Forecasting the Unit Cost of a Product with Some Linear Fuzzy Collaborative Forecasting Models

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1 Algorihms 2012, 5, ; doi: /a Aricle OPEN ACCESS algorihms ISSN Forecasing he Uni Cos of a Produc wih Some Linear Fuzzy Collaboraive Forecasing Models oly Chen Deparmen of Indusrial Engineering and Sysems Managemen, Feng Chia Universiy, aichung Ciy, 407, aiwan; olychen@ms37.hine.ne Received: 6 June 2012; in revised form: 7 Sepember 2012 / Acceped: 17 Sepember 2012 / Published: 15 Ocober 2012 Absrac: Forecasing he uni cos of every produc ype in a facory is an imporan ask. However, i is no easy o deal wih he uncerainy of he uni cos. Fuzzy collaboraive forecasing is a very effecive reamen of he uncerainy in he disribued environmen. his paper presens some linear fuzzy collaboraive forecasing models o predic he uni cos of a produc. In hese models, he expers forecass differ and herefore need o be aggregaed hrough collaboraion. According o he experimenal resuls, he effeciveness of forecasing he uni cos was considerably improved hrough collaboraion. Keywords: fuzzy collaboraive forecasing; uni cos 1. Inroducion Cos forecasing means differen hings a differen sages of he produc life cycle. In produc design, he designer needs o know wheher he produc will be economically produced. Afer a produc goes ino mass producion, forecasing he uni cos is he basis of financial and producion planning aciviies. When a produc eners he marke, he follow-up cusomer service and mainenance coss mus also be aken ino accoun. Accuraely predicing he uni cos of each produc ype is a very imporan ask in any facory. If he uni cos is less han expeced, hen he effors and invesmen of cos reducion are no necessary. Conversely, if he uni cos is more han expeced, hen he profiabiliy of he produc will be over-esimaed, resuling in he wrong invesmen and producion decisions. However, forecasing he uni cos is no an easy ask because of he uncerainy of he uni cos, mainly due o he cos of human operaions in he producion of producs, which is someimes unsable. In addiion, here is no much relevan lieraure in he uni cos forecasing. On he oher hand, several recen sudies (e.g. [1 14])

2 Algorihms 2012, showed ha fuzzy collaboraive forecasing has grea poenial for he predicion of processes wih uncerainy, such as yield learning, changes in price, flucuaions in he cycle ime, and ohers. For hese reasons, he applicaion of fuzzy collaboraive forecasing mehods o improve he performance of he uni cos forecasing is worh a ry. herefore, some fuzzy collaboraive forecasing models are proposed in his sudy, in order o enhance he accuracy and precision of he uni cos forecas. In he proposed mehodology, he sakeholders are a group of domain expers, such as he produc engineer, facory managers and accouning deparmen saff. hese expers apply fuzzy linear regression mehods o predic he uni cos of a produc. A fuzzy linear regression equaion can be convered ino a linear or nonlinear programming problem in a variey of ways. Furhermore, wihin he conversion process some parameers need a subjec seing. As a resul, forecass obained by he expers may be very differen and herefore requires a collaboraive mechanism o deal wih he following issues: (1) How o inegrae hese forecass? (2) How expers can refer o he forecass of ohers o modify heir own? In response o his issue, he mehods presened in his sudy are as follows: (1) Some linear fuzzy regression models for he uni cos forecasing are proposed and compared. (2) Developmen of dedicaed sofware o pass he forecas of each exper o oher expers for heir reference. In he meanime, he sofware can inegrae differen forecass using he hybrid fuzzy inersecion and back propagaion nework approach. (3) In reference o he forecass of ohers, each exper subjecively modifies he parameers in he fuzzy linear regression mehod. he objecives of his sudy are as follows: (1) o enhance he accuracy of he uni cos forecas. In oher words, he forecass obained mus be very close o he acual values. (2) o improve he precision of he uni cos forecasing. Namely, a very small range conaining he acual value can be esimaed. (3) he applicaion of an insance o compare he advanages and disadvanages of differen linear fuzzy collaboraive forecasing models. he organizaion of his sudy is described as follow. Secion 2 firs reviews he lieraure relaed o fuzzy collaboraive forecasing and he uni cos forecasing. he problems faced by he exising mehods are also discussed. hen, in Secion 3, some fuzzy collaboraive forecasing models are proposed o predic he uni cos of a produc. An example is given o illusrae he applicabiliy of hese models. Secion 4 makes conclusions and suggess some direcions for fuure research. 2. Relaed Work Carnes [15] esablished a basic equaion o calculae he uni cos of a wafer. Carnes also compared he long-erm coss of ownership of wo alernaive machines, bu hese coss were no allocaed o he wo machines. Wood [16] defined he lowes cos of all operaions on he same machine as he minimum wafer cos. In Pfizner e al. s view, he recovery of wafers is becoming increasingly imporan in reducing he uni cos along wih he growh in size of a wafer [17].

