Applying Fuzzy ID3 Decision Tree for Software Effort Estimation

Transcription

1 131 Applyig Fuzzy ID3 Decisio Tree for Software Effort Estimatio Ali Idri 1 ad Saaa Elyassami 1 1 Departmet of Software Egieerig ENSIAS, Mohammed Vth Souissi Uiversity, BP. 713, Madiat Al Irfae, Rabat, Morocco Abstract Web Effort Estimatio is a process of predictig the efforts ad cost i terms of moey, schedule ad staff for ay software project system. May estimatio models have bee proposed over the last three decades ad it is believed that it is a must for the purpose of: Budgetig, risk aalysis, project plaig ad cotrol, ad project improvemet ivestmet aalysis. I this paper, we ivestigate the use of Fuzzy ID3 decisio tree for software cost estimatio; it is desiged by itegratig the priciples of ID3 decisio tree ad the fuzzy set-theoretic cocepts, eablig the model to hadle ucertai ad imprecise data whe describig the software projects, which ca improve greatly the accuracy of obtaied estimates. MMRE ad Pred are used as measures of predictio accuracy for this study. A series of experimets is reported usig two differet software projects datasets amely, Tukutuku ad COCOMO 81 datasets. The results are compared with those produced by the crisp versio of the ID3 decisio tree. Keywords: Software cost estimatio, Decisio Tree, Fuzzy ID3, Fuzzy Etropy. 1. Itroductio Estimatio software project developmet effort remais a complex problem, ad oe which cotiues to attract cosiderable research attetio. Improvig the accuracy of the effort estimatio models available to project maagers would facilitate more effective cotrol of time ad budgets durig software project developmet. Ufortuately, may software developmet estimates are quite iaccurate. Molokke ad Jorgese report i recet review of estimatio studies that software projects exped o average 30-40% more effort tha is estimated [13]. I order to make accurate estimates ad avoid gross misestimatios, several cost estimatio techiques have bee developed. These techiques may be grouped ito two major categories: parametric models, which are derived from the statistical or umerical aalysis of historical projects data [5], ad o-parametric models, which are based o a s et of artificial itelligece techiques such as artificial eural etworks [9][4], case based reasoig [19], decisio trees [20] ad fuzzy logic [22][17]. I this paper, we are cocered with cost estimatio models based o fuzzy decisio trees especially Fuzzy Iteractive Dichotomizer 3. The decisio tree method is widely used for iductive learig ad has bee demostratig its superiority i terms of predictive accuracy i may fields [23][10]. The most widely used algorithms for buildig a decisio tree are ID3 [11], C4.5 [12] ad CART [14]. There are three major advatages whe usig estimatio by decisio trees (DT). First, decisio trees approach may be cosidered as white boxes, it is simple to uderstad ad easy to explai its process to the users, cotrary to other learig methods. Secod, it a llows the learig from previous situatios ad outcomes. The learig criterio is very importat for cost estimatio models because software developmet techology is supposed to be cotiuously evolvig. Third, it may be used to feature subset selectio to avoid the problem of cost driver selectio i software cost estimatio model. O the other had, fuzzy logic has bee used i software effort estimatio. It's based o fuzzy set theory, which was itroduced by Zadeh i 1965 [ 15]. Attempts have bee made to rehabilitate some of the existig models i order to hadle ucertaities ad imprecisio problems. Idri et al. [3] ivestigated the applicatio of fuzzy logic to the cost drivers of itermediate COCOMO model while Pedrycz et al. [24] preseted a fuzzy set approach to effort estimatio of software projects. I two earlier works [1][2] we have empirically evaluated the use of crisp decisio tree techiques for software cost estimatio. More especially, the two used crisp decisio tree techiques are the ID3 ad the C4.5 algorithms. The

