An advanced fuzzy collaborative intelligence approach for fitting the uncertain unit cost learning process

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1 hps://doi.org/0.007/s ORIGINAL ARTICLE An advanced fuzzy collaboraive inelligence approach for fiing he uncerain uni cos learning process Yu-Cheng Lin Toly Chen Received: 8 May 08 / Acceped: Augus 08 The Auhor(s) 08 Absrac Esimaing he uni cos of each produc precisely and accuraely is a prerequisie o deermining he profiabiliy of a manufacurer, which is usually addressed by fiing he underlying learning process. However, exising mehods for his purpose ofen deal wih a logarihmic or log-sigmoid value, raher han he original value, of he uni cos. To resolve his problem, in his sudy, a new fuzzy collaboraive inelligence (FCI) approach is proposed by considering he original value of he uni cos direcly. The effeciveness of he new FCI approach is validaed wih a real dynamic random access memory (DRAM) case. The experimenal resuls showed ha he new FCI approach ouperformed wo exising mehods in improving he fiing accuracy in erms of MAE and MAPE and also in reducing he average range of he fied uni coss. Keywords Uni cos Learning process Fuzzy collaboraive inelligence Inroducion The uni cos of each produc is a criical performance measure for a facory []. Compared wih oher performance measures such as yield and produciviy, he uni cos is special for he following reasons:. Since he profi is derived by subracing he price by he uni cos, a lower uni cos immediaely increases he profi.. The uni cos is boh a direc performance measure and an indirec performance measure. Before producion, a reducion in he prices of raw maerials is direcly refleced in he uni cos of he finished goods. Afer producion, a higher yield resuls in more oupu ha drives down he uni cos furher. In addiion, esimaing he uni cos of each produc accuraely is a prerequisie o deermining he profiabiliy of a B Toly Chen olychen@ms7.hine.ne Deparmen of Compuer-Aided Indusrial Design, Overseas Chinese Universiy, Taichung Ciy, Taiwan Deparmen of Indusrial Engineering and Managemen, Naional Chiao Tung Universiy, Universiy Road, Hsinchu 00, Taiwan manufacurer. However, owing o he prevalen cos down philosophy, many manufacurers incorrecly expec ha he uni cos of a produc can be reduced in a linear way. This is a misundersanding because if a linear cos-down works ou, he uni cos will evenually become negaive. For his reason, fiing he reducion in he uni cos of a produc wih a learning process has become a common pracice [8]. The moivaion for his sudy is explained as follows. A problem of he exising mehods is ha hey ofen deal wih he logarihmic or log-sigmoid value, raher han he original value, of he uni cos, as shown in Table. The reason for such a reamen is o simplify he required compuaion or o uilize an exising raining algorihm. However, in his way, he esimaing accuracy or precision wih respec o he original value of he uni cos has no ruly been opimized. To resolve his problem, a new fuzzy collaboraive inelligence (FCI) approach ha can deal wih he original value of he uni cos is proposed in his sudy. This is a homogeneous FCI mehod because all models are buil by solving he same, i.e., mahemaical programming (MP), problems. In he proposed new FCI approach, he logarihmic funcion is approximaed wih a polynomial funcion. Then, he wo nonlinear programming (NLP) models formulaed by Chen [0] are convered ino equivalen polynomial programming (PP) models. Subsequenly, each exper applies eiher PP model o esimae he uni cos of a produc. Finally, he fuzzy uni cos esimaes from he expers are aggregaed

