Validating and Understanding Software Cost Estimation Models based on Neural Networks

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1 Valdatng and Understandng Software Cost Estmaton Models based on eural etworks Al Idr Department of Software Engneerng ESIAS, Mohamed V Unversty Rabat, Morocco E-mal : dr@ensas.ma Alan Abran École de Technologe Supéreure 80 otre-dame Ouest, Montreal, Canada H3C K3 E-mal : aabran@ele.etsmtl.ca Samr Mbark Department of Mathematcs Ibn Tofal Unversty Kentra, Morocco E-mal: Mbark@unv-bntofal.ac.ma Abstract Software development effort estmaton wth the ad of artfcal neural networks (A) attracted consderable research nterest at the begnnng of the nnetes. However, the lack of a natural nterpretaton of ther estmaton process has prevented them from beng accepted as common practce n cost estmaton. Indeed, they have generally been vewed wth skeptcsm by a maorty of the software cost estmaton communty. In ths paper, we nvestgate the use and the nterpretaton of the Radal Bass Functon etworks (RBF) n the software cost estmaton feld. We frst apply the RBF to estmatng the costs of software proects, and then study the nterpretaton of cost estmaton models based on an RBF usng a method whch maps ths neural network to a fuzzy rule-based system, takng the vew that, f the fuzzy rules obtaned are easly nterpreted, then the RBF wll also be easy to nterpret. Our case study s based on the COCOMO 8 dataset.. Introducton Estmatng software development effort remans a complex problem, and one whch contnues to attract consderable research attenton. Improvng the accuracy of the estmaton models avalable to proect managers would facltate more effectve control of tme and budgets durng software development. In order to make accurate estmates and avod gross msestmatons, several cost estmaton technques have been developed. These technques may be grouped nto two maor categores: () parametrc models, whch are derved from the statstcal or numercal analyss of hstorcal proects data [,3], and () non-parametrc models, whch are based on a set of artfcal ntellgence technques such as artfcal neural networks, analogybased reasonng, regresson trees, genetc algorthms and rule-based nducton [6,,0,,4,5]. Although experence has shown that there does not exst a best predcton technque outperformng all the others n every stuaton [4,,6,], non-parametrc models stll have two sgnfcant advantages over parametrc ones: frst, the ablty to model the complex set of relatonshps between the dependent varable requred to predct (cost, effort) and the ndependent varables (cost drvers) collected earler n the lfecycle; second, the ablty to learn from hstorcal proects data, especally for artfcal neural networks. In ths paper, we are concerned wth cost estmaton models based on artfcal neural networks. An artfcal neural network s characterzed by ts archtecture, ts learnng algorthm and ts actvaton functons [7,8,7]. In general, for software cost estmaton modelng, the most commonly adopted archtecture, the learnng algorthm and the actvaton functon are the feedforward mult-layer Perceptron, the Backpropagaton algorthm and the Sgmod functon respectvely [,3,5]. Most of these referenced studes have focused more on the accuracy of the approach relatve to that of other cost estmaton technques. In ths work, we are concerned wth Radal Bass Functon etwork (RBF) models, whch dffer from the Backpropagaton feedforwad Perceptron n the followng ways: ) usually, an RBF archtecture s composed of three layers, as opposed to a Perceptron whch may have more; ) RBF models use a hybrd learnng scheme that combnes supervsed and unsupervsed learnng algorthms, whereas the Backpropagaton s a supervsed learnng algorthm; and 3) the actvaton functon of the mddle-layer neurons of an RBF s usually the Gaussan functon, whereas that of the hdden Perceptron neurons can be Sgmod, Gaussan or other types of functons. However, even though there s varety among neural network archtectures, and most of them have shown ther strengths n solvng complex problems, many researchers n varous felds are hestant to use them because of ther common shortcomng of beng black boxes models. To overcome ths lmtaton, we have studed the use of FRBSs to provde a natural nterpretaton of cost estmaton models based on a Backpropagaton threelayer feedforward Perceptron []. What we have proposed comprsed essentally the use of the Bentez s method to extract the f-then fuzzy rules from ths network []. These fuzzy rules express the nformaton encoded n the archtecture of the network, and the nterpretaton of each fuzzy rule has been determned by analyzng ts premse and ts output. Whle our case study has shown that we can explan the meanng of the output and the propostons composng the premse of /04/$ IEEE.