2 132 two studies are based o the COCOMO' 81 ad a web hypermedia dataset. We have foud that the decisio tree desiged with the ID3 algorithm performs better, i terms of cost estimates accuracy, tha the decisio tree desiged with C4.5 algorithm for the two datasets. The aim of this study is to evaluate ad to discuss the use of fuzzy decisio trees, especially the fuzzy ID3 algorithm i desigig DT for software cost estimatio. Istead of crisp DT, fuzzy DT may allow to exploit complemetary advatages of fuzzy logic theory which is the ability to deal with iexact ad ucertai iformatio whe describig the software projects. The remaider of this paper is orgaised as follows: I sectio II, we preset the fuzzy ID3 decisio tree for software cost estimatio. The descriptio of datasets used to perform the empirical studies ad the evaluatio criteria adopted to measure the predictive accuracy of the desiged models are give i sectio III. Sectio IV focuses o the experimetal desig. I Sectio V, we preset ad discuss the obtaied results whe the fuzzy ID3 is used to estimate the software developmet effort. A compariso of the estimatio results produced by meas of the fuzzy ID3 models ad the crisp ID3 model is also provided i sectio ad sectio V. A coclusio ad a overview of future work coclude this paper. 2. Fuzzy ID3 for Software Cost Estimatio Based o the Cocept Learig System algorithm, Quila proposed a decisio tree called the Iteractive Dichotomizer 3 (ID3). The ID3 techique is based o iformatio theory ad attempts to miimize the expected umber of comparisos. The fuzzy ID3 is based o a fuzzy implemetatio of the ID3 algorithm [16][21]. It's formed of oe root ode, which is the tree top, or startig poit, ad a series of other odes. Termial odes are leaves (effort). Each ode correspods to a split o the values of oe iput variable (cost drivers). This variable is chose i order to reach a maximum of homogeeity amogst the examples that belog to the ode, relatively to the output variable. Figure 1 illustrates a example of fuzzy ID3 decisio tree for software developmet effort where MF represets the membership fuctio used to defie fuzzy sets for each cost driver. The major characteristic of fuzzy ID3 is that a example belogs to a ode to a cer tai degree. The proportio of pk of examples with k classificatio at ode is calculated usig the membership degrees as follows: p N k i i i= 1 k = K N c= 1 i= 1 u ( y ) u ( x ) u ( y ) u ( x ) c i i Where K represets the classes ad N is the umber of examples i the subset. uk( y i) is the membership degree o the web project i that belogs to the class k ad u ( x ) is the membership degree of the web project i at i ode. represets the cojuctio operator. T-orm, which geeralizes itersectio i the domai of fuzzy sets, is usually used for fuzzy cojuctio. The most popular T- orms are miimum ad product. The fuzzy etropy uses the membership degree of examples at a particular ode ad cotributes to ehace the discrimiative power of a attribute, is computed as: H p *log( p ) = (2) k k k The growth of the fuzzy ID3 is realized by expadig a ode of tree characterized by the highest iformatio gai. The iformatio gai is calculated as follows: (1) Fig. 1 A example of fuzzy ID3 decisio tree for software developmet effort

3 133 M j l l l= 1 = (3) G H wh Where H is the etropy i the ode. H l is the etropy of the ode that belogs to the fuzzy set L of the j variable. w is the fuzzy set relative weight. l The ode is split ito as may sub-odes as there are attributes. The algorithm termiated whe all attributes are used for splits, or whe all examples at a ode have the same classificatio. 3. Data Descriptio ad Evaluatio Criteria This sectio describes the dataset used to perform this empirical study ad the evaluatio criteria adopted to measure the estimates accuracy of the desiged software cost estimatio model based o fuzzy ID3 method. 3.1 Data Descriptio I this empirical study, two historical software projects datasets are used: 1- Tukutuku dataset [7] 2- COCOMO'81 dataset [5] The Tukutuku dataset cotais 53 web projects. Each web applicatio is described usig 9 umerical attributes such as: the umber of html or shtml files used, the umber of media files ad team experiece (see Table I). However, each project voluteered to the Tukutuku database was iitially characterized usig more tha 9 software attributes, but some of them were grouped together. For example, we grouped together the followig three attributes: the umber of ew Web pages developed by the team, the umber of Web