TAKAGI-SUGENO AND INTERVAL TYPE-2 FUZZY LOGIC FOR SOFTWARE EFFORT ESTIMATION. Author: Annepu BalaKrishna, RamaKrishna TK

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1 TKGI-SUGENO ND INTERVL TYPE- FUZZY LOGIC FOR SOFTWRE EFFORT ESTIMTION uthor: nnepu BalaKrshna, RamaKrshna TK bstract Software Effort Estmaton carres nherent rsk and ths rsk would lead to uncertanty and some of the uncertanty factors are project complety, project sze etc. In order to reduce the uncertanty, fuzzy logc s beng used as one of the solutons. In ths Chapter nterval type- fuzzy logc s appled for software effort estmaton. Two dfferent methodologes have been dscussed as two models, to estmate effort by usng Takag-Sugeno and Interval Type- fuzzy logc. The Formulas that were used to mplement these models ncludng Regresson nalyss, Takag-Sugeno membershp functons, foot prnt of uncertanty ntervals and de-fuzzfcaton process through weghted average method were outlned along wth analyss. The epermentaton s done wth NS software data set on the proposed models, and the results are tabulated. The measured efforts of these proposed models are compared wth avalable models from lterature and fnally the performance analyss s done based on parameters such as MRE, VRE and VF.. Interval Type- Fuzzy Logc: type- fuzzy set, denoted as, s characterzed by a type- Membershp Functon (MF), µ (, u), where X and u,.e. ~ {((, u), ~ (, u)) X, [0,]} In whch 0 µ (, u), f the unverses of dscourse X and the doman of secondary membershp functon are contnuous, can be epressed as: (, ) /(, ), j [0, ] j Where denotes unon over all admssble and μ. If the unverses of dscourse X and are both dscrete, s replaced by Σ, can also be epressed as: N M k j (, j ) / (, Where Σ denotes unon over and u. In the same way, f X s contnuous and s dscrete or X s dscrete and s contnuous, can be epressed as: M u ( ) / (, ) (or) X j j j j ) N j (, ) / (, ) The frst restrcton that s consstent wth the T constrant 0 μ (). When uncertantes dsappear a type- membershp functon s reduced to a type- membershp functon, n whch case the varable u equals μ ()[,,3,4,5]. The second restrcton that 0 μ (, u) s consstent wth the fact that the ampltude of a

2 membershp functon should le between or equal to 0 and. When μ (, u), s an IT FS, t can stll be epressed as a specal case of general T FS as follow: /(, ) / /, [0,] X If unverses of dscourse X and are both dscrete, the above equaton can be epressed as: ~ / / N N / m / k /... M N / nk / N In the above equaton + denotes unon. Uncertanty n the prmary membershps of an IT FS conssts of a bounded regon named as Footprnt of Uncertanty (FOU). It s the unon of all prmary membershps,.e. FOU ( ) Ths s a vertcal-slce representaton of FOU, because each of prmary membershp s a vertcal slce. The Upper Membershp Functon (UMF) and Lower Membershp Functon (LMF) of are two T MFs that bound the FOU. The UMF s assocated wth the upper bound of FOU () and s denoted as (), X and the LMF s assocated wth the lower bound of FOU() and s denoted as ( ) X,.e. ( ) ( ) FOU ( ) FOU ( ) For an ITFS, [ ( ), ( )], Therefore, the ITFS can be denoted as, Effort= Effort = n [ ( ), j U ( )] / ( n dscreate stuaton) or [ ( ), ( )]/ (n contnuous stuaton). The followng Fgure shows the components of Interval Type- Fuzzy Logc. X X Fuzzy Rule Bases Defuzzfcaton Effort a, b Fuzzfcaton Fuzzy Inference Type Reducer Sze