2 each fuzzy rule, the entre fuzzy rule cannot be easly nterpreted because t uses the -or operator. In ths paper, we explore another mappng method, that s, the Jang and Sun method, to extract f-then fuzzy rules from artfcal neural networks. The use of ths method requres that the archtecture of the network be an RBF [4]. The man advantage of Jang and Sun method s that t uses a more understandable operator, that s, the and logcal conuncton, to combne the propostons composng the premse of each rule. Ths case study s based on the COCOMO 8 hstorcal dataset. Ths paper s organzed as follows: In Secton, we brefly descrbe the RBF archtecture most often used. Secton 3 shows how an RBF can be used to estmate software development effort. We also present the archtecture of the RBF that wll be used n our case study. In Secton 4, we dscuss the results obtaned when the RBF s used to estmate software development effort. In Secton 5, we brefly outlne the prncple of the Jang and Sun method that wll be used to extract the f-then fuzzy rules from our network. In Secton 6, we apply the Jang and Sun method to our RBF and we dscuss the nterpretaton of the obtaned fuzzy rules n software cost estmaton. A concluson and an overvew of future work conclude ths paper.. Radal bass functon networks for software cost estmaton Based on bologcal receptve felds, Moody and Darken proposed a network archtecture called the Radal Bass Functon etwork (RBF) whch employs local receptve felds to perform functon mappngs [5]. It can be used for a wde range of applcatons, prmarly because t can approxmate any regular functon [9]. An RBF s a three-layer feedforward network consstng of one nput layer, one mddle-layer and an output layer. Fgure llustrates a possble RBF archtecture confgured for software development effort. The RBF generates output (effort) by propagatng the ntal nputs (cost drvers) through the mddle-layer to the fnal output layer. Each nput neuron corresponds to a component of an nput vector. The mddle layer contans M neurons, plus, eventually, one bas neuron. Each nput neuron s fully connected to the mddle-layer neurons. The actvaton functon of each mddle neuron s usually the Gaussan functon: f x c ( σ ( x) e ) = (Eq. ) where c and σ are the center and the wdth of the th mddle neuron respectvely.. denotes the Eucldean dstance. Hence, each th hdden neuron has ts own receptve feld n the nput space, a regon centered on c wth sze proportonal to σ.. The Gaussan functon decreases rapdly f the wdth σ s small, and slowly f t s large. The output layer conssts of one output neuron that computes the software development effort as a lnear weghted sum of the outputs of the mddle layer: Effort = M = y y y 3 y M x c ( σ y β wth y = e (Eq. ) Input layer Hdden layer Output layer KDSI c, σ RELY ACAP PCAP CPLX Fgure : A Radal Bass Functon etwork archtecture for software development effort 3. Experment desgn Ths secton descrbes the experment desgn of an RBF on the COCOMO 8 hstorcal proects. We present and dscuss how we set the dfferent parameters of the RBF archtecture accordng to the characterstcs of the COCOMO 8 dataset, especally the number of nput neurons, number of hdden neurons, centers c, wdths σ and weghts β. Before an RBF s ready to make estmates for new software proects, t s traned by a set of combnatons of nputs (cost drvers) and outputs (development efforts) of hstorcal software proects. Our case study conssts of estmatng the software development effort by usng an RBF on the COCOMO 8 dataset. Ths dataset contans 63 software proects []. Each proect s descrbed by 3 attrbutes: the software sze measured n KDSI (Klo Delvered Source Instructons) and the remanng attrbutes are measured on a scale composed of sx lngustc values: very low, low, nomnal, hgh, very hgh and extra hgh. As mentoned earler, the use of an RBF to estmate software development effort requres the determnaton of the mddle-layer parameters and the weghts β. The smplest and most general method for decdng on the mddle-layer neurons s to create a neuron for each tranng software proect. Consequently, the mddle layer of our RBF wll have 63 neurons. Ths approach does not take nto account the exstng smlartes among the 63 hstorcal software proects. As we have shown n varous studes wth the COCOMO 8 dataset, there are some smlartes among certan software proects of the COCOMO 8 dataset [0,,3]. As a consequence, we prefer to frst cluster the 63 proects nto a reasonable number of groups contanng smlar software proects. We most often use the APC-III clusterng algorthm developed by Hwang and Bang [9] to do ths. APC-III s β ) Effort /04/$ IEEE.