pages provided by the customer ad the umber of Web pages developed by a third party (outsourced) i oe attribute reflectig the total umber of Web pages i the applicatio (TotWP). Table I: Software Attributes for the Tukutuku dataset Attributes Descriptio TeamExp DevTeam TotWP TextPages TotImg Average team experiece with the developmet laguage(s) employed Size of developmet team Total umber of web pages Number text pages typed (~600 words) Total umber of images Aim AV TotHigh TotNHigh Number of aimatios Number of audio/video files Total Number of high effort features/fuctios Total Number of low effort features/fuctios The COCOMO'81 dataset cotais 252 software projects which are mostly scietific applicatios developed by Fortra. Each software project is described usig 13 attributes: software size measured i KDSI (Thousads of Delivered Source Istructios) ad the remaiig 12 umerical attributes described i Table II. The 12 umerical attributes describe the eviromet i which the program will be desiged to operate, the relatioship betwee a program ad its host or developmetal platform, selected project maagemet facets of a program such as the experiece of the persoel ivolved i the software project, the time ad storage costraits imposed o the software ad the method used i the developmet. Table II: Software Attributes for the COCOMO'81 dataset Attributes Descriptio SIZE DATA VIRTMIN, VIRTMAJ TIME STOR TURN ACAP AEXP PCAP VEXP LEXP SCED Software Size Database Size 3.2 Evaluatio criteria Virtual Machie Volatility Executio Time Costrait Mai Storage Costrait Computer Turaroud Time Aalyst Capability Applicatios Experiece Programmer Capability Virtual Machie Experiece Programmig Laguage Experiece Required Developmet We employ the followig criteria to measure the accuracy of the estimates geerated by the fuzzy ID3. A commo criterio for the evaluatio of effort estimatio models is the magitude of relative error (MRE), witch is defied as MRE Effort Effort actual estimated = (4) Effortactual

4 134 where dataset, ad Effort actual is the actual effort of a project i the Effort is the estimated effort that was estimated obtaied usig a model or a techique. The MRE values are calculated for each project i the datasets, while mea magitude of relative error (MMRE) computes the average over N projects 1 Effort Effort MMRE 100 N (5) Effort N actual, i estimated, i = i= 1 actual, i The acceptable target values for MMRE are MMRE 25. This idicates that o the average, the accuracy of the established estimatio model would be less tha 25%. Aother widely used criterio is the predictio Pred(p) witch represets the percetage of MRE that is less tha or equal to the value p amog all projects. This measure is ofte used i the literature ad is the proportio of the projects for a give level accuracy [18]. The defiitio of Pred(p) is give as follows: Pred( p) k N = (6) Where N is the total umber of observatios ad k is the umber of observatios whose MRE is less or equal to p. A commo value for p is 25, witch also used i the preset study. The predictio at 25%, Pred(25), represets the percetage of projects whose MRE is less or equal to 25%. The acceptable values for Pred(25) are Pr ed(25) Experimet Desig 9 ad is equal to 13 i the case of COCOMO'81 dataset. Cocerig the maximum umber of fuzzy sets is the maximum partitio size for each variable, is fixed to 7 for all experimets. I the preset paper we are iterested i studyig the impact of the fuzzy cojuctio operators (t-orms) ad the sigificat level parameter (β) o the accuracy of fuzzy ID3. The sigificat level is the membership degree for a example to be cosidered as belogig to the ode. For each dataset, two models of fuzzy ID3 were geerated. The first Fuzzy ID3 effort estimatio model uses the product etropy cojuctio operator to measure the fuzzy etropy (t-orm=product), ad the secod model uses the miimum etropy cojuctio operator to calculate the fuzzy etropy (t-orm=mi). These cojuctio operators are the two commoly used t-orm operators because of their well behaviour ad their computatioal simplicity [6]. The miimum etropy cojuctio operator is defied as: u ( y ) u ( x ) = mi[ u ( y ), u ( x )] (7) k i i k i i Cocerig, the product etropy cojuctio operator is give as: u ( y ) u ( x ) = u ( y )* u ( x ) (8) k i i k i i For each model, a series of experimets is coducted with the fuzzy ID3 algorithm each time usig a differet value of the sigificat level parameter (β). The sigificat level is varied withi the iterval [0, 1]. 