european journal of operational research volume 100 issue 2 1997 [doi...

8/13/2019 European Journal of Operational Research Volume 100 Issue 2 1997 [Doi 10.1016%2Fs0377-2217%2896%290029

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SiwE L S E V I E R European Journal of Operat ional Research 100 (1997) 277-292

EUROPEANJOURNALOF OPERATIONALRESEARCH

m etho do l ogy for the reducti on o f i mp rec i s ion i n theeng i neer i ng process

R o n a l d E . G i a c h e t t i a , , R o b e r t E . Y o u n g a A x e l R o g g a t z b , W a l t e r E v e r s h e i m bG i o v a n n i P e r r o n e c

a W ork gro up for Intelligent Systems in Design and Manufacturing, Dep artment o f Industrial Engineering, No rth Carolina State University,Raleigh, NC 27695-7906, USAb Laboratory for Machine Tools and Production Engineering, Aach en University of Technology, 52056 Aachen, Germany

c Department o f Technology and Production, University of Palermo, Palermo 98128, I taly

A b s t r a c tEng ineering desig n is characterized by a high level of imp recis ion, vag ue parameters, and i l l-defined relat ionships. Indesign, impre cis ion reduct ion m ust occur to arr ive at a f inal prod uct specification. Few d esign system s exis t for adequatelyrepresenting design im precis ion, and form ally reduc ing i t to precise values . F uzz y set theory has considerab le potent ial foraddress ing the imprec is ion in des ign. H ow ever, i t lacks a form al meth odo logy for sys tem deve lopm ent and operation. Onerepercuss ion of this is that imprecis ion reduct ion is , a t present , implemented in a relat ively ad-hoc manner. The main

cont r ibut ion of th i s paper i s to in t roduce a methodology ca l l ed p r e c i s i o n c o n v e r g e n c e for mak ing the transit ion fromimprecise goals and requirements to the precise specif icat ions need ed to manu facture the product . A hierarchy o f fuzz yconstraint netwo rks is presented along w ith a m etho dolo gy for creat ing t ransit ional l inks betw een different levels of thehierarchy. T he solut ion me thod olog y is i llus trated with an examp le within w hich an imprecis ion reduct ion of 98% isachiev ed in only three s tages o f the design process . Th e imp recis ion reduct ion is me asured us ing the c oe f f i c i e n t o fi m pr e c i s i on , a new m easure introduced to quan t i fy imprecision. 1997 Elsevier Science B.V.Keywords: Engineering design; D esign; Decision making; C onstraint processing; Fuzz y set theory; H ierarchical modeling; Concurrentengineering; Precision converg ence; Coefficient of imprecision

1 . I n t r o d u c t i o n

T h e d e s i g n p r o c e s s e n c o m p a s s e s a l l o f th e a c t iv i -t i e s w h i c h a r e p e r f o r m e d t o a r r i v e a t a f i n a l p r o d u c ts p e c i f i c a t io n . I t is d u r i n g t h e d e s i g n p r o c e s s w h e n a ne s t im a t e d 6 0 - 8 5 % o f th e p r o d u c t ' s c o s t i s d e t er -m i n e d ( Z a n g w i l l , 1 9 9 2 ; E v e r s h e i m e t a l ., 1 9 9 5 ) . T h i sha s l e d t o i nc r e a s e d i n t e r e s t i n t he a c t i v i t i e s w h i c hc o m p r i s e t h e d e s i g n p r o c e s s. R a p i d p r o d u c t d e v e l o p -

* Corresponding author.

m e n t c y c l e s r e q u i r e d e s i g n m e t h o d o l o g i e s t o e f f i -c i e n t l y e x p l o r e t h e d e s i g n s p a c e t o b u i l d p r o d u c t sw i t h r e d u c e d c o s t, i m p r o v e d f u n c t i o n a l i ty , a n d i m -p r o v e d q u a l i t y t o c o m p e t e i n g l o b a l m a r k e t s .

O n e o f t h e p h a s e s o f d e s i g n w h i c h i s l ea s t s u p -p o r t e d i s t h e t r a n s i t i o n f r o m e a r l y d e s i g n p h a s e s t ot he f i na l de s i gn s t a ge s ( A be l n , 1990 ) . T h i s i s t het r a n s i ti o n f r o m v a g u e a n d i m p r e c i s e s p e c i f i c a t io n s t op r e c i s e a n d e x a c t v a l u e s a n d i s a m a j o r a c t i v i t y o ft h e d e s i g n p r o c e s s . M a n y s y s t e m s o n l y a t t e m p t t op r o v i d e d e s i g n s u p p o r t i n th e d o m a i n o f w e ll d e f i n e dv a r i a b l e s a n d s p e c i f i c a t i o n s i n w h i c h a l l v a l u e s u s e d

0377-2217 /97/$17.00 1997 Elsevier Science B.V. All rights reserved.PI I S 0 3 7 7 - 2 2 1 7 ( 9 6 ) 0 0 2 9 0 - 1



7 8 R.E. Giachetti et al./ European Journal of Operational Research 100 (1997) 277-292d u r i n g d e s i g n m u s t b e k n o w n w i t h c e r t a i n t y . T h i sre s t r i c t i on t o c e r t a i n t y l i mi t s t he u t i l i t y o f t he sesys t e ms t o t he l a t e r s t a ge s o f t he de s i gn p roc e s s .A l t h o u g h r o u t i n e ly p e r f o r m e d , a l a c k o f a f o r m a lme c ha n i sm fo r qua n t i t a t i ve l y r e p re se n t i ng t he t r a ns i -t i o n h a s h i n d e r e d t h e d e v e l o p m e n t o f t o o l s t o s u p p o r tt he e a r l y pha se s o f de s i gn .

A p r o m i s i n g t e c h n o l o g y f o r r e p r e s e n ti n g t h e i m -p re c i s i on i n de s i gn i s fuz z y s e t t he o ry . A fe w re -se a rc he rs ha ve be gun t o e xp l o re t h i s pos s i b i l i t y .Z i m m e r m a n n a n d S e b a s t i a n ( 1 9 9 4 ) b e l i e v e f u z z yre p re se n t a t i on o f i mpre c i s i on w i l l p l a y a n i nc re a s -i ng l y i mpor t a n t ro l e i n de s i gn sys t e ms . S t a nda rdm a t h e m a t i c a l f u n c t i o n s u s e d i n d e s i g n c a n b e a p p l i e dt o i m p r e c i s e q u a n t i t i e s t h r o u g h u s e o f t h e e x t e n s i o npr i nc i p l e (Za de h , 1965) . The e x t e ns i on p r i nc i p l e de -t e r m i n e s t h e p o s s i b i l i t y f u n c t i o n o f f ( Q I , Q 2 ) t oe n c o m p a s s a ll p o s s ib l e v a l u e s t h a t c o u l d r e s u l t f r o mc om bi n i ng t he ra nge s o f t he a rgum e n t s Q~ a nd Q2(D ubo i s a nd P ra de , 1993) . The sp re a ds , w h i c h a rere p re se n t a t i ve o f t he i mpre c i s i on , e x p a n d i n f u z z yc a l c u l a t i ons . O pe ra t i ons on i mpre c i s e qua n t i t i e s w i l lr e su l t i n e qua l l y i mpre c i s e o r more i mpre c i s e r e su l t s .T h e i m p r e c i s i o n o f t h e r e s u l t s c a n n o t d e c r e a s e . T h ep r o b l e m w i t h s u p p o r t i n g t h e e n t i r e d e s i g n p r o c e s sus i ng fuz z y s e t t he o ry re p re se n t a t i on i s t ha t t heo p e r a t o r s u s e d t o c o m p u t e f u z z y n u m b e r s i n c r e a s et he i mpre c i s i on w h i l e a ne c e s sa ry c ond i t i on t o de -s i gn a n a r t i f a c t w h i c h i s ma nufa c t u ra b l e i s t o d e -c r e a s e t h e i m p r e c i s i o n . A l t h o u g h s o m e w o r k h a sb e e n d o n e o n a n e x t e n s i o n p r i n c i p l e d e f i n e d f o r' o p t i m i s t i c ' f u n c t i o n s , w h e r e t h e i m p r e c i s i o n d o e sno t ne c e s sa r i l y i nc re a se du r i ng c a l c u l a t i ons , i t a p -pe a r s t o ha ve l i mi t e d p ra c t i c a l a pp l i c a b i l i t y (D ubo i sa nd P ra de , 1988) .

T h i s p r o b l e m o f i n c r e a se d i m p r e c i s io n f r o m o p e r -a t i ons on fuz z y s e t s c a n be i l l u s t r a t e d us i ng i n t e rva la r i t h m e t i c . T h e r e l a t i o n s h i p b e t w e e n f u z z y s e t s a n di n t e r v a l s i s w e l l d o c u m e n t e d ( D u b o i s a n d P r a d e ,1988 ; K l i r a nd Y ua n , 1995) . O pe ra t i ons on fuz z yse t s a re e qu i va l e n t t o pe r fo rmi ng i n t e rva l a r i t hme t i ca t e a c h a - l e ve l . F uz z y mul t i p l i c a t i on i s r e l a t e d t oi n t e rva l mu l t i p l i c a t i on , w he re i n t e rva l mu l t i p l i c a t i oni s d e f i n e d as ( M o o r e , 1 9 6 6 )[ a , b ] * [ c , d ] = [ m i n ( a c , a d , b c, b d ) ,

m a x ( a c , a d , b c , b d ) ] .

T h e p r o d u c t w i ll a l w a y s h a v e g r e a t e r im p r e c i s i o nma gn i t ude o f i n t e rva l (K i m e t a l. , 1995) - t ha n the

i npu t va l ue s . Th i s i s r e fe r re d t o a s a ' pe s s i mi s t i c 'c a l c u la t io n b y D u b o i s a n d P r a d e ( 1 9 8 8 ) ( p .4 3 ) . F o r as p e c i fi c e x a m p l e o f t h is p r o b l e m , l e t u s c o m p u t e t h ev o l u m e o f a c u b e w i t h i m p r e c i s e l y d e f i n e d e d g e s .E a c h e d g e i s d e f i n e d b y t h e t r ia n g u l a r f u z z y n u m b e r( 2 , 3, 4 ) a n d t h e v o l u m e i s th e f u z z y p r o d u c t( 2 , 3 , 4 ) ( 2 , 3 , 4 ) ( 2 , 3 , 4 ) = ( 8 , 2 7 , 6 4 ) .I n t u i ti v e l y , m a n y e n g i n e e r s c a n a c c e p t t h e c o n c e p t o fa d im ens ion wi th a to le ran ce 3 ___ 1 ( i .e . a t r i ang ula rf u z z y n u m b e r ( 2 , 3, 4 ) ) y e t f i n d i t u n a c c e p t a b le t h a tt he t o l e ra nc e fo r t he v o l um e i s 27+ ~ 7 ( i. e . a t r i a ngu-l ar f u z z y n u m b e r ( 8 , 2 7 , 6 4 ) ) . W e c a n s ee t ha t t here su l t i s muc h more i mpre c i s e t ha n t he i npu t va l ue sa nd t ha t e x t e nde d a r i t hme t i c c a n p roduc e suc h al a rge sp re a d o f pos s i b l e va l ue s t ha t t he r e su l t s be -c o m e m e a n i n g l e s s . C o n s e q u e n t l y , i n c o r p o r a t i n g are p re se n t a t i on fo r i mpre c i s i on i n t o e x i s t i ng de s i gnsys t e ms a nd u t i l i z i ng e x i s t i ng de s i gn me t hodo l og i e si s i na de qua t e fo r ope ra t i ng w i t h i m pre c i s e qua n t i ti e s ,a n d i t b e c o m e s i m p e r a t i v e t o d e v e l o p m e t h o d o l o g i e sf o r m a n i p u l a t in g f u z z y n u m b e r s t o o b t a in r e a s o n a b l eresul t s .