3 Fgure : Structure of Interval Type- Fuzzy Logc Fuzzfcaton s the process whch Translates nputs (real values) to fuzzy values. Inference System apples a fuzzy reasonng mechansm to obtan a fuzzy output. Knowledge Base contans a set of fuzzy rules, t s of the form R :f s F and. n s F n then Y s G,=, m and a membershp functons set known as the database. Type Reducer transforms a Fuzzy Set nto a Type- Fuzzy Set. The defuzzfcaton traduces one output to precse values. For an nterval type- fuzzy system (ITFS) f PL PL, PR PR ( ), P P ( ) NL NL, NR NR f ( ), ( ) N N are the frng ntervals for the membershp functons postve and negatve respectvely,, are left hand sde uncertanty regon,, are PL PL PR PR rght hand sde uncertanty regon. The followng secton deals the two methodologes that have been used on the proposed models n order to estmate effort. Model Descrpton. Methodology for Model - In ths model the mean of FOU `s as a frng nterval n nterval type-, s consdered to estmate the cost (effort) of the software. Step: Ths step estmate the parameters of a, b (the ampltude a and the eponent b) by regresson analyss (power regresson). Step: The varable sze s then fuzzfed by two nput fuzzy sets named Postve and Negatve respectve. The mean of the szes (L) s nput for determnng the fuzzy membershps. The representaton shown n Fgure3., the membershp value μ P ( ) and μ N ( ) s ether 0 or when s outsde the nterval [-L,L]. Ths process s known as Fuzzfcaton. μ N ( ) μ P ( ) 0.5 L 0 L Fgure : Unverse of Dscourse 3

4 Step 3: Then trangular membershp functon s appled to reduce the uncertanty (FOU-Foot Prnt of Uncertanty).The uncertanty at the left sde (LMF) and rght sde (UMF) are then calculated by usng followng fuzzy rule: If s μ(), s μ() s μ () then μp= [mn (μ(), μ(), μ ())], [ma (μ(), μ(),μ())]. The same logc s used then for Negatve membershp functon. Step 4: fter dentfyng the LMF and UMF for both of these postve and negatve membershp functons, the mean of the LMF and UMF FOU s employed as a frng nterval for convertng Type- nto Type- fuzzy sets, ths s known as Type Reducer. Step 5: Fnally n order to convert Fuzzy values nto output (effort), weghted average defuzzfcaton method s used.. Methodology for Model -: In ths model Takag-Sugeno Fuzzy Controller s consdered, for determnng the membershps, and Interval Type- logc and fuzzy operator for determnng the frng ntervals. Step: Ths step estmate the parameters of a, b (the ampltude a and the eponent b) by regresson analyss (power regresson). Step : The varable sze s fuzzfed by two nput fuzzy sets named Postve and Postve respectvely. The mean and stddev (standard devaton) of sze s used to determne the fuzzy membershps. The (mean+stddev) for the postve and (meanstddev) for postve are the L values. The representaton s shown n Fgure3.3, the membershp value μ p ( ) and μ p ( ) s ether 0 or when s outsde the nterval [- L,L]. Ths process s taken as Fuzzfcaton. Step 3: Then trangular membershp functon s appled, to reduce the uncertanty (FOU). The uncertanty at the left sde (LMF) and rght sde (UMF) are calculated by usng followng fuzzy rule: If s μ(), s μ() s μ () then μp= [mn (μ(), μ(), μ ())], [ma (μ(), μ(),μ())]. The same logc s used smlarly for postve membershp functon. Fgure3.3: Membershp Functons of Fuzzy sets n the Sze s Space Step 4: fter dentfyng the LMF and UMF for both of these postve and postve membershp functons, the mean of the LMF and UMF FOU, fuzzy operator ma(or), Left sde Uncertanty nterval, and Rght sde Uncertanty nterval for Models are used as frng nterval for convertng Type- nto Type- fuzzy sets. Step 5: Fnally n order to convert Fuzzy values nto Output (Effort), weghted average defuzzfcaton method s used. 3. MODEL NLYSIS The secton analyses the proposed models. 4

5 Regresson nalyss: By usng power regresson we calculate [ a, b parameters Y= a b, Where s the varable along the -as. The functon s based on lnear regresson wth both as are scaled logarthmcally. Membershp Functons for Model-, Model- Membershp Functons Used n Model-: The mathematcal defntons of the Postve and Negatve fuzzy sets were dentcal for the nput varable sze are equaton. and.. 0 kloc L P kloc L ( kloc) L kloc L L kloc L and kloc L (.) N kloc L ( kloc) L kloc L L 0 kloc L (.) The value of L affects the control performance and we take the mean of the nput (sze) as the L value. Membershp Functons Used n Model-: The mathematcal defntons of the two Postve fuzzy sets were dentcal for the nput varable sze s equaton.3 and.4. 0 kloc L kloc L ( kloc) L kloc L P L and kloc L (.3) 0 kloc L kloc L ( kloc) L kloc L P L kloc L (.4) The value of L affects the control performance and we taken (mean+stddev), (meanstddev) of the nput (sze) as the L and L value. Fuzzy Trangular Membershp Functon (TMF): The trangular MF n specfed by three parameters (, m, ) as Fgure : 5

