) 1 )2 lD1 lD ') 1 o2Dl ID: DI ID i D I I D i D l ID2ID

Matrix) One Factor at a Time

l

Robttsr Design 273

effects, i1 addition to the prirrary (main) et'fect of each factor on perfoimance. This

type of experiment is generally practical only for a small number of factors and levels

and when experiments are inexpensive (as with fast software-based simulations or very

flexible hardware). F-or an investigarion of k factc-rrs at n let,els bach, the nunrber of trl-

als in the full-factorial experirnent is lA. Full factorial exlreritnentatior' is typically in-

feasible for an experiment with grcater than four to l lve ft lctot's'

. Fract ional factor ia l : This design uses only a smal l t iact ion o i the combir rat ions used

above. In exchange for this efficiency. the abil ity to compute the magnitudes of all the

interact ion ef fects is sacr i l rced lnstead. the in teract ions are col l tbundcd u i th o l i )c i l l l -

teractions or with some of the main f 'actor effects. Note that the fractional factoriallay-

out sti l l maintains balance within the experimental plan. 'Ihis

means that for the sev-

era l t r ia ls at any g iven factor level . each of the other f 'actors is tested xt every level thc

same numtrer of t i tnes.

. Orthogonal array: This design is the smallest fractional factorial plan t[at still allows

the team tcr identify the main effects of each factor'. However, these urain effects are

^coufounded with many interaction eft 'ects. Nevertheless, orthogonal array layouts are-* id . ly

ut i l ized in technical invest igat ions because they are extremely ef f ic ient .

taguctrl popularized the orthogonal array DOE approach, even though statisticials

had developed such plans several decades earlier and the roots of these designs carr be

t-. -

tracecl back many centuries. Orthogonal array plaus are rtauted accordilrg to the num'

ber of rows (exper iments) in the array: I4 , L8, L9,L21, and so on. The appendix to

thrs chapter shows several orthogol)al array erperittretltai plans

, One factor at a tinte: This is an unbalanced experimental plan because each trial is

conducted with all but one of the factors at nominal levels (and the first trialhaving all

the factors at the nominal level). This i-s generall- '- considered to be an ineffective rvay

to explore the factor space, even though the nunber of trials is srnall, I + ft (n - l).

However, for parameter optimization in systems with significant i l l teractions, an adap-

tive versron of the one-at-a-time experirnental pl-an has been shown lr be generally

more efficient than orthogonal array plans (Frey et al., 2003)

The Ford team chose to use the L8 orthogonal array experiment design because this plan

would be an efficient way to explore seven f-actors at two levels each. Subsequeut rout]ds

of experimentation could later be used to explore additional levels of key parameters as

well as interaction effects if necessary. The orlhogonal array experimental plan is shown

in Exh ib i t l 3 -5 .

Testing Noise FactorsSeveral methods are used to explore the effects of ttoise factors in experintents. If some

noise factors can be controlled for the purpose of the experiment, then it may be possible

to tlirectly assess the effect of these noise t'actors.ll 'the noise factors cannot be controlled

during the experiment. we allow the noise to vary naturally and sirnpl5, assess the prod'

uct 's perfbrmance in the presence of noise, Some comnron ways to test l l t l lse l l lctors are:

. Assign addit ionat columns in the orthogonal an'ay or fract ional factor ial la,vout to the

noise factors, essentially treating the noise as another variable. This allows the effects

of the noise factors to be deternrined along rvith the control factors. *i

Factor Description

274 Cl+dpter 13

EXHIBIT 13.5Factorusstgntnentsond tho Lgorthogonul unoyexpeflmsntclesign used fbrthe seat beltexperiment. ThisDOE plan testsseven factors attwo levels each.Each row wasreplic1Lcrl tw,ice.under the hvocornpoundednoise conditions,y ie ld ing l6 testdata points foranalys is ,