2 Table Types of he uni-cos value in exising uni cos esimaion mehods Type of he uni-cos value Uni cos esimaion mehod Logarihmic value Tsuchiya and Kobayashi [8], Chen [0], Thompson [7], Chen and Chiu [] Log-sigmoid value Cavalieri e al. [5], Chen [] Original value This sudy using he fuzzy inersecion (FI)-back propagaion nework (BPN) approach proposed by Chen and Lin [9]. The effeciveness of he proposed mehodology is validaed wih a real case. The differences beween he proposed mehodology and some exising mehods are summarized in Table, showing he originaliy of he proposed mehodology. The conribuion of his sudy is he designing of he PP formulaions ha can deal wih he original value of he uni cos in modeling he uncerain uni cos learning process, which is he firs aemp in his field and has grea poenial of improving he precision and accuracy of esimaing he uni cos. The remainder of his paper is organized as follows. Secion is dedicaed o he lieraure review. In Sec., he models and seps of he proposed new FCI approach are deailed. In Sec., a real dynamic random access memory (DRAM) case is adoped o validae he effeciveness of he new FCI approach. Two exising FCI mehods are also applied o he real case for a comparison. Finally, in Sec. 4, some conclusions are drawn, from which several direcions for fuure invesigaion are also given. Previous work Some of he relaed lieraure on he uni cos esimaion is reviewed as follows. Esimaing he uni cos of a produc is no an easy ask in many indusries. Cohen e al. [5] analyzed he daa of cancellaion coss, holding coss, and delay coss of suppliers in a semiconducor equipmen supply chain. The colleced daa indicaed ha a supplier perceived he cancellaion coss o be abou wo imes higher han he delay coss, and he holding coss o be abou hree imes higher han he delay coss. Such percepions may be misleading and undermine he effeciveness of a uni cos esimaion mechanism. The uni cos is deermined by yield ha is subjec o significan uncerainy caused by human inervenion. To address his, several sudies esimaed he uni cos of a produc wih a fuzzy value. The maximum (minimum) of he fuzzy value indicaes he highes (lowes) possible value of he uni cos, which happens when he yield is a he lowes (highes) level [7]. For example, in Wu e al. [0], he qualiy of raw maerials from suppliers was unsable and was modeled wih a rapezoidal fuzzy number (TrFN). As a resul, he uni cos became a TrFN as well. Based on ha, a fuzzy muli-objecive programming model was esablished for supplier selecion and risk modeling. Chen [0] modeled he uni cos of a DRAM produc wih a riangular fuzzy number (TFN). Lee e al. [0] observed ha he cos per bi of DRAM has been decreasing a a rapid rae wih advances in DRAM processing echnologies ha scale more DRAM cells ino he same die area. Many sudies on producion planning considered his fac by incorporaing in fuzzy uni cos esimaes. For example, Vijayan and Kumaran [9] minimized he oal invenory coss when he uni coss were esimaed wih fuzzy numbers. In Zhao e al. [], he uni cos of each produc was modeled as a fuzzy number, based on which he pricing decisions for subsiuable producs could be made. In addiion o modeling uncerain learning processes, fuzzy logic has been exensively applied o designing conrollers in a noisy environmen [, 4, 6,, 6]. In he pas, he applicaion of fuzzy logic o he uni cos esimaion did no receive much aenion in he manufacuring secor []. However, he grea poenial has been menioned by sudies like Chang e al. [8]. Recenly, some sudies proposed fuzzy mehods, especially FCI mehods, o esimae he uni cos of a