5. Overview of the Empirical Results This sectio describes the experimet desig of the fuzzy ID3 decisio tree o the both Tukutuku ad COCOMO'81 datasets. The use of fuzzy ID3 to estimate software developmet effort requires the determiatio of the parameters, amely the umber of iput variables, the maximum umber of fuzzy sets for each iput variable, the sigificat level value ad the cojuctio operator. The last two parameters play a essetial role i the geeratio of Fuzzy Decisio trees. It greatly affects the calculatio of fuzzy etropy ad classificatio results of Fuzzy Decisio trees. The umber of iput variables is the umber of the attributes describig the historical software projects i the used dataset. Therefore, whe applyig fuzzy ID3 to Tukutuku dataset, the umber of iput variables is equal to This sectio presets ad discusses the results obtaied whe applyig the fuzzy ID3 to the Tukutuku ad COCOMO'81 datasets. The calculatios were made usig Fispro software [8]. We coducted several experimets usig differet cofiguratios of fuzzy ID3. For these experimets, a holdout validatio o the etire datasets was performed. Datasets were radomly split ito two groups: traiig set ad test set. 5.1 Tukutuku dataset The first experimet is performed usig Tukutuku dataset cotaiig 53 historical software projects. Two models of fuzzy ID3 were desiged. The first Fuzzy ID3 effort estimatio model (Model 1) uses the formula of the

5 135 cojuctio operator give i Eq. (8) to compute fuzzy etropy, ad the secod model (Model 2) uses the formula of the cojuctio operator give i Eq. (7). For each model, differet cofiguratios have bee obtaied by varyig the sigificat level (β). The aim is to determie which cofiguratio improves the estimates. We have traied ad tested the two models usig Tukutuku dataset. The results for the differet cofiguratios have bee compared. Figure 2 ad figure 3 show the accuracy of the two fuzzy ID3 models, measured i terms of MMRE ad Pred, o Tukutuku dataset. MMRE Tukutuku dataset Product Miimum Mi sigificat level Fig. 2 Relatioship betwee the accuracy of Fuzzy ID3 (MMRE), the used cojuctio operator ad the SL value Figure 2 compares the accuracy of the two models, i terms of MMRE, whe varyig the sigificat level. We ote that the fuzzy ID3 model usig the product cojuctio operator geerates a l ower MMRE that the other model usig the miimum cojuctio operator for sigificat level value less tha 0.2. For example, for β=0.1 the model 1 geerates a lower predictio error (MMRE=2.45) tha the model 2 (MMRE=5.31). By agaist, model 2 geerates a l ower MMRE tha the model 1 for sigificat level value greater tha or equal to 0.2. For example, for β=0.5 the model 2 geerate a lower predictio error (MMRE=9.09) tha the model 1 (MMRE=34.48). Pred(25) Tukutuku dataset Product Miimum Mi sigificat level Fig. 3 Relatioship betwee the accuracy of Fuzzy ID3 (Pred), the used cojuctio operator ad the SL value Figure 3 shows ad compares the results of the two models, i terms of Pred(25), whe varyig the sigificat level. From this figure, we ote that the accuracy of fuzzy ID3 model usig miimum cojuctio operator performs much better tha fuzzy ID3 model usig product cojuctio operator for sigificat level value greater tha or equal to 0.2. So, model 2 geerates acceptable effort estimates with sigificat level value less or equal to 0.6. We ote that the accuracy of fuzzy ID3 model usig the miimum cojuctio operator performs much better tha fuzzy ID3 model usig the product cojuctio operator for almost every value of sigificat level. Table III summarizes the results obtaied usig differet cofiguratios of fuzzy ID3 for Tukutuku dataset. It shows the variatio of the accuracy accordig to the sigificat level value ad to the used cojuctio operator. Table III: MMRE ad Pred results of differet fuzzy ID3 cofiguratios for Tukutuku dataset Sigificat level (β) Accuracy of Fuzzy ID3 T-orm = Product T-orm = Miimum MMRE Pred(25) MMRE Pred(25) 0.1 2,45 97,73 5,31 97, ,09 95,45 1,82 93, ,7 93,18 3,87 95, ,49 93,18 5,82 90, ,48 77,27 9,09 90, ,08 58, , ,3 39,62 97,41 50, ,66 26,42 111,83 45, ,99 22,64 176,99 20,75