In t h i s pa pe r w e p re se n t a so l u t i on t o t h is p rob l e mu s i n g a n e w a p p r o a c h , t e r m e d p r e c i s i o n c o n v e r -g e n c e , t ha t t a ke s a dva n t a ge o f i nhe re n t p re c i s i ond i v e r g e n c e r e s u l t i n g f r o m f u z z y o p e r a t i o n s t o p r o -d u c e p r e c i s i o n c o n v e r g e n c e a s w e m o v e f r o m c o n -c e p t ua l de s i gn t o de t a i l e d de s i gn . To qua n t i fy t hei mp re c i s i on re duc t i on , a ne w m e a su re i s al so in t ro -duc e d - t he c o e f f i c i e n t o f i m p r e c i s i o n . T h e n e wa pproa c h w i l t be d i s c us se d a nd t he n de sc r i be dt h rough a n e xa m pl e - t he de s i gn o f a c e l l u l a r t e l e-phone . H ow e ve r , p r i o r t o t h i s d i s c us s i on , b r i e f d i s -c us s i ons i n c u r re n t de s i gn t he o ry , fuz z y s e t t he o ry ,a nd c ons t ra i n t ne t w orks w i l l be p re se n t e d .

2 . E n g i n e e r i n g d e s i g n

T h e r e a r e m a n y d e f i n i t i o n s o f d e s i g n w h i c h c a nbe found i n t he l i t e ra t u re . S uh (1990) de f i ne s de s i gna s t he c o n t i n u o u s i n t e r p l a y b e t w e e n w h a t w e w a n tt o a c h i e v e a n d h o w w e w a n t t o a c h i e v e i t . D e s i g ni s a ma pp i ng f rom t he func t i ona l doma i n i n t o t hephys i c a l r e a l m (Tomi ya ma , 1990) , i t i s a de c i s i on -



R.E. Giacheni et al ./ European Journal of Operational Research 100 (1997) 277-292 279ma ki ng p roc e s s (Bra d l e y a nd A gog i no , 1993) , i t i s ase a rc h p roc e s s (G e ro , 1990) , a nd i t i s t he fo rmu l a t i ona n d s a t is f a c ti o n o f c o n s t r a in t s ( S e r r a n o a n d G o d d a r d ,1987) . D e s i gn e xh i b i t s a l l o f t he se t r a i t s a nd c on-f o r m s t o t h e s e d e f i n i t i o n s t o v a r y i n g d e g r e e s . T h er e s e a r ch i n e n g i n e e r i n g d e s i g n i s f o c u s e d o n f i n d i n gsu i t a b l e t he o r i e s o f de s i gn , hope fu l l y l e a d i ng t ome t hods t o a r r i ve a t be t t e r de s i gns (D i xon , 1988) .

I n s t u d y i n g d e s i g n , r e s e a r c h e r s h a v e f o u n d i t c o n -ve n i e n t t o c l a s s i fy the d i f fe re n t t ype s o f de s i gn w h i c ha r e d e s c r ib e d a s a m e t h o d o f d e s i g n ( V D I R i c h t li n i e,1993) . The re a re s e ve ra l de f i n i t i ons o f de s i gn t ype s(Ba u ma nn , 1982 ; Ehr l e nsp i e l , 1985; H i n t z e n e t a l .,1989 ; Eve rshe i m, 1990 ; P a h l a nd Be i t z , 1993) . Inge ne ra l , fou r t ype s o f de s i gn c a n be d i f fe re n t i a t e d ,o r i g i na l de s i gn , a da p t i ve de s i gn , va r i a n t de s i gn , a ndde s i gn by f i xe d p r i nc i p l e s . A n a na l ys i s by H i n t z e n e ta l . (1989) show s t ha t a bou t 5% o f de s i gn a c t i v i t i e sa re o r i g i na l de s i gn , a bou t 55% a re a da p t i ve de s i gn ,a bou t 20% a re va r i a n t de s i gn , a nd t he o t he r 20% i sde s i gn by f i xe d p r i nc i p l e s .

R e g a r d l e s s o f t h e t y p e o f d e s i g n p r o b l e m e n c o u n -t e re d a nd how t he y a re c l a s s i f i e d , i t i s ge ne ra l l ya c k n o w l e d g e d t h a t t h e d e s i g n p r o c e s s c o n s i s t s o fs t a g e s o f p r o g r e s s i v e l y f i n e r d e t a i l e d d e s i g n s(Rode na c ke r , 1991 ; Ko l l e r , 1985 ; P a h l a nd Be i t z ,1993; Roth , 1994; Se i fe r t , 1989; Ehrlenspie l , 1983;H u b k a , 1 9 8 4 ) . F i n g e r a n d D i x o n ( 1 9 8 9 ) p r o v i d e ag o o d r e v i e w o f t h e m a n y a n d v a r i e d p e r s p e c ti v e s o nr e p r e s e n t i n g t h e e n g i n e e r i n g d e s i g n p r o c e s s . H o w -e ve r , t he y no t e t ha t t he i r r e v i e w i s mi s s i ng t he l a rgeb o d y o f r e s e a r c h p u b l i s h e d i n G e r m a n . W e c o n c u ra nd fo r ou r pu rpose s use t he c l a s s i f i c a t i on o f P a h la nd Be i t z (1988) , one o f t he f e w w orks a va i l a b l e i nt ra ns la t i on . Th e y l i st fou r de s i gn s t age s : c l a r i f i c a t i ono f t a s k , c o n c e p t u a l d e s i g n , e m b o d i m e n t d e s i g n , a n dde t a i l de s i gn . Le t u s de sc r i be t he fou r de s i gn s t a ge si n more de t a i l . Clarif ication of task i s a p rob l e mformul a t i on a c t i v i t y w he re t he func t i ona l r e qu i re -me n t s a re spe c i f i e d . Conceptual design i s syn t he s i so f a n a bs t ra c t s t ruc t u re t ha t c a n be a so l u t i on t o t hed e s i g n p r o b l e m . Embodiment design i s t he de ve l op -me n t o f a n a bs t ra c t c onc e p t i n t o a p re l i mi na ry s c a l e de n g i n e e r i n g d r a w i n g . Detailed design i nvo l ve s t hespe c i f i c a t i on o f a t t r i bu t e va l ue s t o t he de s i gn pa ra m-e t e r s . S u f f i c i e n t de ta i l mus t be a d de d t o e a c h s t a gefo r e va l ua t i on a nd a na l ys i s . C l e a r l y , de s i gn i s a ni t e ra t i ve p roc e s s t ha t move s a n i mpre c i s e c onc e p t t o

a p re c i s e de f i n i t i on o f t he p roduc t a nd how i t i s t o bem a n u f a c t u r e d . D e s p i t e t h e d i f f e r en t b r e a k d o w n a n dt e rms use d , t he re a re s i mi l a r i t i e s i n a l l de sc r i p t i onso f t he de s i gn s t a ge s a nd t he de s i gn p roc e s s i n ge n -e ra l .2.1. Imprecision in design

Impre c i s i on i s mos t no t a b l e i n t he e a r l y pha se s o ft he de s i gn p roc e s s a nd ha s be e n de f i ne d a s t hec ho i c e be t w e e n a l t e rna t i ve s (A n t onsson a nd O t t o ,1995) . A t t e mpt s ha ve be e n ma de t o a ddre s s t he i s sueo f i m p r e c i s i o n a n d i n c o n s i s t e n c y i n d e s i g n e m p l o y -i ng i n t e rva l s (D a v i s , 1987 ; K i m e t a l . , 1995 ; Na v i n -c ha nd ra a nd R i nde r l e , 1990). H o w e ve r , t he se w e ref o u n d u n s a t i s f a c t o r y f o r t h e g e n e r a l s i m u l t a n e o u se n g i n e e r i n g p r o b l e m . O t h e r a p p r o a c h e s t o r e p r e s e n t -i ng i mpre c i s i on i n de s i gn i nc l ude us i ng u t i l i t y t he -o ry , i mp l i c i t r e p re se n t a t i ons us i ng op t i mi z a t i onm e t h o d s , m a t r i x m e t h o d s s u c h a s Q u a l i t y F u n c t i o nD e p l o y m e n t , p r o b a b i l i ty m e t h o d s , a n d n e c e s s i t ym e t h o d s . T h e s e m e t h o d s h a v e a l l h a d l i m i t e d s u c c e s si n so l v i ng de s i gn p rob l e ms w i t h i mpre c i s i on . A n t on-s s o n a n d O t t o ( 1 9 9 5 ) p r o v i d e a n e x t e n s i v e r e v i e w o ft he se a pproa c he s a nd t he re a de r i s d i r e c t e d t o t he i rpa pe r fo r a more de t a i l e d d i s c us s i on .

R e u s c h ( 1 9 9 3 ) e x a m i n e d t h e g e n e r a l p r o b l e m o fi mpre c i s i on a nd i nc ons i s t e nc y i n de s i gn , a nd c on-c l ude d t ha t t he p rob l e ms a re w e l l su i t e d t o be so l ve du s i n g f u z z y t e c h n o l o g y . T h e r e a r e t w o a s p e c t s o fi m p r e c i s i o n w h e n m o d e l e d w i t h f u z z y s e ts - a p r e f-e re nc e v i e w a nd a p l a us i b i l i t y v i e w . Impre c i s i on c a nbe de f i ne d a s t he p re fe re nc e a de s i gne r ha s fo r apa r t i c u l a r va l ue bu t w i l l a c c e p t o t he r va l ue s t o al e s se r de g re e . Th i s i n t e rp re t a t i on o f u s i ng fuz z y s e t st o m o d e l p r e f e r e n c e w a s p u t f o r w a r d b y D u b o i s( 1 9 8 7 ) a n d d e m o n s t r a te d i n t h e d o m a i n o f m e c h a n i -c a l e n g i n e e r i n g b y W o o d a n d A n t o n s s o n ( 1 9 8 9 ) .Impre c i s i on c a n a l so be t he p l a us i b i l i t y o f a va l ueunde r a g i ve n pos s i b i l i t y d i s t r i bu t i on . The re e x i s t si m p o r t a n t c o n c e p t u a l d i f f e r e n c e s b e t w e e n t h e t w ob a s e d o n w h e t h e r t h e p a r a m e t e r i s c o n t r o l la b l e o r n o t(Dubois e t a l . , 1995) .