B

cD

Belt webblng st i f fnessr compliance characterist ic of the webbing measured in atensr le load mach ineEdt wrbblng f i lct lonr Cooff lcirnt of fr lct ion, whlch ie a functlon o{ tho b6lt weeveand sur face co6t ingLop bclt forco l lmltor: Al lows control led releasp of the eet belt at a cenain force reverUpper anchorege st l f fness: cgmpliance characterist i of the structure to which theupper anchorage (D- loop) i s mountedBuckle cable st i f fness: compliance characterist ic of the cables by which the bucrreis a t tached to the veh ic le bodyFront seatback bolster: Profi and st i f fness of seat back where the knees mav conracrTongue fr ict ion: coeff icient fr ict ion for the bearing area of the tongue whichs l ides a long the webb ing

FG

1 1 1 1 1 1 1

1 1 1 2 2 2 2

1 2 2 1 1 2 2

1 2 2 2 2 1 1

2 1 2 1 2 1 2

\ 2 1 2 2 j 2 1

2 2 1 1 2 2 1

2 2 1 2 1 1 2

1

2

345

6

7

8

' Use anouter array for the noise factors. This method tests several conrbinations of thenoire factors for each rorv in the main (inner) anay. An example of this approach isshown tn the appenclix, where the outer array consists of an L4 design, testing combi-nations of three noise ractors by replicating each row four times.

' Run replicates of each row, allowing the noise to vary in a natural, uncontrolled man-ner throughout the exper iment . resul t ing in measurable var iance in per formance foreach ron ' Wi th th is approach, i t is par t icu lar iy important to randomize the order of thetrials so that any trends in the noise are unlikely to be correlated with the systematicchanges in the control factors. (See Step 4.)

' Run replicates of each row with compottndecl nQise.ln this method, selected noise fac-tors are combined to create several representative noise conditions or extreme noiseconditions. This approach also yields measurable variance for each row, which can beattributed to the effect of'noise.

The Ford team chosE to ttt i l ize the compounded noise approach in the seat belt experi-ment' The team tested each row using the two combinations of the three noise factors rep-resenting the best- and worst-case conditions. This resulted in l6 experimental runs forthe L8 DOE plan, as shown in Exhibit lJ-5.

-T-

Rctl.tust Design 275

GFEDcBA N- N+ Avg Range

1

2

3

4 "

5

6

7

8

1 1 1 1 1 1 1

1 1 1 2 2 2 2

1 2 2 1 1 2 2

1 2 2 2 2 1 1

2 1 2 1 2 , 1 2

2 1 2 2 1 2 1

2 2 1 1 2 2 1

2 2 1 2 1 ' , \ 2

0 .3403 0 .2915

0 4608 0.3984

0 3682 0 3627

o 2961 0,2647

0,4450 0.4398

0 .3517 0 .3538

0 .3758 0 3580

0 4504 0.407 6

0 .3159 0 .0488

0.4296 0,0624

0 3655 0 0055

0 .2804 0 0314

o 4424 0.0052

0.3528 0.0021

0 .3669 0 0178

0.4290 0.0428

EXHIBIT 13-6 Data obtained from the seat belt experiment.

Step 4: Run tfra e*perimentTo execute the experiment, the product is tested under the various treatment conditions

described by each row in the experimental plan. Randomizing the sequence of the experi-

mental runs ensures that any systematic trend over the duration of the experiment is notcorrelated with the systerhatic changes to the levels of the factors. For example, if the ex-periments of the L8 plan are not randomized. and the test\nditions drift over time, this

ef fect may be incorrect ly at t r ibuted to factor A s ince th ib column changes hal f ivaythrough the experiment. For some experiments. changing certain factors ntay be so diff i-cult that all tr ials at each level of that f actor are rur.l together and only partial ratrdorrriza-t ion uray be achieved. In pract ice, randornize the t r ia ls rvhenever pract ica l . and when not

possib le, va l idate the resul ts rv i th a conf i rmat ion r t tn , (See Step 6. )

In the seat bel t exper iment , each of the e ight factor combinat ions in the L8 design was

tes ted unde r t he nvo compounded no i se , cond i t i ons , The 16 da ta po in t s con ta in in l r t he

back anule data are shorvn in Exhib i t i3-6 rn the columns t i t led N- and N-r .