3 Algorihms 2012, Alhough here have been some lieraure abou fuzzy collaboraive inelligence and sysems, bu very few direcly relaed o fuzzy collaboraive forecasing. Shai and Reich [18,19] defined he concep of infused design as an approach for esablishing effecive collaboraion beween designers from differen engineering fields. Büyüközkan and Vardaloglu [20,21] applied he fuzzy cogniive map mehod o he collaboraive planning, forecasing and replenishmen of a supply chain. he iniial values of he conceps and he connecion weighs of he fuzzy cogniive map are dependen on he subjecive belief of he exper and can be modified afer collaboraion. According o Poler e al. [22], he comparison of collaboraion mehods and he proposing of sofware ools, especially as regards forecasing mehods for collaboraive forecasing, are sill lacking. Pedrycz and Rai [14] discussed he problem of collaboraive daa analysis by a group of agens having access o differen pars of daa and exchanging findings hrough heir collaboraion. A wo-phase opimizaion procedure was esablished, so ha he resuls of communicaion can be embedded ino he local opimizaion resuls. In recen years, Chen [6] used a hybrid fuzzy linear regression-back propagaion nework approach o predic he efficien cos per uni of a semiconducor produc. his mehod firs gahered a group of expers in he field. Each exper hen used a fuzzy linear regression equaion o predic he fuure uni cos. he resul is a fuzzy value, and can be regarded as a non-symmeric inerval forecas. A crisp forecas rarely equal o he acual value. In conras, a fuzzy forecas can conain he acual value. he fuzzy forecass obained by differen expers are aggregaed using a fuzzy inersecion, resuling in a polygon-shaped fuzzy number, which can be defuzzified using a back propagaion nework. Chen [4] considered he case in which each exper has only parial access o he daa, and is no willing o share he raw daa he/she owns. he forecasing resuls by an exper are conveyed o oher expers for he modificaion of heir seings, so ha he acual values will be conained in he fuzzy forecass afer collaboraion. All fuzzy collaboraive inelligence mehods seek consensus of resuls. In his field, Osrosi e al. [23] defined he concep of consensus as he overlapping of design clusers of differen perspecives. Similarly, Chen [2] defined he concep of parial consensus as he inersecion of he views of some expers. Cheikhrouhou e al. [24] hough ha collaboraion is necessary because of he unexpeced evens ha may occur in he fuure demand. In shor, he exising approaches have he following problems: (1) he uni cos forecased by he exising mehods may be lower han he acual value, resuling in over-esimaed profis if he financial plan is based on he forecass. (2) For precision in he uni cos forecasing, he narrowes scope conaining he acual value is required; however, his has rarely been discussed. (3) he peak and average uni coss are forecased separaely, which is problemaic because i is possible ha he forecas becomes invalid in he sense ha he average value may be higher han he peak value [10]. (4) he exising fuzzy linear regression-back propagaion nework mehods seleced paricular fuzzy linear regression mehods, bu did no explain he reasons or compared wih oher fuzzy linear regression mehods.

4 Algorihms 2012, Mehodology he parameers used in he proposed mehodology are defined in advance. (1) b: learning consan. (2) a : normalized uni cos a period. (3) c : acual uni cos a period. (4) c : fuzzy uni cos forecas a period. c ( c1, c 2, c 3) if i is represened wih a riangular fuzzy number. (5) C: uni wafer cos. (6) G: gross die. (7) r(): homoscedasical, serially non-correlaed error erm. (8) : period. (9) : curren period. (10) Y : yield a period. (11) Y 0 : asympoic/final yield. Prior o predic he uni cos of a produc, we emphasize a he ouse ha he reducion in he uni cos follows a learning process, which is he assumpion of his sudy Fuzzy Linear Regression Mehods for Forecasing he Uni Cos According o Gruber [25], he yield of a produc follows a learning process: he uni cos can be calculaed as b / r( ) Y Y e (1) 0 c C /( Y G) C /( Y e G) C / Y Ge (2) b / r( ) b / r( ) 0 0 Obviously, he change in he uni cos is also a learning process, no a usual ime-series. Afer convering o logarihms, ln c ln C lny0 ln G b/ r( ) a b / r( ) (3) where a = lnc lny 0 lng. o consider he uncerainy in he uni cos, parameers in equaion (3) are given in asymmeric riangular fuzzy numbers as follows [26 29]: a ( a1, a 2, a 3 ) (4) herefore, b ( b, b 2, b 3 ) (5) ln c (ln c, ln c, ln c ) a( ) b / r( ) ( a b /, a b /, a b / ) r( ) (6) where (+) represens fuzzy addiion. Equaion (6) is obviously a fuzzy linear regression equaion. A fuzzy linear regression equaion can be fied in various ways. For example, in anaka and Waada [30], a linear programming problem is solved o minimize he fuzziness:

5 Algorihms 2012, subjec o Min Z (ln c3 ln c1) (7) 1 ln c ln c1 s(ln c 2 ln c1) (8) ln c ln c s(ln c ln c ) (9) c1 a1 b1 / (10) c2 a2 b2 / (11) c3 a3 b3 / (12) 0 a1 a2 a3 (13) 0 b1 b2 b3 (14) = 1 ~ (15) where s is he saisfacion level. For he raining daa, he acual values will fall wihin he ranges of he fuzzy forecass. Clearly, he higher value of s leads o wider fuzzy forecass. his model is indicaed wih W(s). he second mehod for fiing a fuzzy linear regression equaion is Peers mehod [31], in which he following linear programming problem is solved, aimed a he maximizaion of he average saisfacion level: Max Z s (16) subjec o (ln c3 ln c1) d (17) 1 ln c ln c1 s (ln c 2 ln c1) (18) ln c ln c3 s (ln c 2 ln c3) (19) s s (20) 1 c1 a / 1 b1 (21) c2 a / 2 b2 (22) c3 a / 3 b3 (23) 0 a a a (24) b b b (25) 0 s 1 (26) = 1 ~ (27)