Y oung e t a l . (1995) c l a s s i fy t he d i f fe re n t sou rc e so f i mpre c i s i on found i n e ng i ne e r i ng de s i gn a s : r e l a -t i onsh i p i mpre c i s i on , da t a i mpre c i s i on , l i ngu i s t i c i m-p re c i s i on , a nd i nc ons i s t e nc y i mpre c i s i on . Relation-ship imprecision i s t he a mbi gu i t y t ha t e x i s t s be t w e e n



280 R.E. Giachetti et al. / European Journal of Operational Research 100 (1997) 277- 292

t h e d e s i g n p a r a m e t e r s . Data impr ec i s ion i s w h e n ap a r a m e t e r ' s v a l u e i s n o t e x p l i c i t l y k n o w n . Lingu is t i cimpr ec i s ion a r i s e s f r o m t h e q u a l i t a t i v e d e s c r i p t i o n so f g o a ls , c o n s t r a in t s , a n d p r e f e r e n c e s m a d e b y h u -m a n s . Incons i s t ency impr ec i s ion a r i s e s f r om t he i n -h e r e n t c o n f l i c t i n g o b j e c t i v e s a m o n g v a r i o u s a r e a s i na p r o d u c t ' s l i f e - c y c l e . R e g a r d l e s s o f t h e d i f f e re n ts o u r c e s o f i m p r e c i s i o n w e b e l i e v e i t c a n b e m o d e l e du s i n g f u z z y s e t t h e o r y .

3 F u z z y se t t h e o r yF u z z y s e t t h e o r y i s a g e n e r a l i z a t i o n o f c l a s si c a l

s e t t h e o r y . I n n o r m a l s e t t h e o r y a n o b j e c t i s e i t h e r am e m b e r o f a s et o r n o t a m e m b e r o f th e s e t. T h e r ea r e on l y t w o s t a t e s . T h i s i s r e f e r r e d t o a s a cr isp set .F u z z y s e t s c o n t a i n e l e m e n t s t o a c e r t a i n d e g r e e .T h u s , i t i s p o s s i b l e t o r e p r e s e n t a n o b j e c t w h i c h h a sp a r t i a l m e m b e r s h i p i n a s e t . T h e m e m b e r s h i p v a l u eo f e l e m e n t x i n a f u z z y s e t i s r e p r e s e n t e d b y / x ( x )a n d i s n o r m a l i z e d s u c h t h a t t h e m e m b e r s h i p d e g r e eo f e l e m e n t x i s i n [ 0 ,1 ] . A l l e l e m e n t s x s uc h t ha t/ x ( x ) >__ a de f i n e a n ' a - l e ve l ' w h i c h i s a bo un de di n t e r v a l . O p e r a t i o n s o n f u z z y s e t s a r e p e r f o r m e da c r os s t he s e a - l e v e l s , e i t he r by d i s c r e t i z i ng / x i n t o as e t o f a - l e v e l s w i t h a ~ [0 , 1 ] o r by t r e a t i ng / z a s ac o n t i n u u m , a n d t h e n a p p l y i n g t h e e x t e n s i o n p r i n c i -p l e . T h i s f o r m s t h e b a s i s o f f u z z y s e t t h e o r y ( Z a d e h ,1965) . S i nc e t he bou nda r i e s o f i nc l u s i on i n a s e t a r e

f u z z y a n d n o t d e f i n i t e , w e a r e a b l e t o d i r e c t l y r e p r e -s e n t a m b i g u i t y o r i m p r e c i s i o n i n o u r m o d e l s .

F u z z y s e t s c a n r e p r e s e n t l i n g u i s t i c t e r m s a n d i m -p r e c i s e q u a n t i t i e s . L i n g u i s t i c t e r m s a r e u s e d t o m o d e lt h e i m p r e c i s i o n i n n a t u r a l l a n g u a g e s t a t e m e n t s s u c ha s ' t a l l ' o r ' i n e x p e n s i v e ' . A f u z z y q u a n t i t y i s a s e td e f i n e d o n ~ , t h e s e t o f r ea l n u m b e r s . I t r e p r e s e n t si n f o r m a t i o n s u c h a s , ' a b o u t 5 i n c h e s ' . T h u s , 4 . 9i n c h e s w o u l d b e a m e m b e r o f th i s s et . 4. 5 i n c h e sm a y a l s o b e a m e m b e r o f th e s e t b u t to a l e s se rd e g r e e . F u z z y n u m b e r s h a v e a m e m b e r s h i p f u n c t i o nt h a t i s n o r m a l , p i e c e w i s e c o n t i n u o u s , a n d c o n v e x .T h e v a l u e w i t h m e m b e r s h i p o f 1 i s c a l l e d t h e m o d a lv a l u e . I f a l l t h re e o f t h e se p r o p e r t i e s h o l d f o r t h em e m b e r s h i p f u n c t i o n , t h e n t h i s i s a n L R f u z z y n u m -b e r ( D u b o i s a n d P r a d e , 1 9 8 8 ; Z i m m e r m a n n , 1 9 8 5 ).S t a n d a r d m a t h e m a t i c a l o p e r a t o r s c a n b e u s e d w i t hf u z z y n u m b e r s t h r o u g h a p p l i c a t io n o f th e e x t e n s i o n

p r i n c i p le , a n d s p e c i f i c i m p l e m e n t a t i o n s o f t h e s e o p -e r a t i o n s a r e f u z z y o p e r a t o r s ( D u b o i s a n d P r a d e , 1 9 8 8;Z a de h , 1965 ) .

4 C o n s t r a i n t s a t is f a c t io nT h e r e p r e s e n t a t i o n o f a c o n s t ra i n t s a t i s f a c t io n

p r o b l e m , d e f i n e d a s a c o n s t r a i n t n e t w o r k p r o b l e m ,c a n b e d e f i n e d a s f o l l o w s ( t h i s i s a d a p t e d f r o mD e c h t e r a nd P e a r l , 1988 ) :F u z z y C o n s t r a in t N e t w o r k P r o b l e m : A f u z z y c o n -s t ra i n t n e t w o r k p r o b l e m c o n s i s t s o f a s e t o f n f u z z yva r i a b l e s , X = { X l , X 2 . . . . . X n}, a nd a s e t o f m c on -s t ra in t s , C = {C l , C z . . . . . C m } . A f u z z y v a r i a b l e X iha s i t s dom a i n , - Qi, w h i c h d e f i ne s t he s e t o f va l ue st ha t t he va r i a b l e c a n ha ve . A c ons t r a i n t C ~ i s a k-aryrelat ion o n

i l ' i 2 , ' ' ' ' ik --i .e . , C i ( X i l , X i z . . . . . ) ~i k) , a nd i s a s ubs e t o f t heC a r t e s i a n p r od uc t g2 /t O i 2 - . . O , . k .

I n t h i s f o r mu l a t i on e a c h c ons t r a i n t i s s a t i s f i e d t o adeg ree , /Xc , ~ [0 , 1], depending on the ins tant ia t ionof the var iables . T h i s i s t h e m e m b e r s h i p v a l u e o f th ec o n s t r ai n t . A s o l u t io n o f t h e n e t w o r k i s d e f i n e d a s a na s s i g n m e n t o f v a l u e s t o a ll t h e v a r i a b l e s s u c h t h a t th ec ons t r a i n t s a r e s a t i s f i e d . T he c ons t r a i n t s a r e s a t i s f i e dw h e n / X c > a s , w h e r e a s i s th e sys tem truththreshold . I t i s a l e ve l o f s a t i s f a c t i on a s o l u t i on mus tf u l f i l w i t h i n t h e e n t i r e n e t w o r k t o b e a c c e p t e d b y t h ede s i gne r . T h i s va l ue i s s e t a p r i o r i by t he u s e r( Y ou ng e t a l. , 1996 ) .

T h e f u z z y c o n s t r a i n t p r o c e s s i n g s y s t e m u s e d t oi m p l e m e n t t h e p r e c i s i o n c o n v e r g e n c e m e t h o d d e -s c r i b e d i n t h i s p a p e r i s F u z C o n . I n Y o u n g e t a l .( 1 9 9 6 ) t h e s y s t e m a n d i t s o p e r a t o r s e t i s d e s c r i b e da l o n g w i th a b r i e f r e v i e w o f p r io r f u z z y c o n s t r a i n tp r o c e s s i n g w o r k . F u z C o n i s t h e l a t e s t g e n e r a t i o n i n as e r ie s o f c o n s tr a i n t p r o c e s s i n g s y s t e m s w h i c h i n c lu d eS P A R K ( Y o u n g e t al ., 1 9 9 2 ), S A T U R N ( F o h n e t al .,1 9 9 3 ) , a n d J U P I T E R ( L i a u e t a l . , 1 9 9 5 ) . T h e a p -p r o a c h t a k e n i n t h e s e s y s t e m s i s t o v i e w e a c h c o n -s t r a i n t a s a l og i c s e n t e nc e w i t h a n a s s oc i a t e d t r u t hv a l u e . ( S e e G r e e f e t a l. ( 1 9 9 5 ) f o r a n e l a b o r a t i o n o ft h e e q u i v a l e n c y o f a lo g i c - b a s e d s y s t e m a n d t h e



R.E. Giachetti et al. / European Journal o f Operational Research 100 (1997) 277-292 281c o n s t r a i n t s a t i s f a c t i o n n e t w o r k p r o b l e m d e f i n e da b o v e . ) T h e s e a r e i n t e r a c t i v e c o n s t r a i n t p r o c e s s i n gs y s t e m s w h i c h a i d t h e u s e r b y p r o p a g a t i n g u s e ra s s i g n m e n t s t h r o u g h t h e n e t w o r k i n a n a t t e m p t t osa t i s fy a l l the cons t ra in t s us ing di rec t ca lcula t ions .Th i s a pp roa c h suppor t s a f e a t u re - r i c h re p re se n t a t i ons c h e m a a n d s u p p o r t s o m n i - d i r e c t i o n a l c o n s t r a i n tp ropa ga t i on . F i nd i ng a so l u t i on t ha t s a t i s f i e s t hec ons t ra i n t s i s suppor t e d by a sub -sys t e m t ha t i de n t i -f i e s t he sou rc e o f a c ons t ra i n t v i o l a t i on a nd a s s is t s i nc o r re c t i ng t he p rob l e m. Th i s t ype o f sys t e m i s c ruc i a lt o s u c c e s s f u l ly m o d e l i n g d e s i g n b e c a u s e :

1 . Re pre se n t i ng de s i gn i n fo rm a t i on re qu i re s a w i deva r i e t y o f da t a t ype s t ha t c ons t ra i n t he de s i gn t h roughc o m p l e x r e l a t io n s h i p s b e y o n d t h e e q u a l i t y r e l at io n .

2 . D e s i gn i s a n i t e ra t i ve p roc e s s t ha t d ra w s upont h e k n o w l e d g e o f t h e u s e r r e q u i r in g t h a t p r o p a g a t i o nno t be r e s t r i c t e d t o a s i ng l e d i re c t i on no r a p re sc r i be ds ta r t ing point .

5 R e l a t e d w o r k

O t h e r r e s e a r c h e r s h a v e a l s o b e g u n t o e x p l o r e t h eu s e o f f u z z y s e t t h e o r y i n d e s i g n . M a n y o f t h e s es y s t e m s a r e f o r c o n d u c t i n g d e s i g n e v a l u a t i o n( K n o s a l a a n d P e d r y c z , 1 9 8 7 ; M i i l l e r a n d T h M i g e n ,1994) . F u z z y se t t he o ry ha s a l so be e n a pp l i e d t os p e c i fi c d e si g n p r o b l e m s s u c h a s d e s i g n f o r a s s e m b l yJa c kson e t al ., 1993 . H ow e ve r , non e o f the se sys t e msp r o f e s s t o s u p p o r t t h e e n t ir e d e s i g n p r o c e s s . T h e y a r ea ll t a rge t e d t o a spe c i f i c t ype o f de s i gn p rob l e m o r t oa know n a nd spe c i f i c pha se o f t he de s i gn p roc e s s .