Step 5: Conduct the AnalysisThere are many ways to analyze the experimental data. For all but the most basic analysis,the team benefits from consulting with a DOE expeft or from refering to a good book onstatistical analysis and experimental design. The basic analyical method is srtmmarized hele.

Computing the Objective FunctionThe team r.ril l have already devised the objective functions for the experiment and will gen-erally have en objectiva relstcd to ths mean perfortnanee and the varianeo in perfotmartce,$gmetimes the mean and varlanee will be ssnrbined arrd expressed as n rlugle objeetivn lnthe form of a signal-to-noise ratlo, The values of the objective function ean lte computed foreach row of the experiment. For the seat belt experiment. the columrs on the Ligltt side o1 thetable in Exhibit l3-6 show the computed objective F.rnction values (average back angle and

range of back angle) f'cr each rcw. Recali that these r:e both objectives to be minrn:rzeC.

476 Chistter 13

EXHIBTT 13.7Factor effectscharts for theseat beltexpenment,

Computing Factor Effeqts by Analysis of MeansThe most straightforward analysis to Jonductrryilt simply yield the rnain effect of eachfactor assigned to a column in ihr .*p.rir*ii, rn"r. -uin .rr*ri, are catteo the factor ef-fecrs' The enal-vsis of'nreans invorves ri;;t averaging ail the computed objective firnc-t ions foreach factor level . In the L8 DoE exampte, ihe ef fect of factor level Al ( factor Al ]1 : : ] l l l , lJ , . .u : . lueeof t r ia ls r ,2 ,3 ,anJ+.s i ,n i rar ly , rheef fectof factor leve l E2is

:ffiil:ir;:iX$:;r.;f#d 7. rhe resulrs of an analysis of n,eun, are conventionalry

Exhibit l3-7 presents the factor effects charts for the seat belt example. These effectsare plotted tbr each of the objective functjons, Exhibit l3_7(a) plots the overage per,for_ntznce at each lactor levet (the first objective tunction), rrris cnan shows rvhich factor lev_els can be used to raise or lower the mean performance. Recall that back angle at peak is tobe minimized, and note that the .i,urt ,rgg.ris urat factor tru.t, fn r-n z c2 Er F l G I I wil

( a )

p o't:

E o.*tec - ' - *

E n 2 4

6

0.34

{ o.sz

o

.c 0,04€o6 0,03

J(J

E 0.02

oP o.orotr

0.00

( b )

Control Factors

n F 1a

A E 2 \ A c z\ f f i t

nF2

control Factors

v I vz f- ' l ftrorfl-

a./t'" zf"

Robust Design 277

minimize the average back angle metric (Factor D appears to have no effect upon firean

performance.) However, these levels wi l l not necessari l - r ' achieve robust performance'-Exhibit

13-7(b) is based ontheraitge of per/brmance ateach factor level (the second ob-

ject ive funct ion). This chart suggests that levels lA2 82 C2 D1 E'1 t '2 Cl l rv i l i nr inirnizc

the range ofback anglc at PcakTaguchi recommends tliat the signal-to-noise ratio for each factor level be plotted in order

to identifo robust sefpoints, Since the signal-to-noise ratio inch'rdes the mean perfontiance in

the numerator ancl the variance in the denominator, it represents a cornbination of these tivo

obiectives or a trade-off between them. Rather than specificaliy plotting the signal-to-

nolse ratio, many engineers and statisticians prefer to sin4rly interpret the tw'o ob.jecttves to-

gether, giving rrrore control over the trade-off. To do so, the factor effects chafis sltou'n in

ixhibit l3-7 can be compared in order to choose a robust setpoint in the next step

Step 6: Select and Confirm Factor Setpoints

Ana[isis of means and the factor effects'charts help the tealn determine which factors

havJb'strong effect on mean performance and variance. and therefore how to achieve ro-

bust perforrnance. These charts help to identify which factors are best able to reduce the

product,s variance (robustness factors; and rvhich factors can be used to improve the per-

fbrmance (scal ing factors) . By choosing setpoints based on these ins ights ' the team

should be able to improve the overall robustness of the product.