3 a one-pass clusterng algorthm, and has a constant radus R 0 defned as follows: R0 = α mn( P P ) (Eq. 3) = where s the number of hstorcal software proects and α s a predetermned constant. R 0 expresses the radus of each cluster and consequently t controls the number of the clusters provded. APC-III generates many clusters f R 0 s small and few clusters f t s large. Accordng to Hwang and Bang, the APC-III s qute effcent at constructng the mddle layer of an RBF, snce t can fnsh clusterng by gong through all the tranng patterns only once; ths s not true wth K-means and SOFM, whch are mult-pass and tme-consumng clusterng algorthms. Concernng the weghts β, we may set each β to the assocated effort of the center of the th neuron. However, ths technque s not optmal and does not take nto account the overlappng that may exst between receptve felds of the hdden layer. Thus, we use the Delta rule to derve the values of β. Fnally, our RBF has 3 nputs (COCOMO cost drvers) and one output (development effort). All the nputs as well as the output of the network are numerc. The RBF s traned by teratng through the tranng data many tmes. The Delta rule uses a learnng rate, and maxmum error equals 0.03 and 0-5 respectvely. 4. Overvew of the emprcal results The followng secton presents and dscusses the results obtaned when applyng the RBF to the COCOMO 8 dataset. We have conducted several experments wth the APC-III algorthm to decde on the number of hdden unts. These experments use the full COCOMO 8 dataset for tranng. Table shows the classfcatons obtaned accordng to dfferent values of α (Eq. 3). Table : umber of clusters accordng to α for the COCOMO 8 dataset α umber of clusters In analyzng the results of Table, we notced that the number of clusters s monotonous decreasng accordng to α. Ths s due to the fact that the radus R 0 s monotonous ncreasng accordng to α. Choosng the best classfcaton of those n Table to determne the number of hdden neurons and ther centers s a complex task. For software cost estmaton, we suggest that the best classfcaton s the one that satsfes the followng two crtera: t mproves the accuracy of the RBF; t provdes coherent clusters,.e. the software proects of a gven cluster have satsfactory degrees of smlarty. Thus, we have conducted several experments wth an RBF, each tme usng one of the classfcatons of the table. These experments use the full COCOMO 8 dataset for tranng and testng. The weghts β are calculated usng the Delta learnng rule. The RBF converges quckly, wth fewer than 000 teratons of learnng. The accuracy of the estmates generated by the RBF s evaluated by means of the Mean Magntude of Relatve Error MMRE and the Predcton level Pred, defned as follows: Effortactual, Effortestmated, MMRE = 00 ; Effort = actual, k Pred(p) = where s the total number of observatons, k s the number of observatons wth an MRE less than or equal to p. A common value for p s 0.5. The MMRE and Pred are calculated for each classfcaton of the table. Fgure shows the MMRE and Pred as functons of the classfcaton (α). We can notce that the accuracy of the RBF s better when α s lower than 0.49 (MMRE 30 and Pred(5) 70). When α s hgher than 0.49, the MMRE s hgh, although the values of Pred(5) are acceptable. Indeed, the MMRE s very senstve to varatons n the estmated values, partcularly when α ncreases, n whch case the classfcaton obtaned may be less coherent,.e. some clusters are composed of software proects that are not suffcently smlar. For those proects, the RBF may generate naccurate estmates MMRE Pred(5) 0,4 0,5 0,6 0,7 0,8 0,9 Fgure : Relatonshp between the accuracy of the RBF (MMRE and Pred) and the used classfcaton (α). In addton to the number of hdden neurons (number of clusters generated by the APC-III algorthm), the accuracy of the RBF depends on the wdths σ used by the hdden neurons (receptve felds). In the prevous experments (Fgure ), we had set the values of σ to 30. However, n the lterature, the wdths σ were usually determned to cover the nput space as unformly as /04/$ IEEE. 3