6 Algorihms 2012, where d is he required range of a fuzzy forecas. Clearly, a larger value of d resuls in a higher average saisfacion level. his model is indicaed wih Peers(d). he hird mehod for fiing a fuzzy linear regression equaion is Donoso s quadraic non-possibilisic mehod [32], in which he quadraic error for boh he cenral endency and each one of he spreads is minimized: subjec o Min (28) Z k ( c c ) k (( c ln c ) ( c ln c ) ) ln c ln c1 s(ln c 2 ln c1) (29) ln c ln c s(ln c ln c ) (30) c1 a1 b1 / (31) c2 a2 b2 / (32) c3 a3 b3 / (33) 0 a1 a2 a3 (34) 0 b1 b2 b3 (35) = 1 ~ (36) where k 1 and k 2 belong o [0 1], and add up o 1. his model is indicaed wih Donoso(k 1, k 2, s). he fourh mehod for fiing a fuzzy linear regression equaion is Chen and Lin s nonlinear programming mehod [8], which changes he objecives and consrains in he wo linear programming models ino nonlinear: Model I subjec o o Min Z (ln c3 ln c1) (37) 1 ln c ln c1 s(ln c 2 ln c1) (38) Model II ln c ln c s(ln c ln c ) (39) c1 a1 b1 / (40) c2 a2 b2 / (41) c3 a3 b3 / (42) 0 a1 a2 a3 (43) 0 b1 b2 b3 (44) = 1 ~ (45) Max Z s (46) subjec o

7 Algorihms 2012, o o (ln c3 ln c1) d (47) 1 ln c ln c1 s (ln c 2 ln c1) (48) ln c ln c3 s (ln c 2 ln c3) (49) s m s m 1 (50) c1 a / 1 b1 (51) c2 a / 2 b2 (52) c3 a / 3 b3 (53) 0 a a a (54) b b b (55) 0 s 1 (56) = 1 ~ (57) where o reflecs he sensiiviy o he uncerainy of he fuzzy forecas; o ranges from 0 (no sensiive) o (exremely sensiive); s indicaes he required saisfacion level; 0 s 1; d is he desired range of every fuzzy forecas; 0 d ; m represens he relaive imporance of he ouliers in fiing he fuzzy linear regression equaion; m Z. When m = 1, he relaive imporance of he ouliers is he highes and is equal o ha of he non-ouliers. o should be se wihin [0 1] if he variaion in he variable is less han 1. Oherwise, i should be greaer han 1. he wo nonlinear programming models are indicaed wih CL1(o, s) and CL2(o, d, m), respecively Aggregaion of Fuzzy Forecass in Fuzzy Collaboraive Forecasing A mechanism is required o combine he fuzzy forecass. he aggregaion mechanism consiss of wo seps. In he firs sep, fuzzy inersecion is applied o aggregae he fuzzy forecass ino a polygon-shaped fuzzy number, in order o improve he precision of forecasing. Every fuzzy forecas conains he acual value. As a resul, he inersecion of he fuzzy forecass also conains he acual value. Besides, he inersecion has a narrower range han hose of he original regions. herefore, he forecasing precision measured in erms of he average range is indeed improved afer inersecion, which is one of he basic mechanisms of fuzzy collaboraive forecasing. Fuzzy inersecion combines n fuzzy forecass in he following manner: I ( c~ (1), c~ (2)..., c~ ( L)) ( x) min( c~ (1) ( x), c~ (2) ( x),..., c~ ( L) ( x)) (58) where I( c~ (1), ~ (2),..., ~ c c ( L) ) indicaes he resul of obaining he fuzzy inersecion of he fuzzy forecass by L expers. If hese fuzzy forecass are approximaed wih riangular fuzzy numbers, hen he fuzzy inersecion is a polygon-shaped fuzzy number (see Figure 1).

8 Algorihms 2012, Figure 1. he fuzzy inersecion of wo riangular fuzzy numbers. he resul of his sep is a polygon-shaped fuzzy number ha specifies he narrowes range of he fuzzy forecas. However, in pracical applicaions a crisp forecas is usually required. herefore, a crisp forecas has o be generaed from he polygon-shaped fuzzy number. For his purpose, a variey of defuzzificaion mehods are applicable [33]. Once he defuzzified value is obained i is compared wih he acual value o evaluae he accuracy. However, among he exising defuzzificaion mehods, no one mehod is beer han all he oher mehods in every case. In addiion, he mos suiable defuzzificaion mehod for a fuzzy variable is ofen chosen from he exising mehods, and hus he opimaliy of he chosen mehod canno be guaraneed. Also, he shape of he polygon-shaped fuzzy number is special. hese phenomena are reasons for proposing a ailored defuzzificaion mehod. In his sudy, a back propagaion nework is applied, because heoreically a well-rained back propagaion nework (wihou being suck in a local minima) wih a good seleced opology can successfully map any complex disribuion. he configuraion of he back propagaion nework used is esablished as follows: (1) Inpus: 2m parameers corresponding o he m corners of he polygon-shaped fuzzy number and he membership funcion values of hese corners. he reason is ha simple aggregaion resuls in a convex domain and each poin in i can be expressed wih he combinaion of corners. he fuzzy inersecion of L fuzzy forecass will have a mos 2 (2 L + 2) corners. All inpu parameers have o be normalized ino a range narrower han [0 1] before hey are fed ino he nework. (2) Single hidden layer: Generally one or wo hidden layers are more beneficial for he convergence propery of he back propagaion nework. (3) he number of neurons in he hidden layer is chosen from 1~4m according o a preliminary analysis, considering boh effeciveness (forecasing accuracy) and efficiency (execuion ime). (4) Oupu: he crisp forecas. (5) Nework learning rule: Dela rule. (6) Nework learning algorihms: here are many advanced algorihms for raining a back propagaion nework, e.g. he Flecher Reeves algorihm, he Broydon Flecher Goldfarb Shanno algorihm, he Levenberg Marquard algorihm, and he Bayesian regularizaion mehod [34]. In his sudy, he Levenberg Marquard algorihm is applied. c