F u z z y c o n s t r a i n t p r o c e s s i n g h a s b e e n s t u d i e d b yK i m a n d C h o ( 1 9 9 4 ) w h o e x t e n d e d t h e d e f i n i t i o n sf o r n u m e r i c a n d i n t er v a l b a s e d c o n s t r a i n t n e t w o r k s t of u z z y c o n s t r a i n t n e t w o r k s . R u t t k a y ( 1 9 9 4 ) d i s c u s s e sd i f fe re n t me t hods fo r de t e rmi n i ng t he j o i n t s a t i s fa c -t i on o f c ons t ra i n t s . F a rg i e r (199 4 ) a nd D ubo i s e t a l .( 1 9 9 5 ) h a v e u s e d f u z z y c o n s t r a i n t n e tw o r k s t o m o d e lf l e x i b l e t e mpora l c ons t ra i n t s i n j ob -shop sc he du l i ng .B o w e n e t a l . ( 1 9 9 2 ) d i s c u s s t h e l i n k b e t w e e n f u z z yc ons t ra i n t s a t i s fa c t i on a nd se ma n t i c s i n c l a s s i c a ll o g i c . O f t h e s e m e t h o d s t h e c o n s t r a i n t m o d e l o fD ubo i s e t a l . (1995) ha s t he g re a t e s t s i mi l a r i t i e s t ot he mode l u se d he re .

R e c e n t w o r k b y C h e n a n d O t t o ( 1 9 9 5 ) u s e d f u z z yse t t he o ry t o so l ve a va r i a t i ona l sys t e m o f c ons t ra i n t s

f o r C A D . H o w e v e r , t h e a p p r o a c h i s l i m i t e d t o n u -me r i c va r i a b l e s a nd e qua l i t y c ons t ra i n t s a nd c a n on l yb e s o l v e d w h e n t h e n u m b e r o f v a r i a b l e s e q u a l s t h en u m b e r o f c o n s t r a i n t s . W o o d a n d A n t o n s s o n ( 1 9 8 9 )d e v e l o p e d t h e Method of Imprecision ( M o I ) w h i c hu s e s f u z z y s e t t h e o r y t o m o d e l t h e p r e f e r e n c e s o f ade s i gne r . In s t e a d o f d i re c t c a l c u l a t i ons t he MoI t e c h -n i q u e a l lo w s a d e s i g n e r t o r a n k t h e c o u p l in g b e t w e e nd e s i g n p a r a m e t e r s a n d p e r f o r m a n c e p a r a m e t e r s .T h o s e w i t h t h e s t r o n g e s t c o u p l in g b e c o m e c a n d i d a te sfo r mod i f i c a t i on i n a n a t t e mpt t o r e duc e t he i mpre c i -s ion of the output . In th i s sense , i t i s re la ted toT a g u c h i ' s m e t h o d ( O t t o a n d A n t o n s s o n , 1 9 9 3 a ) a n dt o u t i l i t y t he o ry (O t t o a nd A n t onsson , 1993b) .

T h e c o m p u t a t i o n s i n t h e s e t w o a p p r o a c h e s a r epe r fo rme d us i ng a d i s c re t i z e d so l u t i on a l go r i t hm de -v e l o p ed b y D o n g a n d W o n g ( 1 9 8 7 ) a n d m o d i f ie d b yW ood e t a l . (1992) . The a l go r i t hm ha s c omput a t i ona lc o m p l e x i t y o f t h e o r d e r O ( M . 2 N - i k ) , w he re N i st h e n u m b e r o f i m p r e c i s e p a r a m e t e r s , M i s t h e n u m -b e r o f a - l e v e l s i n t o w h i c h t h e m e m b e r s h i p f u n c t i o ni s d i v i de d , a nd k i s t he num be r o f mu l t i p l i c a t ionsa nd d i v i s i ons i n t he func t i on , f(d ), conta in ing a l l thec a l c u l a t i ons . The a l go r i t hm w a s i n i t i a l l y l i m i t e d t ope r fo rmi ng c a l c u l a t i ons on a s i ng l e e xpre s s i on , bu th a s b e e n e x t e n d e d t o p e r f o r m c a l c u l a t i o n s f o r asys t e m o f e qua t i ons (Ch e n a nd O t t o , 1995). C l e a r l y ,i n a dd i t i on t o r e p re se n t a t i ona l r e s t r i c t i ons , t he se a p -p r o a c h e s a r e c o m p u t a t i o n a l l y b o u n d e d w h e n a p p l i e dt o e ve n sma l l - s i z e d p rob l e ms a nd i nd i c a t e a ne wa p p r o a c h i s n e e d e d t o s u p p o r t m o d e l i n g i m p r e c i s i o ni n de s i gn us i ng fuz z y s e t t he o ry .

G i a c he t t i a nd Y oung (1996) a nd G i a c be t t i (1996)h a v e d e v e l o p e d a p a r a m e t r i c r e p r e s e n t a t i o n w h i c hfa c i l i t a t e s e f f i c i e n t c omput a t i on o f fuz z y a r i t hme t i con t r i a ngu l a r a nd t r a pe z o i da l fuz z y numbe rs . I t ha sc o m p u t a t io n a l c o m p l e x i t y o f o r d e r O ( k ) , w h e r e k i st he numbe r o f a r i t hme t i c ope ra t i ons i n t he e qua t i ons .A d d i t i o n a l l y , t h e f u z z y c o n s t r a i n t s y s t e m d e v e l o p e dby Y oung e t a l . (1996) i s r i c h i n f e a t u re s w i t h t hea b i l i t y t o r e p re se n t l i ngu i s t i c va r i a b l e s a nd t e rms ,fuz z y va r i a b l e s a nd nu mb e rs , a nd c r i sp va r i a b l e s a ndn u m b e r s i n c o m p l e x r e l a t i o n s t h a t i n c l u d e n o t o n l yc r i sp e qua l i t y , bu t a l so fuz z y e qua l i t y r e l a t ions , fuz z ya nd c r i sp i ne qua l i t y r e l a t i ons , a nd fuz z y a nd c r i spl og i c a l r e l a t i ons . The se de ve l opme n t s s e t t he s t a gef o r t h e d e v e l o p m e n t o f t h e precision convergencem e t h o d o l o g y p r e s e n t e d i n t h is p a p e r .



282 R.E. Giachetti et al. / European Journal of Operational Research 100 (1997) 277-2926 H i e r a r c h i c a l m o d e l o f d e s i g n

T o b e s u c c e s s f u l , a d e s i g n m u s t h a v e a n o v e r a l lm o n o t o n i c a l l y d e c r e a s i n g t r e n d w i t h r e g a r d t o i m p r e -c i s i o n . T h i s i s s h o w n g r a p h i c a l l y i n F i g . 1 . D i f f e r e n ts t a g e s o f t h e d e s i g n p r o c e s s r e q u i r e d i f f e r e n t k n o w l -e d g e r e p r e s e n t a t i o n m e t h o d s . T h e s e a r e i d e n t i f i e da l o n g t h e a b s c i s s a o f t h e f i g u r e . T h e d i f f e re n t s t a g e st h r o u g h w h i c h a d e s i g n p a s s e s c a n b e m o d e l e d b yd e c o m p o s i n g t h e d e s i g n p r o c e s s i n t o a h i e r a r c h y o fc o n s t r a i n t n e t w o r k s . T h i s h i e r a r c h y m a t c h e s t h em o v e m e n t o f d e c i si o n s f r o m p r o d u c t c o n c e p ti o n t op r o d u c t m a n u f a c t u r e . T h e h i e r a r c h y p r o c e e d s f r o mh i g h e r l e v e l , a b s t r a c t m o d e l s o f t h e p r o d u c t t o l o w e rl e v e l , d e ta i l e d m o d e l s o f th e p r o d u c t . A c o n s t r a i n tn e t w o r k i s a n o d e i n t h e h i e r a r c h y . T h e t o p n o d e o ft h e h i e r a r c h y i s t h e r o o t n o d e . A c o n s t r a i n t n e t w o r kc a n b e l i n k e d f r o m a b o v e a n d f r o m b e l o w t o o t h e rn e t w o r k s i n th e h i e ra r c h y . T h e a p p r o a c h o f d e c o m -p o s i n g t h e d e s i g n p r o b l e m i n t o a h ie r a r c h y o f c o n -s t r a i n t n e t w o r k s c o n f o r m s t o t h e f o r m a l h i e r a r c h i c a lm o d e l p r e s e n t e d i n O ' G r a d y e t a l . ( 1 9 9 4 ) .

F i g . 2 s h o w s a h i e r a r c h y o f f u z z y c o n s t r a i n t n e t -w o r k s . T h e c o n s t r a i n t s a r e r e p r e s e n t e d a s c i r c l e s a n dthe va r i a b l e s a r e r e p r e se n te d a s so l id - l i ne l i nks . Thed a s h e d - l i n e l i n k s c o n n e c t d i f f e r e n t l e v e l s o f t he h i e r -a r c h y . T h e y r e p r e s e n t a v a r i a b l e ' s v a l u e i n a h i g h e r

l e v e l o f t h e h i e r a r c h y b e c o m i n g a c o n s t r a i n t i n al o w e r l e v e l o f t h e h i e r a r c h y . T h e s e d a s h e d - l i n e l i n k sare t rans i t ion l inks t h r o u g h w h i c h p r e c i s i o n c o n v e r -g e n c e i s r e a l i z e d . I n t h e e x a m p l e p r e s e n t e d i n t h ene x t s e c t ion , t he c l ar i fi c a t ion o f t a sk c ons t r a in t ne t -w o r k i s t h e r o o t n o d e o f t h e h i e r a r c h y . T h e n e x tl o w e r l e v e l n e t w o r k i s t h e c onc e p t ua l de s i gn ne t-w o r k . T h e l o w e s t l e v e l n e t w o r k i n F i g . 2 i s t h ee m b o d i m e n t d e s i g n n e t w o r k .

A trans i t ion l ink c o n n e c t s a v a r i a b l e ' s v a l u e i none l e ve l o f t he h i e r a r c hy to a c ons t r a in t i n a l owe rl e v e l o f t h e h i e ra r c h y . A s a n e x a m p l e o f l i n k in gb e t w e e n c o n s t r a i n t n e t w o r k s i n d i f f e re n t l e v e l s o f t h ed e s i g n h i e r a r c h y , l e t u s e x a m i n e t r a n s i t i o n l i n k Tc o n n e c t i n g t h e c onc e p t ua l de s i gn n e t w o r k t o t h ee m b o d i m e n t d e s i g n n e t w o r k . I n t h e c onc e p t ua l de -s i gn n e t w o r k , t h e f u z z y v a r i a b l e 7 r e p r e s e n t s t h ep h o n e t h i c k n e s s . T h r o u g h c o n s t r a i n t p r o p a g a t i o n as o l u t i o n i s f o u n d t o t h e c onc e p t ua l de s i gn n e t w o r ktha t s a t i s f i e s i t s c ons t r a in t s . A s pa r t o f t h i s so lu t ion , 7i s a s s i g n e d a v a l u e . W e n o w m o v e d o w n t h e d e s i g nh ie r a r c hy to t he r i e x t s t a ge i n t he de s ign p r oc e s s -e m b o d i m e n t d e s i g n .