6 COCOMO'81 dataset I the secod experimet, we have replicated the previous empirical study usig COCOMO'81 dataset to verify how much the use of a adequate cojuctio operator affects the accuracy of fuzzy ID3 i estimatig software effort. We have coducted several experimets o the two models. To compute fuzzy etropy, model 1 u ses the product cojuctio operator. O the other side, model 2 uses the miimum cojuctio operator for calculatig fuzzy etropy. Pred(25) Cocomo dataset Product Miimum Mi sigificat level For each model, we varied the sigificat level (β) from 0.1 to 0.9 degree. Figure 3 ad figure 4 show the accuracy of the two fuzzy ID3 models, measured i terms of MMRE ad Pred, o COCOMO'81 dataset. MMRE Cocomo dataset Product Miimum Mi sigificat level Fig. 4 Relatioship betwee the accuracy of Fuzzy ID3 (MMRE), the used cojuctio operator ad the SL value Figure 4 compares the accuracy of the two models, i terms of MMRE, whe varyig the sigificat level. From this figure, we ote that the accuracy of fuzzy ID3 model usig the miimum cojuctio operator performs much better tha fuzzy ID3 model usig the product cojuctio operator for each value of sigificat level. Therefore, i terms of MMRE, Model 2 performs better tha Model 1. Figure 5 shows the results of the two models, i terms of Pred(25), whe varyig the sigificat level. From these figures, we cofirm the superiority of the fuzzy ID3 model usig Eq. (7) to compute the fuzzy etropy over that oe usig Eq. (8). Fig. 5 Relatioship betwee the accuracy of Fuzzy ID3 (Pred) the used cojuctio operator ad the SL value Table IV summarizes the results obtaied usig differet cofiguratios of fuzzy ID3 for COCOMO'81 dataset. It shows the variatio of the accuracy accordig to the sigificat level value ad to the used cojuctio operator. Table IV: MMRE ad Pred results of differet fuzzy ID3 cofiguratios for COCOMO'81 dataset Sigificat level (β) Accuracy of Fuzzy ID3 T-orm = Product T-orm = Miimum MMRE Pred(25) MMRE Pred(25) 0.1 1,98 95,93 0,56 99, ,84 95,93 0,98 98, ,11 88,24 1,17 98, ,16 76,92 1,21 98, ,31 69,68 1,34 98, ,08 48,41 60,86 74, ,89 36,51 97,41 57, ,25 22,62 111,83 31, ,32 15,08 176,99 21, Comparisos betwee crisp ad fuzzy ID3 The comparisos betwee the results produced by the two fuzzy ID3 models used i the latest subsectios (A ad B) ad the crisp versio of ID3 decisio tree are show i table V. For the Tukutuku dataset, the crisp ID3 model geerated i [1] is used for the compariso ad i the case of the COCOMO'81 dataset, we used the crisp ID3 model applied i [2]. The best results obtaied by meas of the three models are compared i terms of MMRE ad Pred(25).

7 137 Table V: Result of the differet models used o COCOMO'81 ad o Tukutuku datasets Performace Criteria COCOMO'81 dataset Crisp ID3 Model 1 Model 2 MMRE 28 1,98 0,56 Pred(25) 84 95,93 99,5 Performace Criteria Tukutuku dataset Crisp ID3 Model 1 Model 2 MMRE 24 2,45 1,82 Pred(25) 96 97,93 97,93 The experimetal results show that the fuzzy ID3 models show better estimatio accuracy tha the crisp ID3 model i terms of MMRE ad Pred(25). For example, i the case of the COCOMO'81 dataset, the improvemet is 92% based o the model 1 MMRE ad the crisp ID3 MMRE ad is the 98% based o the model 2 MMRE ad the crisp ID3 MMRE. 4. Coclusios I this paper, we have empirically studied two fuzzy ID3 models for software effort estimatio. Each oe used a differet formula to compute the fuzzy etropy. These fuzzy ID3 models were traied ad tested usig two software projects datasets. The results show that the use of a optimal sigificat level value ad a adequate cojuctio operator for computig the fuzzy etropy improves greatly the estimates geerated by fuzzy ID3 model. The compariso with the crisp versio of ID3 decisio tree shows ecouragig results. Refereces [1]A. Idri, S. Elyassami, Software Cost Estimatio Usig Decisio Trees, I Proceedig of Sixième Coférece sur les Systèmes Itelligets: Théories et Applicatios (SITA'10), Rabat, Morocco, 4-5 Mai, pp [2]A. Idri, S. Elyassami, Web Effort Estimatio Usig Decisio Trees, I Proceedig Iteratioal Symposium o INovatios i Itelliget SysTems ad Applicatios (INISTA 2010), Kayseri, Turkey, Jui, pp [3]A. Idri, L. Kjiri, ad