For example, .onr i , l . , the ef fects of factor A on both average and range of back angl :

iu the exper i rnent . The c l ia(s in Exhib i t l3-7 shoiv that level A1 would rn in i rn ize back

angle, bui level A2 rvould minimize the range of back angle, representing a ttaile-off be-

tween performance and robustness, A sirnilar trade-off is evident in factor F. lJowever,

for fac iors B, C, D, E, a ld G, there is no such t rade-of f , and levels 82,C7, Dl , E l , and

G I min imize both ob. ject ives '

Using factors g, C, D, E, and G to achieve the desired robustness and factors A and F

to incre lse per formarrce, Ford engineers selected the setpoint [A l B2 C2 D1 E1 Fl G1] '

As is usualiy the case, the chosen setpoint is not one of the eight ortl ' togonal array row$

tested in the erperimenr. Given that ihis setpoint has never been tested. a confirmation

run should be used to ensure that the expected robust performance has been achieved'

Step 7: Reflect and RePeat

One round of experiments may be sufficient to identif,v appropriately robust setpoints'

Sometimes, however, further optimization of the product's perfotmance is worthr'vhiie.

and this may require several additional rounds of experimentatton.

Insubsequentexper imenta t ionandtes t i t tg , the teammaychooseto :

. Rcconsider the setpoints chosen for factors displaying a trade-otJ of performance ve r-

sus rob:lstness'. F .xp l r l re i l t t c rac t lons an to l tg some o f the fac to rs in o rder to f i r r ther i t t tDrove t l - t t :

performance.. Fine-tune the paratrteter setpoints using

range.

values between the levels tested or outside this

Robust Design 279

fect performance, such as manufacturing variations, operating conditions, and product

deterioration. We suggest an approach to the development of robust products based on design of ex-

periments (DOE). This seven-step process for robust design is: d.

l. Identify control factors, noise factors, and performailce metrics.

2. Formulate an objective futtction.

3. Develop the experimental plan.

4. Run the exper i tnent .

5. Conduct t l ie analysts.

6. Select aud conf inn facto l 'setpoints.

7. Reflect and repeat.

. Orthogonal array experimental plans provide a very efficient method for exploring the

main effects of each factor chosen for the experiment.

. T4 achieve robust perfonnance, use of objective functions helps in capruring both meanp"rhtman." due to each control factor and variance of performance due to noise factors.

. Analysis of means and f'actor effects charts facilitate the choice of robust paralneter

setpornts.. Because many nuances are involved in successful DOE, most teams applying these

methods wil l benefit from assistance by a DOE expert.

References and BibliographyMany current resources are available on the Intemet vla

www.u lrich-eppin ger.net

Taguchi's metltods for experintental design and details about ot'thogonal array

experimentation plans are explained in several texts, including Taguchi's classic hvo-yolunre text t ranslatccl into Engl ish Phadke provides r ' t t l lerous erarnples and l t r lct icaladvice on application of DOE. Ross emphasizes insights gained through ANOVA analysts.

Tagr"rchi, Genichi, Svstern of Exlterintental Design. Engineering Methods to Optimize

Quutity und lvlinintize'Costs, two volumes, Loutse Watanabe Tr"rng (trans.)' WhiteP la ins , NY, 1987.

Taguchi,,Ge nichi, Introduction to Quality Engineering; Designing Quality into

Prodttcts and Processes, Asian Productivity Organization (trans. and pub.),

T o k y o , 1 9 8 6 .Ptradke, Madhav 5., Qualitt, Engineering Using Robust Design, Prentice Hall,

Englewood Cl i f fs. Ni, 1989.

Ross, Phillip J., Taguchi Techniques.for Qualitv- Engineering, McGrarv-Hill, New

York, 1996.

Grove and Davis present a thorough explanation of experimentai design techniques rn

engineering. inclucl ing planning, running, artd analyzing the experiments. A di f f 'erent

analysis of Ford's seat belt experiment is included in this text, as well as rnany more

automotive applications of robust design.