9 Algorihms 2012, (7) Number of epochs per replicaion: 10,000. (8) Acivaion funcion: Log-sigmoid funcion. (9) Number of iniial condiions/replicaions: Because he performance of a back propagaion nework is sensiive o he iniial condiion, he raining process will be repeaed many imes wih differen iniial condiions ha are randomly generaed. Among he resuls, he bes one is chosen for he subsequen analyses. he Levenberg-Marquard algorihm was designed for raining wih second-order speed wihou having o compue he Hessian marix. I uses approximaion and updaes he nework parameers in a Newon-like way. When raining a back propagaion nework, he Hessian marix can be approximaed as: and he gradien can be compued as: H J J (59) g J e (60) where J is he Jacobian marix conaining he firs derivaives of he nework errors wih respec o he weighs and biases; e is he vecor of he nework errors. he Levenberg-Marquard algorihm uses his approximaion and updaes he nework parameers in a Newon-like way: x 1 ep 1 ] xep [ J J I J e (61) where ep is he epoch number. Newon s mehod is faser and more accurae near an error minimum, so he Levenberg-Marquard algorihm s purpose is o move as quickly as possible o Newon s mehod. hus, μ decreases afer each successful sep and increases only when a enaive sep will increase he performance funcion. Consequenly, he performance funcion is always reduced afer each epoch Performance Evaluaion in Fuzzy Collaboraive Forecasing Some performance measures of fuzzy collaboraive forecasing are defined as follows. Definiion 1. F(p) is a fuzzy forecasing mehod wih parameer p. he fuzzy forecas a period using F(p) is indicaed wih F ( p) ( F 1( p), F 2( p), F 3( p)). he precision and accuracy of F(p) are indicaed wih Prec(F(p)) and Accur(F(p)), respecively. Some of he common funcions for Prec(F(p)) and Accur(F(p)) are described below: (1) he average range (AR): Prec AR (F(p)) = ( F3( p) F1( p)) / (62) 1 (1) Mean absolue error (MAE): Accu MAE (F(p)) = D( F( p)) a / (63) where D() is he defuzzificaion funcion; a is he acual value a period. Mean absolue percenage error (MAPE): 1

10 Algorihms 2012, (2) Roo mean squared error (RMSE): D( F ( p)) a / a Accu MAPE (F(p)) = (64) 1 Accu RMSE (F(p)) = 2 ( D( F( p)) a) (65) 1 All of hese performance indicaors are as small as possible. Definiion 2. FCF(F, G) is a fuzzy collaboraive forecasing mehod on he basis of wo forecasing mehods F and G. he qualiy of collaboraion in he precision and accuracy are indicaed wih QoCp(FCF) and QoCa(FCF), respecively. Some of he common funcions for QoCp(FCF) and QoCa(FCF) are described below: (1) Maximum percenage improvemen (MPI): QoCp MPI (FCF) = QoCa MPI (FCF) = (2) Average percenage improvemen (API): QoCp API (FCF) = QoCa API (FCF) = Prec( F) Prec( FCF) Prec( G) Prec( FCF) Max(, ) 100% Prec( F) Prec( G) (66) Accu( F) Accu( FCF) Accu( G) Accu( FCF) Max(, ) 100% Accu( F) Accu( G) (67) Prec( F) Prec( FCF) Prec( G) Prec( FCF) ( ) / 2 100% Prec( F) Prec( G) (68) Accu( F) Accu( FCF) Accu( G) Accu( FCF) ( ) / 2 100% Accu( F) Accu( G) (69) hese funcions can easily be exended o involve more han wo objecs Some Fuzzy Collaboraive Forecasing Models for he Uni Cos Forecasing An example is given in able 1. he daa of he firs 7 periods were used as raining daa, and he remaining daa are lef for esing. able 1. An example. c (US$)

11 Algorihms 2012, Some linear fuzzy collaboraion forecasing models for used o predic he uni cos. Model 1. FCF(W(s 1 ), W(s 2 )) In his model, he wo objecs use he same fuzzy linear regression mehod (W(s)), bu wih differen values of s o predic he uni cos. In W(s), he mos precise forecas is associaed wih he minimum value of s ha saisfies he consrains. In addiion, he resuls when s is large ofen conain he resuls when s is relaively small, which makes i less effecive for heir collaboraion. Neverheless, he fuzzy collaboraive forecasing mehod, o a cerain exen, improves he precision of W(s). In he previous example, assuming he s values specified by he objecs are 0.3 and 0.6, respecively: Prec AR (W(0.3)) = 0.56 Prec AR (W(0.6)) = 1.14 Prec AR (FCF(W(0.3), W(0.6))) = 0.53 herefore, he qualiy of collaboraion wih respec o he forecasing precision can be evaluaed as QoCp MPI,AR (FCF(W(0.3), W(0.6))) = max(( )/1.14, ( )/0.56) = 54%. QoCp API,AR (FCF(W(0.3), W(0.6))) = (( )/ ( )/0.56)/2 = 29%. In order o evaluae he forecasing accuracy, he forecass by he wo objecs are defuzzified using he cener of graviy (COG) mehod, and hen are compared wih he acual values: Accu MAE (W(0.3)) = 0.16 Accu MAE (W(0.6)) = 0.24 Accu MAPE (W(0.3)) = 10% Accu MAPE (W(0.6)) = 15% Accu RMSE (W(0.3)) = 0.19 Accu RMSE (W(0.6)) = 0.31 while in he fuzzy collaboraive forecasing mehod, he forecass by he wo objecs are aggregaed using he fuzzy inersecion and back propagaion nework approach o generae a single crisp value: Accu MAE (FCF(W(0.3), W(0.6))) = 0.06 Accu MAPE (FCF(W(0.3), W(0.6))) = 4% Accu RMSE (FCF(W(0.3), W(0.6))) = 0.10 herefore, he qualiy of collaboraion wih respec ohe forecasing accuracy can be evaluaed as QoCa MPI,MAE (FCF(W(0.3), W(0.6))) = max(( )/0.16, ( )/0.24) = 71%. QoCa API,MAE (FCF(W(0.3), W(0.6))) = (( )/ ( )/0.24)/2 = 64%. QoCa MPI,MAPE (FCF(W(0.3), W(0.6))) = max((10% 5%)/10%, (15% 5%)/15%) = 67%. QoCa API,MAPE (FCF(W(0.3), W(0.6))) = ((10% 5%)/10% + (15% 5%)/15%)/2 = 58%. QoCa MPI,RMSE (FCF(W(0.3), W(0.6))) = max(( )/0.19, ( )/0.31) = 52%. QoCa API,RMSE (FCF(W(0.3), W(0.6))) = (( )/ ( )/0.31)/2 = 36%. Model 2. FCF(Peers(d 1 ), Peers(d 2 )) In his model, boh objecs use Peers(d), bu wih differen d values o predic he uni cos. In Peers(d), he mos precise forecas is associaed wih he minimum value of d ha saisfies he consrains. In addiion, he resuls when d is large ofen conain he resuls when d is relaively small. As a resul, he benefis of collaboraion are no obvious. In he previous example, assuming he d values specified by he objecs are 0.3 and 0.5, respecively. he forecasing performances of he wo objecs are evaluaed as