A. Abra, COCOMO Cost Model Usig Fuzzy Logic, 7th Iteratioal Coferece o Fuzzy Theory & Techology, Atlatic City, NJ, February, pp [4]A. Idri, ad A. Abra, ad S. Mbarki, "A Experimet o the Desig of Radial Basis Fuctio Neural Networks for Software Cost Estimatio", i 2d I EEE Iteratioal Coferece o Iformatio ad Commuicatio Techologies: from Theory to Applicatios, 2006, Vol. 1, pp [5]B.W. Boehm, Software Egieerig Ecoomics, Place: Pretice-Hall, [6] E. H. Mamdai, "Applicatios of Fuzzy Set Theory to Cotrol Systems: A Survey," i Fuzzy Automata ad Decisio Processes, M. M. Gupta, G. N. Saridis ad B. R. Gaies, eds., North-Hollad, New York, 1977, pp [7]B.A Kitcheham ad E. Medes, A Compariso of Crosscompay ad Withi-compay Effort Estimatio Models for Web Applicatios, Proceedigs of EASE Coferece, 2004, pp [8]Guillaume, S., Charomordic, B., Lablee, J.-L., FisPro: Logiciel ope source pour les systemes d'iferece floue. INRA-Cemagref. [9]G. R. Fiie, ad G. Wittig, ad J.-M. Desharais, "A Compariso of Software Effort Estimatio Techiques: Usig Fuctio Poits with Neural Networks, Case-Based Reasoig ad Regressio Models", Systems ad Software, Vol. 39, No. 3, 1997, pp [10]H. Berger, D. Merkl, ad M. Dittebach, "Exploitig Partial Decisio Trees for Feature Subset Selectio i Categorizatio," i Proceedigs of the 2006 A CM Symposium o A pplied Computig (SAC 2006), Dijo, Frace, 2006, pp [11]J. R. Quila, Iductio o de cisio tree, Machie Learig, Vol. 1, 1986, pp [12]J. R. Quila, C4.5: Programs for Machie Learig, Morga Kaufma Publishers, Sa Mateo, CA, [13]K. Molokke, ad M. Jorgese, "A Review of Surveys o Software Effort Estimatio", i Iteratioal Symposium o Empirical Software Egieerig, 2003, pp [14]L. Breima, J.H. Friedma, R.A. Olse & C.J. Stoe. Classificatio ad Regressio Trees. Wadsworth, [15]L. A. Zadeh, Fuzzy sets, Iformatio ad Cotrol, vol 8, 1965, pp [16]M. Umao, H. Okamoto, I. Hatoo, H. Tamura, F. Kawachi, S. Umedzu, J. Kioshita, Fuzzy Decisio Trees by Fuzzy ID3 algorithm ad Its Applicatio to Diagosis Systems, I Proceedigs of the third IEEE Coferece o Fuzzy Systems, vol. 3, Orlado, 1994, pp [17]M. W. Nisar, ad Y.-J. Wag, ad M. Elahi, "Software Developmet Effort Estimatio Usig Fuzzy Logic A Survey", i 5th Iteratioal Coferece o Fuzzy Systems ad Kowledge Discovery, 2008, pp [18]M. Korte ad D. Port, Cofidece i Software Cost Estimatio Results Based o M MRE ad PRED, PROMISE 08, May 12-13, 2008, pp [19]M. Shepperd ad C. Schofield. Estimatig Software Project Effort Usig Aalogies. Trasactios o S oftware Egieerig, vol. 23, o. 12, 1997, pp [20]R. W. Selby, ad A.A. Porter, "Learig from examples: geeratio ad evaluatio of decisio trees for software

8 138 resource aalysis", IEEE Trasactios o Software Egieerig, Vol. 14, No. 12, 1988, pp [21]R. Weber, Fuzzy ID3: a class ofmethods for automatic kowledge acquisitio, Proceedigs ofthe 2d I teratioal Coferece o Fuzzy Logic ad Neural Networks, Iizuka, Japa, July 17 22, 1992, pp [22]V. Sharma, ad H. K. Verma, "Optimized Fuzzy Logic Based Framework for Effort Estimatio i Software Developmet", Computer Sciece Issues, Vol. 7, Issue 2, No. 2, 2010, pp [23]W. Pedrycz ad Z. A. Sosowski, The desig of decisio trees i the framework of graular data ad their applicatio to software quality models, Fuzzy Sets ad Systems, Vol. 1234, 2001, pp [24]W. Pedrycz, J.F. Peters, S. Ramaa, A Fuzzy Set Approach to Cost Estimatio of Software Projects, Proceedigs of the 1999 IEEE Caadia Coferece o Electrical ad Computer Egieerig Shaw Coferece Ceter, Edmoto Alberta, Caada. 1999, pp A. Idri is a Professor at Computer Sciece ad Systems Aalysis School (ENSIAS, Rabat, Morocco). He received DEA (Master) (1994) ad Doctorate of 3rd Cycle (1997) degrees i Computer Sciece, both from the Uiversity Mohamed V of Rabat. He has received his Ph.D. (2003) i Cogitive Computer Scieces from ETS, Uiversity of Quebec at Motreal. His research iterests iclude software cost estimatio, software metrics, fuzzy logic, eural etworks, geetic algorithms ad iformatio scieces. S. Elyassami received her egieerig degree i Computer Sciece from the UTBM, Belfort-Motbeliard, Frace, i Curretly, she is preparig her Ph.D. i computer sciece i ENSIAS. Her research iterests iclude software cost estimatio, software metrics, fuzzy logic ad decisio trees.