6 a b m c Fgure 3: Trangular Member Functon The parameters (, m, ) (wth < m < ) determne the coordnates of the three corners of the underlyng trangular MF. Fuzzness: Fuzzness of TFN (, m, ) s defned as: m = model value = Left Boundary = s rght boundary β α Fuzzness of TFN (F) =, 0 < F< m The Hgher the value of fuzzness, the more fuzzy s TFN. The value of fuzzness to be taken depends upon the confdence of the estmator. confdent estmator can take smaller values of F. Let (m, 0) dvdes nternally, the base of the trangle n raton K : where K n the real postve number. α Kβ So that m = K β α s per the above defntons, F = m KF F So α * m and β * m K K If we consder F = 0. and K = then 0. α m* 0.9m 0. β m* m 0..m Footprnt of Uncertanty n Unverse Of Dscourse: MODEL-: fter applyng fuzzfcaton process on the sze by usng the postve and negatve member functons, the Footprnt of Uncertanty (FOU) n unverse of dscourse wth uncertanty regons wll be the one as shaded n the followng fgure 3.5. Fgure 4: Footprnt of Uncertanty and Predcton Intervals for Methodology- 6

7 , are left hand sde uncertanty regon, PL PL sde uncertanty regon., PR PR are rght hand MODEL-: fter applyng fuzzfcaton process on the sze by usng two postve member functons, the Footprnt of Uncertanty (FOU) n unverse of dscourse wth uncertanty regons wll be the one as shaded n the followng fgure 3.6. Fgure 5: Footprnt of Uncertanty and Predcton Intervals for Methodology- P L, PL are left hand sde uncertanty regon, P R, PR are rght hand sde uncertanty regon. Frng Intervals : Frng Intervals for Methodology-: Here the means of FOU s are taken as frng strength. P PL PL, PR PR ( ), P P ( ) N NL NL, NR NR ( ), N N ( ) Frng Intervals for Methodology-: Case-I: Here the means two postve member functons of FOU s are taken as frng strength. P P L P L, P R P R ( ), P P ( ) 7

8 P L P L P R P R, ( ), ( ) P P P Case-II: The uncertanty consdered at the left, rght hand sde nterval.e. fuzzy operator OR (ma) s used to determne the frng nterval P ma ( ), P P ( ),ma ( ), P P ( ) Case-III: The uncertanty consdered only at the rght hand sde nterval to determne the frng nterval P ( ), P P ( ) Case-IV: The uncertanty consdered only at the left hand sde nterval to determne the frng nterval P ( ), P P ( ) Defuzzfcaton: In these models weghts average method, whch s of the followng form are consdered. C = N N w w (.5) where w s the weghtng factor and μ s the membershp obtaned from trangular member functon. Performance Measures: Three crterons were consdered and they are outlned below ) Varance ccounted For (VF) var (Measured Effort Estmated Effort) % VF = 00 var (measured effort) ) Mean bsolute Relatve Error (MRE) abs (Measured Effort Estmated Effort) % MRE = mean 00 (measured effort) 3) Varance bsolute Relatve Error (VRE) 8

9 (abs (Measured Effort Estmated Effort) % VRE = var 00 (measured effort) The followng secton descrbes the epermentaton part of the work, and to conduct the study and n order to establsh the effectvely of the models dataset of 0 projects from NS software project data [] were used. 4. MODEL EXPERIMENTTION pplcaton on Model-: The membershp functon defntons and the membershps are shown here usng equaton. and.; the L value s the mean of the nput szes.e. 46. By applyng power regresson analyss ( for the nput szes and effort the obtaned values are : a=.7 and b= The membershp functons are defned as follows usng equaton 3. and 3.. N kloc 46 kloc L ( kloc) 46 kloc 46 L 0 kloc 46 By applyng Trangular membershp functon for the above membershp functons the left and rght boundares obtaned are shown below. Foot prnt of uncertanty ntervals for the μ P s [ to 0.575] for left hand sde.e. LMF and [0.9 to.] for rght hand sde.e. UMF. Foot prnt of uncertanty ntervals for the μ N s [0.377 to ] for left hand sde.e. LMF and [0.495 to 0.549] for rght hand sde.e. UMF. The means of FOU ntervals s taken as frng strength. = ( ), ( ) P P P = [0.58, ] = N ( ), N N ( ) = [0.364, 0.477] The type reducer acton by usng the trangular membershp functon whch s appled to the uncertanty regon as a secondary member functon and the results obtaned are shown n Table and Table. Defuzzfcaton process s done through weghted average method. Table :Trangular Fuzzy Number of djusted Sze and Effort Estmaton for Postve Membershp Functon S.No Sze(m) α = 0.58m m β =m aα b am b aβ b E p