12 Algorihms 2012, Prec AR (Peers(0.3)) = 0.48 Prec AR (Peers(0.5)) = 0.68 Accu MAE (Peers(0.3)) = 0.16 Accu MAE (Peers(0.5)) = 0.20 Accu MAPE (Peers(0.3)) = 10% Accu MAPE (Peers(0.5)) = 12% Accu RMSE (Peers(0.3)) = 0.19 Accu RMSE (Peers(0.5)) = 0.25 Afer collaboraion, he forecasing precision and accuracy are boh improved: Prec AR (FCF(Peers(0.3), Peers(0.5))) = 0.48 Accu MAE (FCF(Peers(0.3), Peers(0.5))) = 0.07 Accu MAPE (FCF(Peers(0.3), Peers(0.5))) = 6% Accu RMSE (FCF(Peers(0.3), Peers(0.5))) = 0.13 he qualiy of collaboraion in he wo aspecs can be evaluaed as QoCp MPI,AR (FCF(Peers(0.3), Peers(0.5))) = 29%. QoCp API,AR (FCF(Peers(0.3), Peers(0.5))) = 15%. and QoCa MPI,MAE (FCF(Peers(0.3), Peers(0.5))) = 65%. QoCa API,MAE (FCF(Peers(0.3), Peers(0.5))) = 61%. QoCa MPI,MAPE (FCF(Peers(0.3), Peers(0.5))) = 50%. QoCa API,MAPE (FCF(Peers(0.3), Peers(0.5))) = 45%. QoCa MPI,RMSE (FCF(Peers(0.3), Peers(0.5))) = 48%. QoCa API,RMSE (FCF(Peers(0.3), Peers(0.5))) = 40%. respecively. Model 3. FCF(Donoso(k 11, k 21, s 1 ), Donoso(k 12, k 22, s 2 )) In his model, boh objecs use Donoso(k 1, k 2, s), bu wih differen parameer values o predic he uni cos. his mehod has more parameers ha can be adjused, so here is a greaer degree of freedom, which provides a space for coordinaion. Assuming in he previous example, he parameer values specified by he wo objecs are (k 11, k 21, s 1 ) = (0.2, 0.8, 0.2) (k 12, k 22, s 2 ) = (0.7, 0.3, 0.3) hen heir forecasing performances are Prec AR (Donoso(0.2, 0.8, 0.2)) = 0.48 Prec AR (Donoso(0.7, 0.3, 0.3)) = 0.57 Accu MAE (Donoso(0.2, 0.8, 0.2)) = 0.15 Accu MAE (Donoso(0.7, 0.3, 0.3)) = 0.14 Accu MAPE (Donoso(0.2, 0.8, 0.2)) = 9% Accu MAPE (Donoso(0.7, 0.3, 0.3)) = 9% Accu RMSE (Donoso(0.2, 0.8, 0.2)) = 0.17 Accu RMSE (Donoso(0.7, 0.3, 0.3)) = 0.18 Comparaively, he forecasing performance of he fuzzy collaboraive forecasing mehod is

13 Algorihms 2012, Prec AR (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3))) = 0.48 Accu MAE (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3))) = 0.07 Accu MAPE (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3))) = 5% Accu RMSE (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3))) = 0.10 herefore, he qualiy of collaboraion can be evaluaed as QoCp MPI,AR (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3)))) = 16%. QoCp API,AR (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3)))) = 8%. QoCa MPI,MAE (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3)))) = 53%. QoCa API,MAE (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3)))) = 52%. QoCa MPI,MAPE (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3)))) = 44%. QoCa API,MAPE (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3)))) = 44%. QoCa MPI,RMSE (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3)))) = 44%. QoCa API,RMSE (FCF(Donoso(0.2, 0.8, 0.2), Donoso(0.7, 0.3, 0.3)))) = 43%. I is worh noing ha he performance of his collaboraion model is no as good as expeced. Model 4. FCF(CL1(o 1, s 1 ), CL1(o 2, s 2 )) In his model, he wo objecs use he same mehod CL1(o, s), bu wih differen values of o and s o predic he uni cos. CL1(o, s) is an exension of W(s) by considering a nonlinear objecive funcion insead. o make a comparison, he parameer values of he wo objecs are se o (o 1, s 1 ) = (3, 0.3) (o 2, s 2 ) = (2, 0.6) he s values are he same wih hose in he original W(s) mehods. Afer forecasing he uni cos, he performances of he wo objecs are evaluaed as Prec AR (CL1(3, 0.3)) = 0.56 Prec AR (CL1(2, 0.6)) = 1.12 Accu MAE (CL1(3, 0.3)) = 0.16 Accu MAE (CL1(2, 0.6)) = 0.23 Accu MAPE (CL1(3, 0.3)) = 10% Accu MAPE (CL1(2, 0.6))) = 14% Accu RMSE (CL1(3, 0.3)) = 0.19 Accu RMSE (CL1(2, 0.6)) = 0.30 hen, he forecasing performances are compared wih hose of he linear mehods W(0.3) and W(0.6). he resuls are summarized in able 2. able 2. Comparison of he performances of CL1(o, s) and W(s). Prec AR Accu MAE Accu MAPE Accu RMSE W(0.3) CL1(3, 0.3) W(0.6) CL1(2, 0.6) Obviously, he use of a nonlinear objecive funcion may change he opimal soluion. he qualiy of collaboraion is evaluaed as follows:

14 Algorihms 2012, QoCp MPI,AR (FCF(CL1(3, 0.3), CL1(2, 0.6))) = 50%. QoCp API,AR (FCF(CL1(3, 0.3), CL1(2, 0.6))) = 25%. QoCa MPI,MAE (FCF(CL1(3, 0.3), CL1(2, 0.6))) = 71%. QoCa API,MAE (FCF(CL1(3, 0.3), CL1(2, 0.6))) = 65%. QoCa MPI,MAPE (FCF(CL1(3, 0.3), CL1(2, 0.6))) = 67%. QoCa API,MAPE (FCF(CL1(3, 0.3), CL1(2, 0.6))) = 60%. QoCa MPI,RMSE (FCF(CL1(3, 0.3), CL1(2, 0.6))) = 66%. QoCa API,RMSE (FCF(CL1(3, 0.3), CL1(2, 0.6))) = 57%. hen, he qualiy of collaboraion in FCF(CL1(3, 0.3), CL1(2, 0.6)) is compared wih ha in FCF(W(0.3), W(0.6)). he resuls are shown in able 3. able 3. Comparison of FCF(W(0.3), W(0.6)) and FCF(CL1(3, 0.3), CL1(2, 0.6)). FCF(W(0.3), W(0.6)) FCF(CL1(3, 0.3), CL1(2, 0.6)) QoCp MPI,AR 54% 50% QoCp API,AR 29% 25% QoCa MPI,MAE 71% 71% QoCa API,MAE 64% 65% QoCa MPI,MAPE 67% 67% QoCa API,MAPE 58% 60% QoCa MPI,RMSE 52% 66% QoCa API,RMSE 36% 57% As can be seen from his able, he use of a nonlinear model CL1(o, s) insead of he linear model W(s) can indeed achieve a beer qualiy of collaboraion, especially in he forecasing accuracy. Model 5. FCF(CL2(o 1, d 1, m 1 ), CL2(o 2, d 2, m 2 )) his model assumes ha boh of he wo objecs use CL2(o, d, m), bu wih differen values o predic he uni cos (o 1, d 1, m 1 ) = (3, 0.3, 2) (o 2, d 2, m 2 ) = (2, 0.5, 3) he performances of he wo objecs are evaluaed as Prec AR (CL2(3, 0.3, 2)) = 0.48 Prec AR (CL2(2, 0.5, 3)) = 0.66 Accu MAE (CL2(3, 0.3, 2)) = 0.16 Accu MAE (CL2(2, 0.5, 3)) = 0.16 Accu MAPE (CL2(3, 0.3, 2)) = 10% Accu MAPE (CL2(2, 0.5, 3))) = 10% Accu RMSE (CL2(3, 0.3, 2)) = 0.19 Accu RMSE (CL2(2, 0.5, 3)) = 0.20 hrough he collaboraion of he wo objecs, FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3)) achieves a beer forecasing performance: Prec AR (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 0.35 Accu MAE (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 0.05

15 Algorihms 2012, Accu MAPE (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 4% Accu RMSE (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 0.09 he qualiy of collaboraion is assessed as follows: QoCp MPI,AR (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 46%. QoCp API,AR (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 37%. QoCa MPI,MAE (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 69%. QoCa API,MAE (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 68%. QoCa MPI,MAPE (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 63%. QoCa API,MAPE (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 63%. QoCa MPI,RMSE (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 54%. QoCa API,RMSE (FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3))) = 53%. Model 6. FCF(CL1(o 1, s 1 ), CL2(o 2, d 2, m 2 )) In his model, one of he wo objecs uses CL1(o, s), and he oher uses CL2(o, d, m). (o 1, s 1 ) = (3, 0.3) (o 2, d 2, m 2 ) = (2, 0.5, 3) he forecasing performance of FCF(CL1(3, 0.3), CL2(2, 0.5, 3)) is evaluaed as Prec AR (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 0.39 Accu MAE (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 0.07 Accu MAPE (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 5% Accu RMSE (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 0.11 he qualiy of collaboraion is assessed as follows: QoCp MPI,AR (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 41%. QoCp API,AR (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 30%. QoCa MPI,MAE (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 59%. QoCa API,MAE (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 59%. QoCa MPI,MAPE (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 52%. QoCa API,MAPE (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 52%. QoCa MPI,RMSE (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 46%. QoCa API,RMSE (FCF(CL1(3, 0.3), CL2(2, 0.5, 3))) = 45% Comparison of he Performances of he Fuzzy Collaboraive Forecasing Models In his secion, he performances of he fuzzy collaboraive forecasing models are compared. Firs, he forecasing accuracy considering he average range of forecass, he performances of differen models are compared in Figure 2. Obviously, in erms of he forecasing accuracy, FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3)) is much beer han he oher models. However, his is parly because he seings of he parameers in he models are differen and subjecive. In order o confirm he effecs of differen models for he forecasing accuracy, he qualiy of collaboraion, especially QoCp API,AR, is considered o be a beer indicaor. he comparison resuls are shown in Figure 3. As can be seen from his figure, FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3)) is indeed he mos precise fuzzy collaboraive forecasing model in his case.