produc. The FCI mehods gaher several expers ha apply fuzzy mehods o esimae he uni cos of a produc [4]. Then, he fuzzy uni cos esimaes from he expers are aggregaed [5]. For example, Chen Table The differences beween he proposed mehodology and some exising mehods Mehod Cohen e al. [5] Uncerainy considered Collaboraion involved Type of he uni-cos value Esimaion mechanism No No Original value Saisical analyses Collaboraion mechanism Chen [0] Yes Yes Logarihmic MLP FI-BPN value Lee e al. [0] Yes No Original value Saisical analyses The proposed mehodology Yes Yes Original value PP FI-BPN

3 Table Advanages and disadvanages of wo ypes of FCI mehods FCI mehods Advanages Disadvanages Exper-based Agen-based The parameer seing is meaningful o he pracice I is no necessary o gaher expers The consensus among agens is guaraneed I is efficien o reach a consensus I is no easy o gaher enough expers ha are available I is ime-consuming o reach a consensus among expers A consensus is no guaraneed Parameers are arbirarily specified and may no be meaningful o he pracice [0] proposed an FCI mehod in which expers fied fuzzy linear regression (FLR) equaions o esimae he effecive cos per die of a semiconducor produc. Then, he fuzzy uni cos esimaes from he expers were aggregaed using he FI- BPN approach. Hsiao e al. [8] applied fuzzy neural nework (FNN) and geneic algorihm joinly o esimaing he hookup consrucion cos of a semiconducor fabricaion plan. Chen [] proposed anoher FCI mehod in which expers configured FNNs o esimae he uni cos and hen a radial basis funcion (RBF) nework was used o aggregae he fuzzy uni cos esimaes. Recenly, he FCI mehod proposed by Chen and Chiu [] used sofware agens o quicken he collaboraion process. In sum, he exising FCI mehods can be divided ino exper-based and agen-based ypes depending on wheher real expers or sofware agens are involved. The advanages and disadvanages of he wo ypes of FCI mehods are summarized in Table. In addiion, an FCI mehod can be homogeneous if he expers apply he same ypes of fuzzy mehods, or heerogeneous if expers apply differen ypes of fuzzy mehods. As a consequence, heoreically, he exising mehods can be classified ino four caegories, as shown in Fig.. Mos of he exising FCI mehods, e.g., Chen [0], Chen [], and Chen and Chiu [], fall ino he Type I caegory. The proposed mehodology also belongs o he Type I caegory. The new FCI approach A fuzzy uni cos learning model Wihou loss of generaliy, all fuzzy variables or parameers in he new FCI approach are given in TFNs ha are defined as follows. Definiion (TFN). A TFN Ã (A, A, A ) is a fuzzy number wih he following membership funcion: ( ( ) ( )) x A x A μã(x) min max, 0, max, 0. A A A A () A fuzzy uni cos learning model can be described as []: C C 0 e b, () where C (C, C, C ) is he fuzzy uni cos esimae wihin ime period ; 0 C ; ~T. C 0 (C 0, C 0, C 0 ) is he asympoic (or final) uni cos; 0 C 0. b (b, b, b ) is he learning consan; b 0. According o he arihmeic on TFNs [], C C0 e b, () C C0 e b, (4) C C0 e b, (5) Convering he parameers on boh sides of () (5) o heir logarihmic values gives: ln C ln C0 + b /, (6) ln C ln C0 + b /, (7) ln C ln C0 + b /, (8) Chen s FCI mehod Chen [0] proposed an FCI mehod for DRAM uni cos esimaion. In he FCI mehod, each exper formulaes eiher of he following NLP models o esimae he uni cos of a DRAM produc: (Model NLPI) Min Z NLPI subjec o (ln C ln C ) o(k), (9) ln C ln C + s(k)(ln C ln C ), (0) ln C ln C + s(k)(ln C ln C ), () ln C ln C 0 + b /, () ln C ln C 0 + b /, () ln C ln C 0 + b /, (4) ln C 0 ln C 0 ln C 0, (5) 0 b b b, (6) T