10 Table : Trangular Fuzzy Number of djusted Sze and Effort Estmaton for Negatve Membershp Functon S.No Sze(m) α = 0.364m M β = 0.477m aα b am b aβ b E N The followng Table 3 shows the Measured Effort, Estmaton Effort, bsolute Error and Relatve Error. Table3 : Error Calculatons S.No Sze Measured Effort Estmated Effort(E p +E n /) bsolute Error Relatve Error pplcaton on Model-: The membershp functon defntons and the membershps shown here are obtaned usng equaton 3.3 and 3.4, the L value s the (mean + stddev) of the nput szes for Postve s ( ) 84.8, and L value s the (mean stddev) of the nput szes for Postve s ( ) 7.7. By applyng power regresson ( analyss for the nput szes and effort the obtaned values of a, b are a=.7 and b=0.853 P kloc L 0 kloc 84.8 ( kloc) 84.8 kloc 84.8 L kloc

11 and P kloc L 0 kloc 7.7 ( kloc) 7.7 kloc 7.7 L kloc 7.7 Case-: The means of FOU ntervals s taken as frng strength. By applyng Trangular membershp functon for the above membershp functons the left and rght boundares obtaned are shown n the followng Table3.8 and Table3.9 [μ P (, m, ), μ P (, m, )] Foot prnt of uncertanty ntervals for the μ P s [0.574 to 0.9] and Foot prnt of uncertanty ntervals for the μ P s [ to.]. The means of FOU ntervals s taken as frng strength. = P ( ), P P ( ) = [0.736, ] The type reducer acton by usng the trangular membershp functon and assocated results are shown n Table3.0. Defuzzfcaton process s done through weghted average method. Table 4: Trangular Fuzzy Number of djusted Sze and Effort Estmaton for Case- The followng Table 5 shows the Measured Effort, Estmated Effort, bsolute Error and Relatve Error. S.No β Sze α= M = (m) 0.736m m aα b am b aβ b Effort Table 5: Error Calculatons for Case- S.No Sze Measured Effort Estmated Effort bsolute Error Relatve Error

12 Case : Usng Fuzzy Operator for frng strength Ths case deals wth uncertanty at the left, rght hand sde nterval.e. fuzzy operator OR (ma) s used here to determned the frng nterval P ma ( ), P P ( ),ma ( ), P P ( ) Foot prnt of uncertanty ntervals for the μ P s [0.574 to 0.9], Foot prnt of uncertanty ntervals for the μ P s [ to.].the fuzzy operator ma of FOU ntervals s taken as frng strength. = ( ), ( ) P P P = [0.9,.] The Table 6 shows the Effort estmaton usng above frng ntervals. Table 6: Trangular Fuzzy Number of djusted Sze and Effort Estmaton for Case- S.No Sze(m) α = 0.9m m β =.m aα b am b aβ b Effort The Table 7 shows the Measured Effort, bsolute Error, Estmated Effort and Relatve Error. Table 7: Error Calculatons for Case- S.No Sze Measured Effort Estmated Effort bsolute Error Relatve Error