T h e e m b o d i m e n t d e s i g n s t a ge f o r t he c e l lu l a rp h o n e i s m o d e l e d b y t h e e m b o d i m e n t d e s i g n c o n -s t r a in t ne twor k . V ia t r a ns i t i on l i nk T , t he va luea s s ign e d to } a s pa r t o f t he so lu t ion to t he c onc e p t ua l

L e v e l o fI m p r e c i s i o no r F u z z i n e s s

C l a r if i ca t io n C o n c e p t u a l E m b o d i m e n t D e t a i le do f T a s k D e s i g n D e s i g n D e s i g n

De s ign p r og r e s s ion

I IL i n g u i s t ic V a r i a b l e sI rF u zz y N u m b e r s

R e a l N u m b e r sFig. 1. Design stages versus imprecision level and the ty pe of v ariables o u se.



R . E . G i a c h e t ti e t a l . / E u r o p e a n J o u r n a l ( ~ ' O p e ra t i o n a l R e s e a r c h 1 0 0 ( 1 9 9 7 ) 2 7 7 - 2 9 2 283

d e s i g n n e t w o r k d e f i n e s c o n s t r a i n t C 7 i n t h e e m b o d i -m e n t d e s i g n n e t w o r k . A s o l u t i o n t o t h e e m b o d i m e n td e s i g n c o n s t r a i n t n e t w o r k m u s t s a t i s f y t h e r e l a t i o n -s h i p d e f i n e d b y c o n s t r a i n t C 7 . S i n c e o p e r a t i o n s o nf u z z y n u m b e r s o n l y p r o d u c e v a l u e s w i t h i n c r e a s e dim pr e c i s ion , t he f uz z y se t s s e l e c t e d f o r u se i n t h i sl o w e r - l e v e l c o n s t r a i n t n e t w o r k m u s t h a v e s u f f i c i e n t l yr e d u c e d i m p r e c i s i o n s u c h t h a t t h e i r c o m b i n e d e f f e c tw i l l n o t e x c e e d t h e i m p r e c i s i o n o f T w i t h i n c o n -s t r a in t C 7 . Th i s bounds the so lu t ion in t he e m b o d i -m e n t d e s i g n p h a s e a n d h a s t h e a f f e c t o f f o r c i n g t h ed e s i g n e r t o w o r k b a c k w a r d s f r o m i m p r e c i s e q u a n t i -t i e s t o m o r e p r e c i s e q u a n t i t i e s . S u c c e s s i v e a p p l i c a -t i ons o f t h i s p r inc ip l e - h igh e r l e ve l s s e t r e qu i r e -m e n t s f o r l o w e r l e v e ls - r e d u c e s t h e o v e r a l l i m p r e c i -s i o n a s t h e d e s i g n p r o g r e s s e s t h r o u g h t h e d e s i g n

s t a ge s . We c a l l t he r e su l t o f t h i s p r inc ip l e p r e c i s ionc o n v e r g e n c e .

7 . P r e c i s i o n c o n v e r g e n c e u s i n g f u z z y c o n s t r a i n ts a t is f a c ti o n A n e x a m p l e

I n t h i s e x a m p l e t a k e n f r o m Y o u n g e t a l . ( 1 9 9 5 ) ,w e d e s i g n a c e l l u l a r t e l e p h o n e i n w h i c h w e b a l a n c ep h o n e w e i g h t , b a t t e r y l i f e , b a t t e r y t e c h n o l o g y , p h o n es e l li n g p r i c e , m a n u f a c t u r i n g c o s t , a n d t i m e - t o - m a r k e t .T h e d e s i g n p a r a m e t e r s a r e :

P h o n e w e i g h t r a ng e s b e t w e e n 0 .2 a n d 1 .0kg .B a t t e r y l i f e - r a ng e s b e t w e e n 2 a n d 1 0hour s .

C l a r i f i c a t i o n o f T a s k ?~ t I O I I C W e i q # l [ / /

{I.e, MO SS) Brlttt 'rV /lmoqo-mork~,t- - / . , , e f ~ - ' g . . . . . fl T C l u l o b J f j t / , ._ , ~ . . . .. . .. . . .. . .. . . .. . ~ 1 . .. . / // d T ] h . 4 i > . . . . . . . . dT;s ; // ~ ........................T x . . . . . . . . . . . . . . . . ~ * I> ' l l l l t t f ' r l / L i[7> ' l ~f l f / ( ~t / / / // 7 ....... ... ~, ................ // i/ , ~D l l i f tq l r i c f - - - 2 //{ C o n c e p t u a l D e s i g n/ . . . . . . . . . . . . . . . . . . . :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 i f _ . . . .. ./ - ( 6 ; // I

/ ~_Th .... /......................................................................... 2 _ .................. ..........................................' W T

E m b o d i m e n t D e s iK n/ ... ..... . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . /? ,~ /5/.;i .... , ..... . /

/ /

/ .....~ - ~ s . , ~ / I // y c 4 j c , f ; y qfi~J // /

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / /

Fig. 2. Design hierarchy for the cel lular phone example.

h n p r e c i s i o n



284 R . E . G i a c h e t t i e t a l. / E u r o p e a n J o u r n a l o f O p e r a t i o n a l R e s e a r c h 1 0 0 ( 1 9 9 7 ) 2 7 7 - 2 9 2

T i m e - t o - m a r k e t - r an g es b e tw e e n 9 a n d 2 4m o n t h s .S e l l i n g p r i c e - r a ng e s b e tw e e n $ 5 0 a n d$ 2 0 0 .

M a n u f a c t u r i n g c o s t - r an g e s b e tw e e n $ 2 0a n d $ 9 0 . a t t e r y t e c h n o l o g y i s o n e o f t h e f o l l o w -i n g : n e w t e c h n o l o g y , n i c k e l - c a d m i u m t y p e 1 ,n i c k e l - c a d m i u m t y p e 2 , a l k a l i n e .A l t h o u g h t h e s e v a r i a b l e s a r e s t a t e d a s q u a n t i t a t i v e

r a n g e s , v a l u e s f o r t h e m a t t h e c l ar i fi c a t i on o f t a sks t a g e a r e d i s c u s s e d i n te r m s s u c h a s unde s i rab l e ,l e s s de s i rab l e , more o r l e s s sa t i s f ac t o ry , more de s i r -ab l e , a n d very des i rable . C h e n a n d H w a n g ( 1 9 9 2 )e x a m i n e d t h is t y p e o f r e p r e s en t a t io n a n d c o n c l u d e dt h a t t y p i c a l l y d e s c r i p t i v e t e r m s a r e r a r e l y l e s s t h a nt h r e e , r a r e l y m o r e t h a n t e n , a n d t y p i c a l l y f i v e . T a b l e1 r e s t at e s t h e p r o b l e m v a r i a b l e s u s i n g d e s c r i p t i v et e r m s a n d s h o w s h o w e a c h v a r i a b l e ' s t e rm i s m a p p e di n t o a c o m m o n s e t o f te r m s . F i g . 3 s h o w s a n e x a m -p l e m a p p i n g f o r p h o n e w e i g h t f r o m i ts q u a n t it a ti v er a n g e o n t o f i v e t r i a n g u l a r f u z z y n u m b e r s w h o s ec o m b i n e d r a n g e i s n o r m a l i z e d b e t w e e n z e r o a n d o n e .

NormalizedRangeundersirable / 10 /\\ /~ /

/

desirable /'satisfactory /

ore . J ' : /desirableverydesirable4{ I I I ~

1.0 0.5 0.0 0.2 0.5 1.0Degree of Phone Weight (kg.)Membership

Fig. 3. An example mapping for the phone weight to a normalizedrange of linguistic terms.

T h e m a p p i n g o n t o a n o r m a l i z e d r a n g e a l l o w s u s t ot r a n s f o r m t h e d i f f e r e n t d a t a t y p e s i n t o a c o m m o nf u z zy l i ngu i s t i c r e p r e s e n t a t i o n s o w e c a n m o r e e a s i l yc o m p u t e w i t h t h e m . T h e y a r e n o w e a c h a f u z z y

Table 1Typical terms used to describe the range for each variable for the c l a r i fi c a t i o n o f t a s k constraint network and their mapping onto a commonterm setBattery technology(not a fuzzy number)

Variable

one of the following: new technology,nickel-cadmium type 1,nickel-cadmium type 2,alkaline

Satisfaction rangeu n d e s i r a b l ebut notunacceptable

l e s sd e s i r a b l e

m o r e r l e s ss a t i s f a c t o r y

m o r ed e s i r a b l e

v e r yd e s i r a b l e

Phone weight

Battery life

T i m e - t o - m a r k e t

Selling price

Manufacturing cost

high weight

short life

long

high

high

above averageweightbelow averagelifelonger thanaveragehigher thanaveragehigher thanaverage

average weight

average life

average

average

average

below averageweightabove averagelifeshorter thanaveragelower thanaveragelower thanaverage

low weight

long life

short

low

low



R . E . G i a c h e tt i e t a l . / E u r o p e a n J o u r n a l o f O p e r a t i o n a l R e se a r c h 1 0 0 ( 1 9 9 7 ) 2 7 7 - 2 9 2 285

l i n g u i s t i c v a r i a b l e w h o s e p o s s i b l e v a l u e s a r e o n e o ft h e d e s c r i p t i v e t e r m s i n T a b l e 1 . A s a n e x a m p l e , t h ev a ri a bl e P h o n e w e i g h t c a n n o w o n ly b e a s si g ne do n e o f t h e f i v e d e s c r i p t i v e t e rm s { h i g h w e i g h t , a b o v ea v e r a g e w e i g h t , a v e r a g e w e i g h t , b e l o w a v e r a g ew e i g h t , l o w w e i g h t } . T h e s o l u t i o n t o t h e p r o b l e mw i l l b e a d e s c r i p t i v e t e r m f o r e a c h v a r i a b l e s e l e c t e df r o m t h e p o s s i b l e d e s c r i p t i v e t e rm s l i st e d i n T a b l e 1 .I t i s t h e n m a p p e d b a c k i n t o i t s q u a n t i t a t i v e r a n g e t oide n t i f y a m or e r e s t r i c t i ve r a nge ( se e F ig . 3 ) . I n t h i sw a y , a s o l u t i o n r e d u c e s t h e i m p r e c i s i o n f o r e a c hva r i a b l e .

C o n t i n u i n g t h e p r o b l e m s t a t e m e n t , t h e v a r i a b l e sa r e r e l a t e d t o e a c h o t h e r t h r o u g h t h e f o l l o w i n g t y p e so f r e l a t i onsh ips :

Ma rket ing requirements: The r e l a t i o n s h i p o fP h o n e w e i g h t a n d B a t t e r y l i f e t o S e l l -i n g p r i c e .

Produ ct ion requirements: The r e l a t i o n s h i p o fT i me t o m a r k e t to M a n u f a c t u r i n g c o s t .Co st requirements. The r e l a t i o n s h i p o f S e l l i n gprice to Man ufa ctu rin g cost.1st Tech nolog y restrictions. Th e r e l a t i o n s h i p o fB a t t e r y t e c h n o l o g y to P h o n e w e i g h t a n dto Ba tt er y life.2nd Technology restr ic t ions: The r e l a t i o n s h i p o fB a t t e r y t e c h n o l o g y to T i m e t o m a r k e tand Ma nuf act uri ng cost.