4 Fig. Caegories of exising FCI mehods Paricipans Fuzzy Mehods Applied Homogeneous Heerogeneous Real Expers Type I Type II Sofware Agens Type III Type IV The objecive funcion minimizes he higher-order sum of ranges of he uni cos esimaes. Consrains (0) and () reques ha he membership of he acual value in he fuzzy uni cos esimae o be greaer han s(k). Consrains (5) and (6) define he sequence of he hree corners of he corresponding TFN. (Model NLPII) Max Z NLPII m(k) T s m(k), (7) x ln(x) approximaion subjec o: (ln C ln C ) o(k) Td(k) o(k), (8) ln C ln C + s (ln C ln C ), (9) ln C ln C + s (ln C ln C ), (0) ln C ln C 0 + b /, () ln C ln C 0 + b /, () ln C ln C 0 + b /, () ln C 0 ln C 0 ln C 0, (4) 0 b b b, (5) 0 s, (6) T, where o(k), s(k), m(k), and d(k) are he opimizaion parameers specified by exper k; k ~K. o(k), m(k) Z + ; s(k) [0, ]; d(k) 0. In Model NLPI, The objecive funcion maximizes he generalized mean of he saisfacion levels ha are explained below. Consrains (9) and (0) reques ha he membership of he acual value in he fuzzy uni cos esimae o be greaer han s. Consrains (4) and (5) define he sequences of he hree corners of he TFNs. Definiion (The saisfacion level). The saisfacion level (s) is he minimal membership in which an acual value (A i ) belongs o he corresponding fuzzy esimae (Ẽ i ): s min i μẽi (A i ) (7) Fig. Polynomial approximaion of he logarihmic funcion The objecive funcion (7) can be replaced by Max Z NLPII s m(k) (8) In Chen s mehod, he logarihmic values of parameers were derived insead of heir original values o simplify he compuaion. As a resul, he opimizaion resuls apply only o he logarihmic values, and no o he original values, of he uni cos. The new FCI approach In he new FCI approach, firs, a polynomial funcion is used o approximae he logarihmic funcion. However, he problem is ha i is difficul o approximae he logarihmic funcion wihin a wide range wih a limied-order polynomial funcion o a sufficien degree of precision. To resolve his difficuly, he range of he uni cos is deermined. For example, if x.5, hen ln x x.94x x 0.080x 4 (9) The absolue error of such an approximaion is less han 0.0, as shown in Fig. : ln x ( x.94x x 0.080x 4 ) 0.0; x.5 (0) The approximaion formulae for some oher ranges are provided in Table 4.

5 Table 4 Approximaion formulae for some oher ranges Range Approximaion formula Absolue error < x.0 ln x x.5x + 0.4x 0.048x 4.0 x.0 ln x x x + 0.x 0.00x 4.0 x.5 ln x x x + 0.x 0.00x 4.5 x.5 ln x x 95x x x 4.5 x 4.0 ln x x 0.467x x x 4 Subsiuing (0) ino() gives:.64c.94c C 0.080C C 0.94C C C4 0 + b /. () Similarly, () and (4) can also be approximaed as:.64c.94c C 0.080C4.64C 0.94C C C4 0 + b /, ().64C.94C C 0.080C4.64C 0.94C C C4 0 + b /. () As a resul, he following PP models are buil o replace he wo NLP models: (Model PPI) Min Z PPI subjec o: (C C ) o(k) (4) C C + s(k)(c C ), (5) C C + s(k)(c C ), (6).64C.94C C 0.080C4.64C 0.94C C C4 0 + b /, (7).64C.94C C 0.080C4.64C 0.94C C C4 0 + b /, (9) C 0 C 0 C 0, (40) 0 b b b, (4) T Compared o model NLP I, he objecive funcion remains unchanged. Only he consrains involving logarihmic erms, i.e., (0) (5), are replaced wih polynomial consrains, i.e., (5) (40). (Model PPII) Max Z PPII subjec o s m(k) (4) (C C ) o(k) Td(k) o(k), (4) C C + s (C C ), (44) C C + s (C C ), (45).64C.94C C 0.080C4.64C 0.94C C C4 0 + b /, (46).64C.94C C 0.080C4.64C 0.94C C C4 0 + b /, (47).64C.94C C 0.080C4.64C 0.94C C C4 0 + b /, (48) C 0 C 0 C 0, (49) 0 b b b, (50) 0 s, (5) T The objecive funcion is he same wih ha of model NLP II, while he consrains involving logarihmic erms, i.e., (8) (4), are replaced wih polynomial consrains, i.e., (4) (49). Unlike he wo NLP models, he PP models are more racable because he KKT condiions for he PP problems are also polynomials ha can be analyically solved using any mahemaics or opimizaion sofware []. For example, he Lagrangian funcion for he PPI model is:.64c.94c C 0.080C4.64C 0.94C C C4 0 + b /, (8)