13 Case 3: Rght-hand Sde Uncertanty Interval as frng nterval In ths case the uncertanty consdered only at the rght hand sde nterval.e. Frng nterval P ( ), P P ( ) Foot prnt of uncertanty ntervals for the μ P s [0.574 to 0.9] Foot prnt of uncertanty ntervals for the μ P s [ to.].the more uncertanty s on the rght hand sde. = ( ), ( ) P P P = [0.6996,.] The Table 8 shows the Effort estmaton usng above frng ntervals. Table 8: Trangular Fuzzy Number of djusted Sze and Effort Estmaton Case-3 S.No Sze(m) α = β M m =.m aα b am b aβ b Effort The Table 9 shows the Measured Effort, bsolute Error, Estmated Effort and Relatve Error. Table 9: Error Calculatons for Case-3 S.No Sze Measured Estmated bsolute Relatve Effort Effort Error Error Case 4: Left-hand Sde Uncertanty Interval The uncertanty n ths case s consdered only at the left hand sde nterval.e. frng nterval 3

14 P ( ), P P ( ) Foot prnt of uncertanty ntervals for the μ P s [0.574 to 0.9] Foot prnt of uncertanty ntervals for the μ P s [ to.].the more uncertanty s on the left hand sde. = P P ( ), P( ) = [0.574, 0.9] The Table 0 shows the Effort estmaton usng above frng ntervals. Table 0: Trangular Fuzzy Number of djusted Sze and Effort Estmaton for Case-4 S.No Sze(m) α = 0.574m M β =0.9m aα b am b aβ b Effort The Table shows the Measured Effort, bsolute Error, Estmated Effort and Relatve Error. Table : Error Calculatons for Case-4 S.No Sze Measured Estmated bsolute Relatve Effort Effort Error Error RESULTS ND DISCUSSIONS One of the objectve of the present work s to employ Interval Type- fuzzy logc for tunng the effort parameters and test ts sutablty for software effort estmaton. Ths methodology s then tested usng NS dataset provded by Boehm. The results are then compared wth the models n the lterature such as Baly-Basl, laa F. Sheta and Harsh. Comparson wth other models: 4

15 The Table compares effort estmaton of TSFC- Interval Type- Models wth other avalable models. The resultng data ndcate that the appromaton accuracy of the type- fuzzy systems methodology whch s used n ths chapter s comparable wth the Baley-Basl, laaf. Sheta, Harsh models. The fuzzy systems approach to effort estmaton has an advantage over the other models as the Interval Type- fuzzy systems archtecture determnes the frng ntervals for nputs whch reduces the factors of uncertanty, and the fuzzy rules be etracted from numercal data, whch may easly be analyzed and the mplementaton s also relatvely easy. Table : Effort Efforts n Man-Months of Varous Models wth Interval Type- Models S.No Sze Measured effort Baley Basl Estmate laa F. Sheta G.E.model Estmate laa F. ShetaModel Estmate Harsh model Harsh model Interval Type- Model- I TSFC Model Case-I Case-II Case-III Case-IV ssessment through Graph Representaton of Measured Effort Vs Estmated Effort: The Fgure 6 shows measured effort Vs estmated effort of nterval type- models and one can notce that the estmated efforts are very close to the measured effort. 5

Fgure 6: Measured Effort Vs Estmated Effort of TSFC models Fgure 7: Effort Estmatons of Varous models Vs TSFC Models PERFORMNCE NLYSIS: Parameters such as VF, MRE, and VRE are employed to

16 Fgure 6: Measured Effort Vs Estmated Effort of TSFC models Fgure 7: Effort Estmatons of Varous models Vs TSFC Models PERFORMNCE NLYSIS: Parameters such as VF, MRE, and VRE are employed to asses as well as to compare the performance of the estmaton models. The ntegraton of Takag-Sugeno and Interval Type- fuzzy logc can be powerful tool when tacklng the problem of 6

effort estmaton. It can be seen from the resultng data that the Fuzzy logc models for Effort estmaton outperform the Baly-Basl, laa F. Sheta and Harsh models.

17 effort estmaton. It can be seen from the resultng data that the Fuzzy logc models for Effort estmaton outperform the Baly-Basl, laa F. Sheta and Harsh models. The computed MRE, VRE and VF for all the models are ndcated n Table 3. Model Table 3 Summary Results of VF, MRE and VRE Varance ccounted For (VF%) Mean bsolute Relatve Error(MRE%) Varance bsolute Relatve Error(VRE%) Baley Basl Estmate laa F. Sheta G.E.Model Estmate laa F. Sheta Model Estmate Harsh model Harsh model Interval Type- Model TSFC Model Case-I TSFC Model Case-II TSFC Model Case-III TSFC Model Case-IV Fgure 8: Varance ccounted For Of Varous Models Vs TSFC Models 7