T h e s e r e q u i r e m e n t s c o n s t i t u t e d e s i g n r e s t r i c t i o n sa n d c a n b e r e p r e s e n t e d a s a f u z z y c o n s t r a i n t n e t w o r ki n w h i c h t h e r e q u i r e m e n t s a r e c o n s t r a i n t s i n t e r c o n -n e c t e d t h r o u g h s h a r e d v a r i a b l e s. T h e f u z z y c o n s t r a i n tn e t w o r k f o r t h i s p r o b l e m i s t h e r o o t n o d e s h o w n i nF i g . 2 . T h e m a r k e t i n g c o n s t r a i n t is s h o w n i n T a b l e 2 .I n t h e m a r k e t i n g c o n s t r a i n t w e s e e t h a t o n e o f f o u rc o n d i t i o n s m u s t b e m e t t o s a t i s f y t h e c o n s t r a i n t .E x a m i n i n g t h e f i r s t c o n d i t i o n i n t h e c o n s t r a i n t w eh a v e t h e f o l l o w i n g :

P h o n e w e i g h t is l o w w e i g h t a n dB a t t e r y l i f e is l ong l i f e a ndS e l l i n g p r i c e is h igh .T h i s c o n d i t i o n e x p r e s s e s m a r k e t i n g ' s v i e w t h a t

c u s t o m e r s w i l l p a y m o r e f o r a l i g h t w e i g h t p h o n etha t l a s t s a l ong t im e . I n t he c ons t r a in t , ' i s ' is af u z z y l i n g u i s t i c o p e r a t o r t h a t c o m p a r e s a l i n g u i s t i c

T a b l e 2T h e m a r k e t i n g c o n s t r a i n t e x p r e s s e d u s i n g f u z z y l i n g u i s t i c v a r i -a b l e s , d e s c r i p t iv e t e r m s a n d l i n g u i s t ic o p e r a t o r sP h o n e w e i g h t is l o w w e i g h t a n d B a t t e r y l i f e is long l i f eand Sell ing price s highorP h o n e w e i g h t s low weight nd Ba tt er y lif e is abovea v e r a g e l i f e a n d S e l l i n g p r i c e i s ave rageo rP h o n e w e i g h t is b e l ow a v er a ge w e i g ht a n d B a t t e r y l i f eis l on g l if e a n d S e l l i n g p r i c e is a v e r a g eo rP h o n e w e i g h t is b e lo w a v er a ge w e i g ht a n d B a t t e r y l i f eis a b o ve a v e ra g e l if e a n d S e l l i n g p r i c e is l o w e r t h a na v e r a g e

v a r i a b l e t o a d e s c r i p t i v e t e r m a n d i s a n e x a m p l e o fm o d e l i n g r e l a t i o n s h i p i m p r e c i s i o n . P h o n ew e i g h t B a t t e r y l i f e a n d S e l l i n g p r i c ea r e l i n g u i s t i c v a r i a b l e s , a n d ' l o w w e i g h t ' , ' l o n g l i f e ' ,a n d ' h i g h ' a r e d e s c r i p t iv e t e r m s f r o m T a b l e 1 . T h e s ea r e e x a m p l e s o f m o d e l i n g linguistic imprecision.T h e r e m a i n i n g c o n s t r a i n t s h a v e s i m i l a r r e l a t i o n s h i p sb a s e d o n t h e c o n c e r n s o f t h e p r o d u c t i o n , m a r k e t i n g ,m a n a g e m e n t , a n d t e c h n o l o g i c a l d e p a r t m e n t s w i t h i nt h e c o m p a n y . U s i n g f u z z y l i n g u i s t i c v a r i a b l e s , d e -s c r i p t i v e t e r m s a n d l i n g u i s t i c o p e r a t o r s a l l o w s u s t ob u i l d a m o d e l u s i n g i m p r e c i s e d e s c r i p t i o n s t y p i c a l l ye m p l o y e d i n r e q u i r e m e n t s p e c i f i c a t i o n s w h i l e s u p -p o r t i n g t h o s e d e s c r i p t i o n s w i t h a w e l t d e f i n e d m a t h e -m a t i c a l b a s i s a n d p r o c e s s i n g a p p r o a c h . I n t h i s w a y ,t h e h i g h e s t l e v e l c o n s t r a i n t n e t w o r k o f F i g . 2 i s af u z zy cons t ra in t ne twork a n d a d d r e s s e s t h e ' c l a r i f ic a -t io n o f ta s k p h a s e ' i n P a hl a n d B e i t z ' s 0 9 8 8 ) m o d e l .T h e f u z z y c o n s t r a i n t n e t w o r k i s d e s c r i b e d i n Y o u n ge t a l . ( 1 9 9 5 ) a n d s o l v e d u s i n g f u z z y c o n s t r a i n t s a t i s -f a c t i o n t e c h n i q u e s . I t w a s s h o w n t o h a v e a s o l u t i o no n l y w h e n a c o m p r o m i s e c o u l d b e f o u n d b e t w e e nt w o c o n f l i c t i n g c o n s t r a i n t s - t h e 2 n d t e c h n o l o g yr e s t r i c t i o n s a n d t h e p r o d u c t i o n r e q u i r e m e n t s . C o n -f l i c t i n g c o n s t r a i n t s a r e c o m m o n i n d e s i g n a n d a r e a ne x a m p l e o f inconsistency imprecision. The abi l i ty tof i n d c o m p r o m i s e s o lu t i o n s t h ro u g h a s o u n d m a t h e -



286 R.E. G iachet t i e t a l . / Europe an Journal o f Operat ional Research 100 (1997) 27 7-2 92ma t i c a l ba s i s i s a r e c e n t l y i de n t i f i e d s t r e ng t h o ffuz z y c ons t ra i n t s a t i s fa c t i on (Ma r t i n -C l oua i re , 1993 ;La ng a nd S c h i e x , 1993 ; Y oung e t a l . , 1995) .

Th e so l u t i on t o t he c l a r i f i c a t i on -o f - t a sk is a s fo l -l ow s (Y oung e t a l . , 1995) :

P h o n e w e i g h t : b e l o w a v e ra g e ( 0 .2 8 t o 0 .6 k g );B a t t e r y l i f e : a b o v e a v er a ge ( 6 t o 9 .2 h o ur s) ;B a t t e r y t e c h n o l o g y : n ic ke l- ca dm i um ty pe2 ;T i m e - t o - m a r k e t : a v e ra g e (1 3 .5 t o 1 9 .5mon t hs ) ;S e l l i n g p r i c e : l o w e r t ha n a v er ag e ( $6 5 to$125) .W e c a n s e e i n t he so l u t i on a l l t he va r i a b l e s ha ve a

r e d u c e d r a n g e a n d a r e m o r e p r e c i s e t h a n w h e n w ebe ga n . Th i s i s t he ob j e c t i ve o f s t r a t e g i c p l a nn i ng a ndi s n o t u n e x p e c t e d . H o w e v e r , b y u s i n g f u z z y t e c h n o l-o g y w e n o w h a v e a f o r m a l b a s i s f o r w h a t h a s

p r e v i o u s l y b e e n a n a d - h o c p r o c e d u r e . T h i s a p p r o a c h ,i n a nd o f i t s e l f , i s u se fu l . H ow e ve r , t he i s sue i s howt o ve r t i c a l l y i n t e g ra t e t h i s so l u t i on w i t h t he ne x ts ta g e t o c o n t i n u e t h e p r o g r e s s i o n t o w a r d s m o r e p r e -c i s e r e s u l t s . T o s h o w h o w t h i s m i g h t b e a c c o m -p l i sh e d , l e t u s f o c u s u p o n o n e o f t he p a r a m e t e r s f r o mt he so l u t i on o f t he s t r a t e g i c p l a nn i ng p rob l e m - t hephone w e i gh t . In t he so l u t i on , t he phone ' s w e i gh t i sd e t e r m i n e d t o b e ' b e l o w a v e r a g e ' w h i c h m a p s o n t o ar e d u c e d r a n g e o f a c c e p t a b l e w e i g h t s d e f i n e d b y t h ei n t e rva l 0 .28 t o 0 .6 kg w h i c h i s r e p re se n t e d by t hef u z z y n u m b e r ( 0 . 2 8 , 0 .4 4 , 0 . 6 0 ) . ( 0 . 2 8 , 0 . 4 4 , 0 . 6 0 ) i sa t r ip l e d e s c r i b in g t h e t r ia n g u l a r f u z z y n u m b e r s h o w ni n F i g . 4 a nd i s a n e xa mpl e o f data imprecision. Int h e n e x t l o w e r l e v e l o f t h e d e s i g n h i e r a r c h y t h ep h o n e ' s w e i g h t b e c o m e s t h e u p p e r b o u n d f o r t h ep h o n e ' s m a s s i n c o n s t r a i n t C 2 o f t h e conceptualdes ign c ons t ra i n t ne t w ork . A so l u t i on t o t he concep-tual des ign pha se mus t s a t i s fy c ons t ra i n t C2 .

Table 3Fuzzy constraints for the conceptual design constraint network

Fuzzy constraint DescriptionC1C2 ffl ~ M

C3 1= 1.5/7C4 4



R.E. Giachetti et al . / European Journal of Operational Research 100 (1997) 277-2 92 287

h s tmembership 0.5

of M0.0

.28 .44 .60

Fuzzy numberFig. 4. Triangular fuzzy set for the phone 's weight (i.e., mass),/~ = (0.28,0.44,0.60).

7.1. Conceptual design networkT h e s e c o n d n e t w o r k i n t h e d e s i g n h i e r a r c h y r ep r e -

sents the conceptual design pha se . The so l u t i on t othe clarification of task c o n s t r a i n t n e t w o r k b e c o m e s

a r e q u i r e m e n t f o r c o n c e p t u a l d e s i g n . T o s p e c i f y t h ere qu i re m e n t w e s t a t e it a s a fuz z y c ons t ra i n t in w h i c ht h e m a s s o f t h e c e l l u l a r p h o n e m u s t b e l e s s t h a n o re q u a l t o t h e t r ia n g u l a r f u z z y n u m b e r , Q ( r e p re s e n t e dby t he t r ip l e (0 .2 8 , 0 .4 4 , 0 .60 ) ) . Thu s , ,Q i s a n uppe rb o u n d o n t h e m a s s o f t h e p h o n e a n d r e c e i v e s i t sva l ue v i a t r a ns i ti on li nk M show n c onne c t i ng t he set w o c ons t ra i n t ne t w orks i n F i g . 2 . Ta b l e 3 l i s t s t hec o n s t r a i n t s f o r t h i s l e v e l o f t h e e x a m p l e p r o b l e m .The conceptual des ign c ons t ra i n t ne t w ork i s t hese c ond l e ve l i n F i g . 2 . The ob j e c t i ve a t t h i s s e c ondl e v e l is t o d e t e r m i n e t h e p h o n e ' s d i m e n s i o n s ( l e n g t h ,w i d t h , a nd t h i c kne s s ) sub j e c t t o t he c ons t ra i n t s .