280 Chapter 13

Exercises

Grot'e, Daniel lvl., and Tinrothy P. Davis, Eng,ineering, Qtrulit.r.und [i.t1tc't.itttatrtulDesign, Addison Wesley Longman, Edinburgh Gate, UK, l9r)2

$evernl excel lent texts provide cletni led u*plrnat lons of the use of stat ist icul rncrrhods,t lact ional fnctol ial experimentnl pluns, analyt ieal and graphieal lnterpretat ions, 0rrcl

, response surface methods,Box, George E, P., J. Stuart Hunter, and wi l l iam G. Hunter, Stat ist icsforExperimenters' An IntrodLrction to De,sign, Data Anab)sis, qncl Model Bttilding, JohnWiley and Sons; New york, 197g.Box, George E. P., and Norman R. Draper, Empirical Moetet Builcling ancl Re.sponseSwtaces, John Wiley and Sons, New york, 19g7.Montgomery, Douglas C., Design lncl Anabt.sis of Experimenrs, fifth edition, JohnWiley and Sons, New York, 2001.

Rcccnt rcscarch h ls rc t tc *cc l in tc fcs t in one-a t -a - t i rne DOE p lans . i \n a r l lp t i vc onc-factor-at'a-1ime approach has been shown to yield better performarlce optimization thanthe conesponding orthogonal array clesign for systems where the interaction effects aremore significant than the noise and error efl'ecrs.

Frey, Daniel D., Fredrik Engelhardt, and Edward M. Greitzer, ..A Role for one-Factor-at-a-Time Experimentation in Parameter Design," Research in EngineeringDesign,2003.

DoE can be used in rnany aspects of product development. Almqr.rist and wyner explainhow careful ly planned experinrents are effect ive in evaluat ing and tunrng parameters otsales canrpaigns.

Almquist, Eric, and Gordon wyner, "Boost your Marketing Rol with ExperimentalDesign," Hervard Bu-siness Review, y or. 79,No, 9, octob.i zott t, pp. 1 35-1 4 1,

I' Design an experiment to determine a robust process for makilg coffee.2' Explain why the l/4-fractional-factorialand orthogonalanay plan\ shown in Exhibit l3-4

are balanced.

3' Fonnulate an appropriate signal- to-noise rat io for the seat belt expepment. Analyzethe experiniental data using this metr ic. Is signal- to-noise rat io a useful object ive func-t ion in this case? Why or whv not?

Thought Q,uestions

2 .

J .

t . If you are able to afford a larger experiment (with more runs), how might you best uti,I ize the additional runs?when would you choose not to randomize the order of the experiments? How wouldyou guard against bias?Explain the importance of balance in an experimental plan.

ry

Appendix

Robust Design 281

Orthogonal ArraysDOE, texts provide several orthogonal array plans for experitnents. The sitnplest arraysare for two-level and three-levei factor experiments. Using advanced tcchniques. DOEplans can also be created for mixed two-. three-. and/or four-level factor experiments andmany other special situations. This appendix shows some of the basic orthogonal anaysfrom Taguchi's text Introduction to Quality Engineering (1986). These plans are shownin row/column format, with the factor level assignments in the columns and the experi-nrental runs in the rows, The numbers 1" 2. and 3 in each cell indicate the f'actor levels.(Alternat ively, factor levels can be labeled as - and * for t rvo- level factors or *,0, and +

for three levels.) Recall that the orthogonal arrays are named according to the number ofrows in the design. lncluded here are the two-levelanays L4, L8, and Ll6 and the three-l * ,e l a r rays L9 anc l L27 A lso shown is a DOE p la t r us i r tg thc L8 innu l l r ray ' l i r r :ever tcoritrol factors and the L4 outer anay fbr three noise factors. This plan allows analysis ofthe effects of the three noise factors.