18 Fgure 9: Mean bsolute Relatve Error of Varous models Vs TSFC Models Fgure 0: Varance bsolute Relatve Error of varous models Vs TSFC Models 8

19 6. CONCLUSION : In ths study we proposed new model structures to estmate the software cost (Effort) estmaton. Interval Type- fuzzy sets s used for modelng uncertanty and mpresson to better the effort estmaton. Rather than usng a sngle number, the software sze can be regarded as a fuzzy set yeldng the cost estmate also n the form of a fuzzy set. These proposed models were able to provde good estmaton capabltes as per the as per the epermental study takng parameters lke VF, MRE, and VRE. The work of Interval Type- fuzzy sets can be appled to other models of software cost estmaton. However n these models only the sze s used as nput for estmatng the effort. But there are so many Cost Drvers whch have to be consdered for measurng effort. In fact the man dffculty s to determne whch cost drver really capture the reason for dfferences n estmated effort among the projects. Therefore for large projects of sze>00 KDLOC the estmaton process requres data to be more accurate, consstent wth approprate cost drvers. It s reasonable to assume that one should specfy cost drvers for large projects as they are essental to calbrate the estmaton model. References [] Km ohnson, Dept of Computer Scence, Unversty of Calgary, lberta, Canada, Software Cost Estmaton: Metrcs and Models (pages to 7). [] Hareton Leung,Zhang Fan Department of Computng The Hong Kong Polytechnc unversty, Software Cost Estmaton. [3] Robert W. Zmud. n Emprcal Valdaton of Software Cost Estmaton Models. Chrs F.Kemerer, May 987,Volume 30 Number 5. [4] Krshna murthy and Douglas Fsha, machne learnng approaches to estmatng software development effort, 995. [5] Lawrence H.putnam, general emprcal soluton to the macro software szng and estmatng problem, 978. [6] Xsh Huang, Danny Ho, ng Ren, Luz F. Capretz,Improvng the COCOMO model usng a neuro-fuzzy approach. ppled Soft Computng 7 (007) 9 40 [7] Petr Musflek,Wtold Pedrycz,Gancarlo Succ,Marek Reformat, Software cost estmaton wth fuzzy, models,000. [8] Harsh Mttal, Pradeep Bhata Optmzaton Crtera for Effort Estmaton usng Fuzzy Technque. CLEI ELECTRONIC OURNL, VOLUME 0, NUMBER, PPER, UNE 007. [9] laa F. Sheta, Estmaton of the COCOMO Model Parameters Usng Genetc lgorthms for NS Software Projects ournal of Computer Scence (): 8-3, 006, ISSN [0] Hao Yng, The Takag-Sugeno fuzzy controllers usng the smplfed lnear control rules are nonlnear varable gan controllers, automatc, Vol:34,No.,pp [] Zehua Lv, Ha n, Pngpeng Yuan The Theory of Trangle Type- Fuzzy Sets. IEEE Nnth Internatonal Conference on Computer and Informaton Technology.009. [] erry M. Mendel, erry M. Mendel, Robert I. ohn] Standard Background Materal boutinterval Type- Fuzzy Logc Systems That Can Be Used By ll uthors. 9

20 [3] Shgeo be, Mng- Shong Lan, Fuzzy rules etracton drectly from numercal data for functon appromaton,ieee transactons on systems,mn,and cybernetcs,vol.5,no.,995. [4] erry M. Mendel Interval Type- Fuzzy Logc Systems Made Smple, 006. [5] Masaharu Mzumoto,Kokch Tanaka Fuzzy sets of type under algebrac product and algebrac sum, 006 [6] Babak Rezaee] New pproach to Desgn of Interval Type- Fuzzy Logc Systems. Eghth Internatonal Conference on Hybrd Intellgent Systems [7] Roberto Sepulveda, Patrca Meln, ntono Rodrguez, lejandra Manclla, Oscar Montel, nalyzng the effects of the Footprnt of Uncertanty n Type- Fuzzy Logc Controllers,006. [8] Sakat Maty Color Image Segmentaton usng Type- Fuzzy. Internatonal ournal of Computer and Electrcal Engneerng, Vol., No. 3, ugust 009. [9] uan R. Castro, Oscar Castllo, Lus G. Martínez Interval Type- Fuzzy Logc Toolbo

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