T h e d e s i g n e r h a s a c o n c e p t o f a t h i n , s l e n d e rphone a nd se l e c t s va l ue s a ppropr i a t e l y . The va l ue s

Table 4Fuzzy constraints for the embodiment design constraint networkConstraint No. Fuzzy constraint DescriptionCIC2

C3C4C5C6C7

C8

= 2ST MP w ~ 27 w Width calculation~, ~, W Upper bound of width (from the solution of the conceptual design network

via transition link W)MP(PN, MP L MP T, MP w, MPwT) Relation that defines the structure of the mou thpiece inventory data?w >- 0.250.2 < ST



288 R . E . G i a c h e tt i e t a l . / E u r o p e a n J o u r n a l o f O p e r a t io n a l R e s e a r c h 1 0 0 ( 1 9 9 7 ) 2 7 7 - 2 9 2

s e l e c te d f o r w i d t h a n d t h i c k n e s s a r e ~ = ( 5 , 7 , 9 ) c ma n d 7 = ( 1 . 5 , 2 , 2 . 5 ) c m . T h e m a s s i s c a l c u l a t e d inc o n s t r a i n t C 1 a u t o m a t i c a l l y b y c o n s t r a i n t p r o p a g a -t i o n , a n d i s fi t = ( 2 5 3 . 5 , 8 8 2 , 2 2 9 5 ) g . T h i s r e s u l t i sa u t o m a t i c a l l y c h e c k e d b y c o n s t r a i n t p r o p a g a t i o na g a i n s t c o n s t r a in t C 2 u s i n g p o s s i b i l i t y t h e o r y ( D u b o i sand Prade , 1988) :/X c 2 = P o s s ( ~ l z , M ) = 0 . 5 7 .

S inc e mc2 > as , w h e r e a s i s t h e s y s t e m t h r e s h -o l d f o r d e t e r m i n i n g a c c e p t a b i l i t y , t h e r e l a t i o n s h i pd e f i n e d i n c o n s t r a i n t C 2 i s s a t i s f i e d . T h e p r o b l e ms o l u t io n i s a n a p p r o x i m a t e s i z e f o r th e c e l l u l a r p h o n et h a t s a t i s f i e s t h e r e q u i r e m e n t s f r o m t h e clarificationof task pha se . I n f i nd ing a so lu t ion to t he conceptualdesign c o n s t r a i n t n e t w o r k , t h e c o n s t r a i n t p r o p a g a t i o np r o c e s s i n f e r s f u z z y v a l u e s t o a s s i g n t o t h e f u z z yv a r i a b l e s s u c h t h a t t h e a p p l i c a t i o n o f f u z z y m u l t i p l i -c a t i o n i n c o n s t r a i n t C 1 p r o d u c e s a v a l u e t h a t d o e sn o t v i o l a t e t h e u p p e r b o u n d f o r t h e p h o n e ' s m a s s i nc o n s t r a i n t C 2 . T h e s o l u t i o n i s p r o d u c e d b y s a t i s f y i n gc o n s t r a i n t C 2 , t h r o u g h s u c c e s s i v e a p p l i c a t i o n s o ff u z z y a r i t h m e t i c o p e r a t o r s . B e c a u s e t h e a p p l i c a t i o no f a f u z z y a r i t h m e t i c o p e r a t o r a l w a y s r e s u l t s i n a ni n c r e a s e in i m p r e c i s i o n , a s i d e e f f e c t o f u s in g c o n -s t r a i n t p r o c e s s i n g t e c h n o l o g y i s t o e f f e c t i v e l y s o l v et h e p r o b l e m b a c k w a r d s s o t h a t t h e f u z z y v a l u e sa s s igne d to va r i a b l e s r , ~ , a nd 7 ( r e p r e se n t ing l e ng th ,w e i g h t a n d t h i c k n e s s ) a r e m o r e p r e c i s e t h a n t h ef u z z y v a l u e f o r t h e p h o n e ' s m a s s . I n t h e n e x t l o w e rl e v e l o f t h e d e s i g n h i e r a r c h y t h e p h o n e ' s w i d t h a n dt h i c k n e s s b e c o m e u p p e r b o u n d s o n t h i c k n e s s a n dw i d t h w h i c h t h e e m b o d i m e n t d e s i g n m u s t s a t i s f y .7.2. Embodiment design of the cellular phone mouth-piece

T h e s o l u t i o n t o t h e conceptual design c o n s t r a i n tn e t w o r k s e t s t h e f u z z y v a l u e s f o r t h e d i m e n s i o n s o ft h e p h o n e . I t s s o l u t i o n w a s d e t e r m i n e d b y s a t i s f y i n gc o n s t r a i n t s w h i c h m o d e l e d t h e i n t e n d e d u s e o f t h ep h o n e . I n t h e embodiment design p h a s e , t h e d e s i g n e rs e l e c t s c o m p o n e n t s w h i c h w i l l b e c o m e p a r t o f t h ep h o n e a s m a n u f a c t u r e d . T h e s e l e c t i o n o f t h e s e c o m -p o n e n t s a s w e l l a s t h e r e l a t i o n s h i p a m o n g t h e m a n do t h e r v a r i a b l e s m u s t f u r t h e r r e d u c e t h e i m p r e c i s i o ni n v o l v e d i n t h e d e s i g n p r o b l e m t o c o r r e s p o n d a sc l o s e a s p o s s i b l e t o t h e s p e c i f i c a t i o n s o f s t a n d a r d

Tab le 5M o u t h p i e c e i n v e n t o r y d a t a u s e d b y c o n s t r a i n t C 3P a rt n u m b e r ( P N ) M P L ( c m ) M P w ( c m ) M P T ( c m ) M P w - ( g )MP -01 4 4 1 .25 100M P-0 2 5 5 1 .25 125M P-0 3 6 6 1 .5 150MP -04 6 .5 6 .5 1 .75 150MP -05 7 7 1 .75 200

c o m p o n e n t s . T h e s t a n d a r d c o m p o n e n t ' s s p e c i f i c a -t i o n s a r e t h e l o w e r l i m i t o n t h e l e v e l o f i m p r e c i s i o na n d r e p r e s e n t t h e p r o d u c t i o n p r o c e s s v a r i a t i o n s h o w ni n F i g . 1 . W h e n t h i s le v e l o f i m p r e c i s i o n i s a c h i e v e d ,t h e p r e c i s i o n c o n v e r g e n c e p r o c e s s c a n t e r m i n a t e .

I n o u r e x a m p l e , w e c o n c e n t r a t e o n a s u b c o m p o -n e n t o f t h e c e l l u l a r p h o n e , t h e m o u t h p i e c e d e s i g na n d t h e s e l e c t i o n o f a p a r t i c u l a r m o u t h p i e c e f r o mt h o s e a v a i l a b l e i n i n v e n t o r y . T a b l e 4 s h o w s t h ec ons t r a in t s f o r t he embodiment design f u z z y c o n -s t r a i n t n e t w o r k . T a b l e 5 s h o w s t h e i n v e n t o r y d a t a o nwhic h the r e l a t i on c ons t r a in t , C 3 , ope r a t e s . Th i sc o n s t r a i n t n e t w o r k i s t h e b o t t o m n o d e i n t h e d e s i g nh i e r a r c h y o f F i g . 2 . T h e v a l u e ' s a s s i g n e d t o v a r i a b l e si n s o l v i n g t h e conceptual design c o n s t r a i n t n e t w o r kb e c o m e c o n s t r a i n t s i n t h e embodiment design c o n -s t r a i n t n e t w o r k v i a t r a n s i t i o n l i n k s W a n d T . T h e s et r a n s i t i o n l i n k s b e c o m e c o n s t r a i n t s C 2 a n d C 7 i n t h eembodiment design n e t w o r k .

T h e d e s i g n e r s e l e c t s th e f u z z y s e t ( 0 . 2 0 , 0 . 2 5 , 0 . 3 0 )a s t h e f u z z y v a l u e f o r t h e s h e l l t h i c k n e s s , S t . T h ed e s i g n e r a l s o s e l e c t s t h e f u z z y s e t ( 0 . 1 5 , 0 . 2 , 0 . 2 5 )a s t he f uz z y va lue f o r t he c l e a r a nc e , va r i a b l e ? T ,b e t w e e n t h e m o u t h p i e c e a n d t h e p h o n e ' s o u t e r s h e l li n t he t h i c kne ss d i r e c t ion . The de s igne r s e l e c t s pa r tn u m b e r s f r o m t h e m o u t h p i e c e i n v e n t o r y d a t a t a b l et h a t a r e a u t o m a t i c a l l y p r o p a g a t e d t o c o n s t r a i n t s C 1a nd C 6 , i n f e r r ing va lu e s f o r ~ a nd 7 . C on s t r a in tp r o p a g a t i o n a u t o m a t i c a l l y c h e c k s r e q u i r e m e n t s c o n -s t r a in t s , C 2 a nd C 7 . A pa r t i a l o r de r ing c a n be p r o -d u c e d b a s e d o n / Z c 2 a n d / Z c7 w h i c h s h o w s h o ww e l l t h e m o u t h p i e c e s s a t i s f y t h e s e c o n s t r a i n t s . T h ed e s i g n e r i s f r e e t o c h o o s e a n y m o u t h p i e c e w h i c hsa t i s f i e s t he c ons t r a in t s , wh e r e / x > a s . The p a r t i a l l yo r d e r e d s e t o f m o u t h p i e c e s p r o v i d e s t h e d e s i g n e rw i t h k n o w l e d g e a s t o w h i c h m o u t h p i e c e b e s t s a t i s -f i e s t h e p r e f e r e n c e s e t i n p r i o r d e s i g n s t a g e s . A s a ne x a m p l e , m o u t h p i e c e M P - 0 1 s a t i s f i e s c o n s t r a i n t s C 2



R.E. Giachetti et al. / European Journal of Operational Research 100 (1997) 277-292 2 8 9

a n d C 7 a t f u z z y s e t m e m b e r s h i p v a l u e s o f / X c 2 = 1 .0a n d / Z c7 = 0 . 7 5 w i t h a n o v e r a l l w i d t h o f @ =( 4 . 7 , 4 . 9 , 5 .1 ) a n d t h ic k n e ss } ' = ( 1 . 9 5 , 2 . 1 5 , 2 . 3 5 ) .A l t e r n a t i v e l y , t h e d e s i g n e r c a n s e a r c h f o r a s y s t e mt r u th t h r e sho ld l e ve l , a s , a t wh ic h the c ons t r a in tne tw or k i s s a t i s f i e d . Th e c ~ va lue i s i nd i c a t ive o ft h e s o l u t i o n ' s q u a l i ty . A s a s a p p r o a c h e s o n e t h es o l u t i o n q u a l it y i s b e t t e r a n d a s a s a p p r o a c h e s z e r othe so lu t ion qua l i t y i s wor se .7.3. Detailed design

T h e d e t a i l e d d e s i g n s t a g e c o u l d b e a c c o m p l i s h e du s i n g a p a r a m e t r ic C A D s y s t e m . T h e p h y s i c a l la y o u to f t h e p r o d u c t h a s b e e n d e t e r m i n e d , t h e p a r a m e t e r sh a v e c o n v e r g e d t o w i t h i n t h e c a p a b i l i t i e s o f t h ep r o d u c t i o n p r o c e s s s o t h e n o m i n a l d i m e n s i o n s a n dt o l e r a n c e s a r e d e f i n e d b y t h e f u z z y n u m b e r s i n t h es o l u t i o n s e t . T h e p r o c e d u r e f o l l o w e d i n t h e p r e v i o u ss t a g e s c a n b e c o n t i n u e d w h e r e t h e s o l u t i o n v a l u e sf r o m e m b o d i m e n t d e s i g n b e c o m e c o n s t r a i n t s i n d e -t a i l e d d e s i g n . U s i n g a p a r a m e t r i c C A D s y s t e m , t h e s ec o n s t r a i n t s c a n b e r e p r e s e n t e d , t h u s e n s u r i n g t h ec o n t i n u a t i o n o f t h e o r i g i n a l d e s i g n o b j e c t i v e s i n to t h ed e t a i l e d d e s i g n s t a g e .