6 L C,λ (C C ) o(k) + (λ (C + s(k)(c C ) C )) + (λ (C C s(k)(c C ))) + (μ (.64C.94C C 0.080C4.64C C C C4 0 b /)) + (μ (.64C.94C C 0.080C4.64C C C C4 0 b /)) + (μ (.64C.94C C 0.080C4.64C C C C4 0 b /)) + λ (C 0 C 0 )+λ 4 (C 0 C 0 )+λ 5 (b b )+λ 6 (b b ). (5) The KKT equaions for he PPI model can be derived as: (Equaliy consrains for PPI) C 0 (μ ( C 0.888C C 0 )) + λ 4 0, (5) (μ ( C 0 C 0.888C C 0 )) + λ λ 4 0, (54) (μ ( C 0 C 0 b b b.888c C 0 )) λ 0, (55) (μ ( /)) + λ 6 0, (56) (μ ( /)) + λ 5 λ 6 0, (57) (μ ( /)) λ 5 0, (58) (Inequaliy consrains for PPI) C C + s(k)(c C ), (59) C C + s(k)(c C ), (60) C 0 C 0 0, (6) C 0 C 0 0, (6) b b 0, (6) b b 0, (64) λ,λ,λ λ 6,μ μ 0, (65) T. Obviously, all hese equaions or consrains are polynomials. Afer esimaion, he expers fuzzy uni cos esimaes are fuzzily inerseced o deermine he narrowes range of he uni cos: μ FI( C (),..., C (K ))(x) min (μ k C (k)(x)), (66) where C (k) is he fuzzy uni cos esimae by exper k. Then, a BPN is consruced wih he following configuraion o derive a single represenaive value from he fuzzy uni cos esimaes:. Inpus he normalized values of C (k) C (k) for each k: N(C (k)) N(C (k)) N(C (k)) C (k) min C (k) C (k) min max C (k), (67) C (k) min C (k) C (k) min C (k), (68) C (k) min C (k) max max C (k) min C (k). (69) As a resul, here are K inpus in oal.. A single hidden layer wih 6 K nodes.. Oupu (o ) he normalized value of he uni cos esimae. To conver i back o he unnormalized value: ) U(o ) o (max C min C +minc, (70) U(o ) is compared wih C o evaluae he esimaing performance. 4. Training algorihm he Gradien-Descen algorihm []. Alhough he Levenberg Marquard algorihm is quick and yields a very small error, i ends o overfi, resuling in a poor generalizaion for he esing daa [6]. Applicaion o a real DRAM case The real DRAM case discussed by Chen [0] was adoped o illusrae he new FCI approach. As menioned in Chen and

7 C Fig. A real case Tsai [4], esimaing he uni cos is one of he mos criical asks o a DRAM manufacurer. The real case, shown in Fig., conains he average uni coss of he DRAM produc during 0 periods. Two expers collaboraed o esimae he uni cos of he DRAM produc using he new FCI approach. Their seings of he parameers were Exper I: PPI (o(), s() ) Exper II: PPII (o(), m(), d() 0) The daa of he firs five periods were used o build he PP models. Then, he remaining daa were used o evaluae he esimaing performance. The fied uni cos learning models were (547, 547, 547) C () (0.9990,.460,.460)e (7) and (547, 547, 547) C () (.0879,.460,.460)e, (7) respecively. The fuzzy uni cos esimaes generaed by (7) and (7) were fuzzily inerseced o deermine he narrowes range of he uni cos during each period. The resuls are shown in Fig. 4. For a comparison, Chen s FCI mehod was also applied o he real case. For a fair comparison, he parameer seings by he expers were no varied. The resuls are shown in Fig. 5. According o he experimenal resuls,. The new FCI approach was more precise han Chen s FCI mehod because he average range of he fuzzy uni cos esimaes generaed using he new FCI was % narrower han ha generaed using Chen s FCI mehod.. In addiion, boh mehods achieved a hi rae of 80% for he esing daa.. Anoher disincion beween he wo mehods was he uni cos a he ninh period. The acual value fell near C Fig. 4 Deermining he narrowes range