Two-Level Ofthogonal Arrays

L4: 3 Factors at 2 Levels Each

A B c1 1 1 1

2 1

3 2 1 a

4 2 2

L8: 7 Factors at 2 Levels Each

A B c D E F G

1 1 I 1 1

2 1 1 1 2 a 2 2

3 1 2 2 1 2 2

4 1 a 2 2 2 1 1

5 2 ,14 2 1 z

6 1 I 1 t 1

7 4. 2 1 z z

8 z J 2 1 2

282 Chaprer 13

L161 16 Fnctorr i l € Levclr Each

A B c D E F G H I J K L M N o1 1 1 1 1 1 1 1 1 12 1 1 1 1 2 2 2 ) 2 a 1 23 1 1 ) 2 2 2 'l

2 ) 2 24 1 2 2 a 2 ) 2 1

Z 1 1 1 15 1 2 2 I ) ) 2 1 ,] 2 26 2 2 2 2 2 2 1 2 2 17 2 a 2 2 1 1 1 ) 2 a a 1I 2 2 2 2 I 2 2 1 1 1 2 29 2 1 2 1 2 ,1 2 1 ) 1 2 1 2 1 2

1 0 ')1 ) 1 2 1 a

a 1 2 1 , 'l 2 11 1 1 2 2 1 2 1 a 1 1 21 2 1 I 1 2 1 2 1 z 11 3 z 2 1 2 2 ,l

2 1 2 2'14z 1 2 z 2 1 2 1 1 a

1 5 2 2 1 2 2 2 ')1 2 1 1 2

1 6 2 2 1 2 1 ) 2 1 1 a ,l 2 2 1

Three-Level Orthogonal Arrays

L9: 4 Factors at 3 Levels Each

A B c D1 1 12 1 a 2 25 3 3 34 2 1 2 3

1 2 3 1

6 2 3 1 27 3 1 ?

8 3 2 1 3I 3 3 2

Robu.st Design 283

L27t 13 Factors at 3 Levels Each

A B c D E F G H I J K L M

1 1 ,] 1 1 1 1 1

2 1 I 1 2 2 2 2 2 2 2 2 2

3 1 1 3 3 3 3 3 3 3 3 3

4 1 2 2 2 1 1 1 2 2 2 3 3 3

5 2 2 2 2 2 2 3 3 3 1, 1 1

6 1 2 2 2 3 3 3 1 1 1 2 2 2

7 1 3 3 3 1 1 1 ) 3 3 2 2 2

I 1 3 3 ) 2 2 1 1 1 3 3 J

9 3 3 3 3 ? 2 2 a

1 0 1 I ? 1 2 1 2

1 1 z a, 1 z z 1

"1,2 a z 3 3 Z 3 1 z 1 I

1 3 2 2 3 ,l 1 2 3 2 3 1 3 1 2

1 4 2 2 3 1 2 3 1 3 2 ,1 2 3

1 5 a a 3 1 3 1 ) 1 2 3 az 3 1

1 6 2 3 1 2 1 2 3 3 1 2 2 3 ,1

1 7 2 3 1 2 ) 3 1 ,1 2 3 1 2

1 8 a 3 1 2 J 1 ) 3 1 2 3

1 9 3 1 3 2 3 2 1 3 2 3 2

20 3 3 2 2 1 3 2 1 3 ) ?

2 1 3 3 a 3 2 1 3 2 3 2

22 3 1 3 3 z I 3

23 3 z 2 1 3

24 ? 2 3 3 I 1 3 2 2 ,lJ

25 3 Ja 1 1 3 z 3 2 1 2 1 3

26 3 ? 1 ') 1 3 1 3 3 2 1

27 3 J 2 1 3 2 2 1 3 1 3 2

284 Chdpter 13

Combined Inner and Outer Arrays

L8 x L4: 7 control Factors and 3 Noise Faqtors at 2 Levels Each

iI

lII

1 1 2 2 Na

1 2 1 2 Nb

A B c D E F G 1 2 ; 2 1 Nc1 1 1 1 ,l

1 1

2 1 1 1 2 2 a

3 1 z z 1 z

4 1 z z I z 1 15 I a 1 2 1 z

6 2 z 2 1 2 17 z 1 1 z 2 1

I Z 1 1 1