8 . A m e a s u r e f o r t h e r e d u c t i o n o f d e s i g n impr e c is i onC o m m o n m e a s u r e s o f f u z z i n es s a r e d e f in e d a s t h e

l a c k o f d i s ti n c t i o n b e t w e e n a f u z z y s e t a n d i t s c o m -p l e m e n t . T h e l e s s a s e t d i f f e r s f r o m i t s c o m p l e m e n t ,t h e f u z z i e r i t i s ( K l i r a n d Y u a n , 1 9 9 5 ) . W o o d a n dA n t o n s s o n ( 1 9 8 9 ) s h o w e d t h a t t h e s e m e a s u r e s a r ei n a d e q u a t e f o r q u a n t i f y i n g d e s i g n i m p r e c i s i o n a n d

d e v e l o p e d a n e w m e a s u r e o f th e d e s i g n i m p r e c i s i o no f a f u z z y s e t . T h i s n e w m e a s u r e , c a l l e d t h e g a m m af u n c t i o n , q u a n t i f i e s t h e s p r e a d o f t h e m e m b e r s h i pf u n c t i o n a b o u t t h e m o d e v a l u e a n d i s g i v e n b y t h ee x p r e s s i o n

IxlD ( C ) = Y '~ ( e ~e (x ') - 1 ) . ( 1 )i = l

H o w e v e r , t h i s m e a s u r e i s n o t a d e q u at e t o c o m p a r ei m p r e c i s e q u a n t i t i e s o f d i f f e r e n t u n i t s a n d s c a l e b e -c a u s e i t o n l y a c c o u n t s f o r t h e a b s o l u t e s p r e a d o f t h ef u z z y s e t . I t d o e s n o t a c c o u n t f o r t h e r e l a t i v e d i f f e r -e n c e i n m a g n i t u d e d i f f e r e n t i m p r e c i s e q u a n t i t i e s m a yh a v e . A s i m i l a r p r o b l e m e x i s t s i n s t a t i s t i c s i n c o m -p a r i n g t h e v a r i a b i l i t y o f s a m p l e s d r a w n f r o m p o p u l a -t i ons t ha t d i f f e r i n s c a l e o r un i t s . I n suc h c a se s t hes t a n d a r d d e v i a t i o n s a r e s c a l e d b y d i v i d i n g e a c h b y i t so w n m e a n t o p r o d u c e t h e c o e f f i c i e n t o f v a r i a t io n a s ar e l a t iv e m e a s u r e o f v a r i a b i l it y . I n a n a n a l o g o u s m a n -n e r , w e c a n c o m p a r e t h e r e l a t i v e i m p r e c i s i o n o fd i s p a ra t e f u z z y s e t s b y m o d i f y i n g t h e g a m m a f u n c -t i on to a c c oun t f o r t he r e l a t i ve s c a l e o f t he f uz z y se t .W e d o t h i s b y s c a l i n g e x p r e s s i o n ( 1 ) b y t h e m o d e o ft h e f u z z y s e t , d e n o t e d b y b . W e c a l l t h i s t h e coeffi-cient of imprecision a nd de f ine i t a sc ( , ~ ) = b f , , ~ (e ~A ~ x)- 1 ) d x . ( 2 )

A c r i s p n u m b e r h a s a m e a s u r e o f f u z z i n e s s e q u a lt o 0 . T h e h i g h e r t h e c o e f f i c i e n t o f i m p r e c i s i o n , t h em o r e i m p r e c i s e th e u n d e r l y i n g m e m b e r s h i p f u n c ti o n .W h e n a p p l i e d t o c r i s p i n t e r v a l s , t h e mode p a r a m e t e ri s d e t er m i n e d b y b = ( c - a ) . T h i s m e a s u re o ff u z z i n e s s a p p e a r s t o c o r r e c t l y r a n k a m i x t u r e o ft r i a n g u l a r f u z z y n u m b e r s a n d c r i s p i n t e r v a l s r e g a r d -l e s s o f t h e m o d e v a l u e b .

Table 6Coefficient of imprecision for design param eters n the exa mp le problemDesign param eterand its corresponding fuzzy set D esi gn tage in which the parameterap pe ars Coefficient f imprecision, c(A )P h o n e w e i g h t c l a r i f i c a t i o n o f t as k [ 0 . 2 , 0 . 1 ]F~/Clarification of task = (0.28, 0.44, 0.60>lconceptua l des ign = ( 16 . 9 , 21 ,2 5 . 5>I~'Conceptual design = (5 , 7, 9>}'Concept,adesign= W Em bodim ent des ign = (4.7 , 4.9, 5.1 >~Embodiment design = < 1.95, 2.15, 2.35>

Clarification of task 2,29Clarification of task 0,522Concep tual design 0.287Conceptual design 0.410Concep tual design 0.359Embod iment design 0.059Embod iment design 0.032



290 R.E. Giachetti et al. / European Journal of Operational Research 100 (1997) 277-292T h e c o e f f i c i e n t s o f i m p r e c i s i o n f o r t h e d e s i g n

p a r a m e t e r s i n t h e s o l u t i o n s t o t h e e x a m p l e c o n s t r a i n tn e t w o r k s a r e c o m p u t e d a n d c o m p i l e d i n T a b l e 6 . I nthe clarification o f task c o n s t r a i n t n e t w o r k t h e o r i g i -n a l r a n g e f o r t h e m a s s w a s r e d u c e d t o r ~ =( 0 . 2 8 , 0 .4 4 , 0 . 6 0 ) , w i th c ( f f t ) = 0 . 5 2 2 . T h e concep-tual des ign c o n s t r a i n t n e t w o r k r e d u c e d t h i s m a s s i n t ot h r e e s e p a r a t e v a r i a b l e s w i t h l o w e r c o e f f i c i e n t s o fi m p r e c i s i o n a s l i s t e d i n T a b l e 6 . A l l t h r e e v a l u e ss h o w a r e d u c t i o n i n i m p r e c i s i o n f r o m t h e i m p r e c i -s i o n i n t h e m a s s . T h e h i e r a r c h y o f c o n s t r a i n t n e t -w o r k s r e d u c e d t h e i m p r e c i s i o n f r o m 2 . 2 9 t o a l o wv a l u e o f 0 .2 8 7 f o r a 8 7 % t o t a l r e d u c t i o n i n i m p r e c i -s i o n f r o m t h e clarification o f task s t a ge to t heconceptual des ign s t a g e . F u r t h e r p r e c i s i o n c o n v e r -g e n c e i s r e a l iz e d b y m a k i n g e a c h o f t h e th r e e p r o d -u c t d i m e n s i o n s a c o n s t r a i n t i n a l o w e r l e v e l n e t w o r k .A n e x a m p l e o f embodiment des ign i s p r e s e n t e d u s -i n g v a r i a b l e s f o r t h i c k n e s s a n d w i d t h . T h i s n e t w o r kd e m o n s t r a t e s a m e t h o d f o r r e d u c i n g t h e n u m b e r o ff e a s ib l e a l t e r na t ive s , a nd l e a ds t o t he s e l e c t ion o f ad i s c r e t e p a r t f r o m a c o m p o n e n t d a t a b a s e . T h e n e wt h i ck n e s s a n d w i d th ar e ( 1 . 9 5 , 2 . 1 5 , 2 . 3 5 ) a n d( 4 . 7 , 4 . 9 , 5 .1 ) . Th i s r e du c e s t he im pr e c i s ion f u r the r ,a n d w h e n m e a s u r e d a s a p e r c e n t a g e o f t h e i n i t i a lm a s s , a 9 8 % r e d u c t i o n i n i m p r e c i s i o n i s r e a l i z e d a st h e d e s i g n p r o g r e s s e d f r o m t h e c lar i f icat ion of tasks t a ge to t he embodiment des ign s t a g e . T h e s o l u t i o nto t he embodiment des ign c o n s t r a i n t n e t w o r k r e d u c e st h e i m p r e c i s i o n t o a m a g n i t u d e a p p r o a c h i n g t h e t o l -e r a n c e s o f t h e m a n u f a c t u r i n g p r o c e s s e s . T h i s s i g n a l st h e c r o s s o v e r f r o m a f u z z y a n a l y s i s o f i m p r e c i s i o n t ot h e u n c e r t a i n t y f r o m t h e s t o c h a s t i c v a r i a t i o n t h a t i sa n i n h e r e n t p r o p e r t y o f t h e m a n u f a c t u r i n g p r o c e s s e s .T h i s t e r m i n a t i o n i s s h o w n i n F i g . 1 w h e r e t h e i m p r e -c i s ion l i ne i n t e r se c t s t he unc e r t a in ty l i ne .

9 . Conc l u s i o n sI m p r e c i s i o n i s a n i n h e r e n t c h a r a c t e r i s t i c o f e n g i -

n e e r i n g d e s i g n . A l l d e s i g n s b e g i n w i t h g o a l s a n dr e q u i r e m e n t s t h a t a r e d e s c r i p t i v e s t a t e m e n t s o f w h a tf u n c t i o n s a p r o d u c t s h o u l d p e r f o r m . T h e s e i m p r e c i s ep r o d u c t s p e c i f i c a t i o n s a r e t r a n s f o r m e d b y t h e d e s i g np r o c e s s t o t h e p r e c i s e s p e c i f i c a t i o n s n e c e s s a r y f o rm a n u f a c t u r in g . A m e t h o d o l o g y w a s o u t l i n ed w h e r e b ya c o n c u r r e n t e n g i n e e r i n g d e s i g n p r o b l e m c o u l d b e

s t r u c tu r e d s o t h a t t h e e f f e c t o f c o n s t r a i n t p r o p a g a t i o nw o u l d b e a r e d u c t i o n i n d e s i g n i m p r e c i s i o n . A t o t a lr e d u c ti o n o f 9 8 % w a s a c h i e v e d i n t h e e x a m p l e p ro b -l e m . U s i n g a h i e r a r c h y o f f u z z y c o n s t r a i n t n e t w o r k sw e d e c o m p o s e d t h e d e s i g n p r o c e s s a c c o r d i n g t ol e v e l s o f d e c i s i o n m a k i n g a p p r o p r i a t e t o t h e c h r o n o -l o g i c a l d e s i g n s t a g e s . T h e h i e r a r c h y m o d e l s i n f o r m a -t i o n a t d i f fe r e n t l e v e l s o f a b s t r a c t i o n a n d c o r r e s p o n d st o a c t u a l d e s i g n p r o c e s s e s . I t f a c i l i t a t e s t e a m b a s e dd e s i g n s i n c e c o n c e p t u a l d e s i g n d e c i s i o n s a r e u s u a l l ym a d e b y d i f f e r e n t p e o p l e t h a n e m b o d i m e n t d e s i g nd e c i s i o n s . T h i s p r e c i s i o n c o n v e r g e n c e a p p r o a c h i sq u a n t i f i a b l e . W e c a n d e t e r m i n e w h e n t h e i m p r e c i s i o ni n a d e s i g n h a s b e e n s u f f i c i e n t l y re d u c e d t o t he p o i n tt h a t i t m a t c h e s t h e i m p r e c i s i o n i n h e r e n t i n t h e p r o -d u c t i o n p r o c e s s . T h i s g i v e s u s a t e r m i n a t i o n m e a s u r ef o r t h e d e s i g n i t e r a t i o n c y c l e , a n d a l s o a m e a n s t om a p i n h e r e n t p r o d u c t i o n p r o c e s s v a r i a t i o n t o i t s i m -pa c t on the de s ign .

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