of he uni cos C Fig. 5 The narrowes range of he uni cos deermined using Chen s FCI mehod he border of he fuzzy uni cos esimae generaed using he new FCI approach, bu was considerably ouside ha generaed using Chen s mehod, showing he superioriy of he new FCI approach over Chen s FCI mehod. Subsequenly, a BPN was consruced o derive he represenaive value of he uni cos from he fuzzy uni cos esimaes. MATLAB 07a was applied o implemen he BPN on a PC wih an Inel Core i ghz CPU wih 8 GB of RAM. The BPN was considered o be converged if he mean squared error (MSE) became less han 0 4. The maximal number of epochs was se o 000. The esimaion resuls, i.e., he represenaive values of he uni cos esimaes during he periods, are summarized in Fig. 6. Chen s mehod was also applied and he resuls are shown in Fig. 7. The esimaing accuracy was measured in erms of he mean absolue error (MAE), mean absolue percenage error (MAPE), and roo mean squared error (RMSE). The performances of he wo mehods in improving he esimaing accuracy are compared in Table 5. Obviously, he new FCI approach surpassed Chen s FCI mehod in erms of MAE or

8 C Fig. 6 The represenaive values during various periods C Fig. 7 The esimaing resuls using Chen s mehod Table 5 The esimaing accuracy achieved by wo mehods esimae acual value upper bound lower bound esimae acual value upper bound lower bound Measure Chen s FCI DS-FCI MAE MAPE 8.% 7.% RMSE 0 MAPE. The performances of he wo mehods wih regard o RMSE were quie close. These resuls revealed ha he esimaing accuracy could be ruly opimized only if he original value of he uni cos was considered direcly. Subsequenly, he FCI mehod proposed by Chen []was also applied o he real case for a second comparison, in which wo expers consruced wo FNNs o esimae he uni cos of he DRAM produc wih a TFN. The inpus o eiher FNN were he uni coss wihin he L previous periods, and he oupu was he fuzzy uni cos esimae wihin he curren period. All of hem were normalized values. The FNNs ried o fi he log-sigmoid value, insead of he original or logarihmic value, of he uni cos. Eiher exper deermined he average saisfacion level on he lef-hand side (s L (k)), he average saisfacion level on he righ-hand side (s R (k)), and he average range of he fuzzy uni cos esimaes (ψ(k)), as summarized in Table 6. The esimaion resuls from he wo expers are summarized in Fig. 8. Table 6 Parameers specified by expers Exper No. s L (k) s R (k) ψ(k) C C (Exper #) (Exper #) esimae acual value upper bound lower bound esimae acual value upper bound lower bound Fig. 8 The esimaing resuls by wo expers using Chen s [] mehod C Fig. 9 The aggregaion resuls esimae acual value upper bound lower bound Subsequenly, FI and an RBF were applied o deermine he narrowes range and o derive he represenaive value of he uni cos, respecively. The resuls are shown in Fig. 9. The esimaing performance using Chen s [] mehod was evaluaed as: Average range 54 Hi rae 80% MAE 0.0 MAPE 7.4% RMSE 0.4

9 The new FCI approach surpassed Chen s [] mehod in improving he average range, MAE, and MAPE. Conversely, Chen [] performed slighly beer han he new FCI approach wih regard o RMSE, which was no unexpeced since he FNNs used in Chen s [] mehod aimed o minimize he RMSE. To ascerain wheher he advanage of he new FCI approach over he exising mehods was saisically significan, he performances of all mehods along each dimension were ranked. Then, he ranks of each mehod along all dimensions were summed up. The resuls are shown in Table 7.The proposed mehodology surpassed he oher mehods. The FCI mehod proposed by Chen and Chiu [] was based on agens and