16 FCF(W(0.3),W(0.6)) FCF(Peers(0.3),Peers(0.5)) FCF(Donoso(0.2,0.8,0.2 ),Donoso(0.7,0.3,0.3)) FCF(CL1(3,0.3),CL1(2, 0.6)) FCF(CL2(3,0.3,2),CL2( 2,0.5,3)) FCF(CL1(3, 0.3), CL2(2, 0.5, 3)) FCF(W(0.3),W(0.6)) FCF(Peers(0.3),Peers(0.5)) FCF(Donoso(0.2,0.8,0.2 ),Donoso(0.7,0.3,0.3)) FCF(CL1(3,0.3),CL1(2, 0.6)) FCF(CL2(3,0.3,2),CL2( 2,0.5,3)) FCF(CL1(3, 0.3), CL2(2, 0.5, 3)) Prec AR (US$) Algorihms 2012, Figure 2. he forecasing accuracy of he fuzzy collaboraive forecasing models Secondly, in order o compare he forecasing accuracy of he models, hree indicaors MAE, MAPE, and RMSE are considered. he comparison resuls in he hree indicaors are shown in Figures 4-6, respecively. For all indicaors of he forecasing accuracy, FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3)) is he bes fuzzy collaboraive forecasing model. Nex, he qualiy of collaboraion is compared, and he resuls are shown in Figure 7. he qualiy of collaboraion in FCF(CL1(3, 0.3), CL1(2, 0.6)) is he bes if RMSE is aken ino accoun, and FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3)) achieves he highes qualiy of collaboraion if MAE or MAPE is considered. Figure 3. he qualiy of collaboraion of he fuzzy collaboraive forecasing models. QoCp API,AR 40% 35% 30% 25% 20% 15% 10% 5% 0%

17 FCF(W(0.3),W(0.6)) FCF(Peers(0.3),Peers(0.5)) FCF(Donoso(0.2,0.8,0.2),Donoso(0.7,0.3,0.3)) FCF(CL1(3,0.3),CL1(2,0.6)) FCF(CL2(3,0.3,2),CL2(2,0.5,3)) FCF(CL1(3, 0.3), CL2(2, 0.5, 3)) RMSE (US$) FCF(W(0.3),W(0.6)) FCF(Peers(0.3),Peers(0.5)) FCF(Donoso(0.2,0.8,0.2),Donoso(0.7,0.3,0.3)) FCF(CL1(3,0.3),CL1(2,0.6)) FCF(CL2(3,0.3,2),CL2(2,0.5,3)) FCF(CL1(3, 0.3), CL2(2, 0.5, 3)) MAPE FCF(W(0.3),W(0.6)) FCF(Peers(0.3),Peers(0.5)) FCF(Donoso(0.2,0.8,0.2),Donoso(0.7,0.3,0.3)) FCF(CL1(3,0.3),CL1(2, 0.6)) FCF(CL2(3,0.3,2),CL2( 2,0.5,3)) FCF(CL1(3, 0.3), CL2(2, 0.5, 3)) MAE (US$) Algorihms 2012, Figure 4. he forecasing accuracy (MAE) of he fuzzy collaboraive forecasing models Figure 5. he forecasing accuracy (MAPE) of he fuzzy collaboraive forecasing models. 7% 6% 5% 4% 3% 2% 1% 0% Figure 6. he forecasing accuracy (RMSE) of he fuzzy collaboraive forecasing models

18 FCF(W(0.3),W(0.6)) FCF(Peers(0.3),Peers(0.5)) FCF(Donoso(0.2,0.8,0.2 ),Donoso(0.7,0.3,0.3)) FCF(CL1(3,0.3),CL1(2, 0.6)) FCF(CL2(3,0.3,2),CL2( 2,0.5,3)) FCF(CL1(3, 0.3), CL2(2, 0.5, 3)) Algorihms 2012, Figure 7. he qualiy of collaboraion of he fuzzy collaboraive forecasing models. 80% 70% 60% 50% 40% 30% 20% 10% 0% QoCa_API_MAE QoCa_API_MAPE QoCa_API_RMSE 4. Conclusions Forecasing he uni cos of every produc ype in a facory is an imporan ask. Afer he uni cos of every produc ype in a facory is accuraely forecased, several managerial goals (including pricing, cos down projecing, capaciy planning, ordering decision suppor, and guiding subsequen operaions) can be simulaneously achieved. However, i is no easy o deal wih uncerainy in he uni cos. his paper presens some fuzzy collaboraive forecasing models based on a few well-known fuzzy linear regression mehods o predic he uni cos of a produc. An example is used o illusrae he applicabiliy of he proposed mehodology. According o he experimenal resuls, (1) he effeciveness of he uni cos forecasing was grealy improved hrough he collaboraion of he expers, especially when using FCF(CL2(o 1, d 1, m 1 ), CL2(o 2, d 2, m 2 )). (2) Wih respec o he qualiy of collaboraion on he forecasing precision, only one performance measure is proposed and he proposed performance measure can effecively compare he differences among he models. (3) Wih respec o he forecasing accuracy on he forecasing accuracy among he performance measures, he one ha considers MAPE can effecively compare he differences among he models. he conribuion of his sudy includes he following: (1) Six fuzzy collaboraive forecasing models for he uni cos forecasing are invesigaed. From his, he mos effecive one can be idenified. (2) More performance measures on he qualiy of collaboraion have been proposed. Acknowledgemens his work is parially suppored by Naional Science Council of aiwan.

19 Algorihms 2012, References 1. Chen,. A fuzzy-neural knowledge-based sysem for job compleion ime predicion and inernal due dae assignmen in a wafer fabricaion plan. In. J. Sys. Sci. 2009, 40, Chen,. A hybrid fuzzy and neural approach wih virual expers and parial consensus for DRAM price forecasing. In. J. Innov. Compu. Inf. Conrol 2012, 8, Chen,. An online collaboraive semiconducor yield forecasing sysem. Exper Sys. Appl. 2009, 36, Chen,. An applicaion of fuzzy collaboraive inelligence o uni cos forecasing wih parial daa access by securiy consideraion. In. J. echnol. Inel. Plan. 2011, 7, Chen,. Applying a fuzzy and neural approach for forecasing he foreign exchange rae. In. J. Fuzzy Sys. Appl. 2011, 1, Chen,. Applying he hybrid fuzzy c means-back propagaion nework approach o forecas he effecive cos per die of a semiconducor produc. Compu. Ind. Eng. 2011, 61, Chen,. Some linear fuzzy collaboraive forecasing models for semiconducor uni cos forecasing. In. J. of Fuzzy Sys. Appl. 2012, in press. 8. Chen,.; Lin, Y.C. A fuzzy-neural sysem incorporaing unequally imporan exper opinions for semiconducor yield forecasing. In. J. Uncerain. Fuzziness Knowl. Based Sys. 2008, 16, Chen,.; Lin, C.W.; Wang, Y.C. A fuzzy collaboraive forecasing approach for WIP level esimaion in a wafer fabricaion facory. Appl. Mah. Inf. Sci. 2012, in press. 10. Chen,.; Wang, Y.C. A fuzzy-neural approach for global CO 2 concenraion forecasing. Inel. Daa Anal. 2011, 15, Chen,.; Wang, Y.C. A hybrid fuzzy and neural approach for forecasing he book-o-bill raio in he semiconducor manufacuring indusry. In. J. Adv. Manuf. echnol. 2011, 54, Pedrycz, W. Collaboraive archiecures of fuzzy modeling. Lec. Noes Compu. Sci. 2008, 5050, Pedrycz, W. Collaboraive fuzzy clusering. Paern Recogni. Le. 2002, 23, Pedrycz, W.; Rai, P. A mulifaceed perspecive a daa analysis: A sudy in collaboraive inelligen agens. IEEE rans. Sys., Man. Cybern., Par B: Cybern. 2008, 38, Carnes, R. Long-erm cos of ownership: beyond purchase price. In Proceedings of 1991 IEEE/SEMI Inernaional Semiconducor Science Symposium, Burlingame, CA, USA, May 1991; pp Wood, S.C. Cos and cycle ime performance of fabs based on inegraed single-wafer processing. IEEE rans. Semi. Manuf. 1997, 10, Pfizner, L.; Benesch, N.; Öchsner, R.; Schmid, C.; Schneider, C.; schafary,.; runk, R.; Dudenhausen, H.M. Cos reducion sraegies for wafer expendiure. Microelecr. Eng. 2001, 56, Shai, O.; Reich, Y. Infused design I. heory. Res. Eng. Design Res. 2004, 15, Shai, O.; Reich, Y. Infused design II. Pracice. Res. Eng. Design Res. 2004, 15,

20 Algorihms 2012, Büyüközkan, G.; Vardaloglu, Z. Analyzing of Collaboraive Planning, Forecasing and Replenishmen Approachusing Fuzzy Cogniive Map. In Proceeding of 2009 Inernaional Conference on Compuers and Indusrial Engineering, Universiy of echnology of royes, France, 6 8 July 2009; pp Büyüközkan, G.; Feyzioglu, O.; Vardaloglu, Z. Analyzing CPFR supporing facors wih fuzzy cogniive map approach. World Academy Sci. Eng. echnol. 2009, 31, Poler, R.; Hernandez, J.E.; Mula, J.; Lario, F.C. Collaboraive forecasing in neworked manufacuring enerprises. J. Manuf. echnol. Manag. 2008, 19, Osrosi, E.; Haxhiaj, L.; Fukuda, S. Fuzzy modelling of consensus during design conflic resoluion. Res. Eng. Design 2012, 23(1), Cheikhrouhou, N.; Marmier, F.; Ayadi, O.; Wieser, P. A collaboraive demand forecasing process wih even-based fuzzy judgemens. Compu. Ind. Eng. 2011, 61, Gruber, H. Learning and Sraegic Produc Innovaion: heory and Evidence for he Semiconducor Indusry; Elsevier: Amserdam, he Neherlands, Chen,. A fuzzy back propagaion nework for oupu ime predicion in a wafer fab. Appl. Sof Compu. 2003, 2, Chen,. A hybrid fuzzy and neural approach for evaluaing he cos compeiiveness of a semiconducor produc. In. J. echnol. Inel. Plan. 2010, 6, Chen,. A SOM-FBPN-ensemble approach wih error feedback o adjus classificaion for wafer-lo compleion ime predicion. In. J. Adv. Manuf. echnol. 2008, 37, Chen,.; Wang, M.J.J. A fuzzy se approach for yield learning modeling in wafer manufacuring. IEEE rans. Semicond. Manuf. 1999, 12, anaka, H.; Waada, J. Possibilisic linear sysems and heir applicaion o he linear regression model. Fuzzy Ses Sys. 1988, 272, Peers, G. Fuzzy linear regression wih fuzzy inervals. Fuzzy Ses Sys. 1994, 63, Donoso, S.; Marin, N.; Vila, M.A. Quadraic programming models for fuzzy regression. In Proceedings of Inernaional Conference on Mahemaical and Saisical Modeling in Honor of Enrique Casillo, Ciudad Real, Spain, June Liu, X. Parameerized defuzzificaion wih maximum enropy weighing funcion anoher view of he weighing funcion expecaion mehod. Mah. Com. Mod. 2007, 45, Eraslan, E. he esimaion of produc sandard ime by arificial neural neworks in he molding indusry. Mah. Probl. Eng. 2009, aricle ID by he auhors; licensee MDPI, Basel, Swizerland. his aricle is an open access aricle disribued under he erms and condiions of he Creaive Commons Aribuion license (hp://creaivecommons.org/licenses/by/3.0/).