herefore was no compared in his sudy. Subsequenly, he new FCI approach was applied o anoher case o furher elaborae is effeciveness. The second case conained he uni cos daa of a DRAM produc wihin periods ha are shown in Fig. 0. The colleced daa were cu in half for model building and esing. Two expers collaboraed o esimae he uni cos of he produc. The models chosen by he wo expers were Exper I: PPII (o(), m(), d() 0.64) Exper II: PPI (o(), s() 0.) The solving of he PP problems led o he following fuzzy uni cos learning models: (0.8499, , ) C () (.080,.96,.557)e, (7) (0.4770, , ) C () (.68,.985,.6664)e, (74) The uni cos esimaes by he wo expers were aggregaed using he FI-BPN approach. The aggregaion resuls were compared wih acual values in Fig.. The esimaion performance was evaluaed as follows: MAE MAPE.6% MAPE The average range which showed a very good fi ha suppored he effeciveness of he new FCI approach. C Fig. 0 The uni cos daa of anoher case C Fig. The aggregaion resuls Conclusions esimae acual value Reducing he uni cos of each produc is a goal ha is pursued by every facory. However, he resuls of cos reducion acions are ofen overesimaed, resuling in unreliable and misleading uni cos esimaes ha undermine financial or producion planning. To resolve his problem and o enhance he effeciveness of uni cos esimaion, a new FCI approach is proposed in his sudy. The new FCI approach is novel because i handles he original value, raher han he logarihmic or log-sigmoid value, of he uni cos direcly, unlike Table 7 A comparison of he performances of various mehods along all dimensions Mehod Rank (MAE) Rank (MAPE) Rank (RMSE) Rank (average range) NLP I NLP II PP I PP II FCI [0] 4 4 FCI [] 6 The new FCI approach 8 Sum of ranks

10 exising mehods. Such a reamen is believed o be a viable way o ruly opimize he precision or accuracy of he uni cos esimaion. The proposed new FCI approach and wo exising mehods were applied o a real DRAM case, from which he following observaions were made:. The esimaing precision, measured in erms of he average range for he esing daa using he new FCI approach, was % beer han ha using Chen s [0] mehod, and 0% beer han ha using Chen s [] mehod, respecively.. The new FCI approach also ouperformed he wo exising mehods in improving he esimaing accuracy, especially wih regard o MAE and MAPE. In fuure research, he reamens aken in his sudy can be applied o modify oher FCI mehods. The new FCI approach should also be applied o oher ypes of producs or learning processes in oher fields, such as produciviy learning [], energy efficiency learning [7], ec., o furher elaborae on is effeciveness. These learning processes are ofen subjec o uncerainy. The applicaion of he proposed mehodology is able o deal wih such uncerainy and model he learning process precisely. Furher, here are various ypes of learning curve models, such as log-linear and non-log-linear learning curve models [9]. The proposed mehodology can be modified o be suiable for modeling oher ypes of learning curve models. Open Access This aricle is disribued under he erms of he Creaive Commons Aribuion 4.0 Inernaional License (hp://creaivecomm ons.org/licenses/by/4.0/), which permis unresriced use, disribuion, and reproducion in any medium, provided you give appropriae credi o he original auhor(s) and he source, provide a link o he Creaive Commons license, and indicae if changes were made. References. Bazaraa MS, Sherali HD, Shey CM (99) Nonlinear programming: heory and algorihms. Wiley, New York. Blance RS (00) Learning from produciviy learning curves. 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