4 possble [8,9]. Coverng the hstorcal software proect space unformly mples that the RBF wll be able to generate an estmate for a new proect even though t s not smlar to any hstorcal proect. In such a stuaton, we prefer that the RBF does not provde any estmate than one t may easly lead to wrong manageral decsons and proect falure. As a consequence, we have adopted a smple strategy based prmarly on two gudelnes. Frst, we assgn one value to all wdths σ. Second, ths value depends on the radus R 0. Indeed, R 0 s used n the APC-III algorthm to delmt the area of each cluster. Ths looks lke a reasonable way to set the wdths σ to a value closer to R 0. As we notced earler, n addton to the accuracy of the RBF, the am s to explan ts process by mappng the RBF to a fuzzy rule-based system. The above emprcal experments allow us to choose whch RBF archtecture that wll be used n the mappng process. The method developed by Jang and Sun wll be used to extract the f-then fuzzy rules from the chosen RBF archtecture[4]. Ths method s presented below. 5. Equvalence between Radal Bass Functon networks and fuzzy rule-based systems: Background Snce ts foundaton by Zadeh n 965 [6], fuzzy logc has been the subect of many nvestgatons. One of ts man contrbutons to solvng complex problems s undoubtedly the Fuzzy Rule-Based Systems [7]. Essentally, an FRBS s based on a set of f-then fuzzy rules. A fuzzy rule s an f-then statement where the premse and the consequence consst of fuzzy propostons, whereas n a classcal producton rule the premse and the consequence are crsp. An example of a fuzzy rule n cost estmaton may be f the competence of the analysts s hgh then the effort s low. The man advantage of fuzzy rules over classcal rules s that they are easer for humans to understand and may be easly nterpreted. Ths s because fuzzy rules, unlke classcal rules, use lngustc values nstead of numercal data n ther premses and consequences. As a result, some researchers have nvestgated the equvalence between neural networks and FRBSs [,5,4,8], ther obectve beng to translate the knowledge embedded n the neural network nto a more understandable language, that s, the f-then fuzzy rules. Furthermore, establshng equvalence between neural networks and FBRSs can be used to apply advances and new developments of one model to the other and vce versa. Jang and Sun have developed a method whch proves the functonal equvalence between an RBF, such as the one used n the prevous sectons, and a fuzzy rulebased system whch uses Takag-Sugeno rules [4]. Hence, we use ther method to provde a natural nterpretaton of cost estmaton models based on the RBF. Jang and Sun fxed fve condtons to the establshment of a functonal equvalence between an RBF and a FBRS usng Takag-Sugeno rules. Under these condtons, the RBF s equvalent to an FBRS whch uses a set of fuzzy rules R assocated wth all hdden neurons: R : f x s A and x s A and.. and x s A n n then y = β where x are the nputs of the RBF, β are the weghts of the hdden unts to the output unt, and A s a fuzzy set wth a Gaussan membershp functon defned by: µ A ( x c( )) σ ( x) e A = (Eq. 4) where c( A ) s the th component of the th center of the classfcaton used by the RBF. 6. Valdaton and nterpretaton of the fuzzy rules In ths secton, we apply the method developed by Jang and Sun on the RBF presented and dscussed n Sectons, 3 and 4. The RBF that we consder adopts the classfcaton assocated to α=0.45 because of the accuracy of ts estmates (Fgure ). Hence, the RBF has 59 hdden unts and therefore the equvalent FBRS has 59 fuzzy f-then rules. Each fuzzy rule contans 3 fuzzy propostons n ts premse and one numercal output. These 59 fuzzy rules express the knowledge encoded nto the synaptc weghts of our RBF. The obectve s to gve a more comprehensble nterpretaton of these fuzzy rules n software cost estmaton. Each fuzzy rule wll nterpreted by nterpretng ts premse, ts output and ts mplcaton. For smplcty, we dscuss only three fuzzy rules (Table ). The premse of each th fuzzy rule s composed of 3 fuzzy propostons combned by the and operator. Each s assocated wth one cost drver (nputs of the RBF). For nstance, the frst proposton of the rule R DATA s approxmately equal to 7.74 concerns the cost drver DATA, whch represents, wth the other three cost drvers, the effect of the sze and complexty of the database on the software development effort. Each th proposton s expressed as x s A where A s a fuzzy set wth the membershp functon of Equaton 4. It can be understood as x s approxmately equal to c ( A ), where c( A ) s the th element of the th center. The lngustc qualfcaton approxmately equal to s represented here by a fuzzy set wth a Gaussan membershp functon of Equaton 4. The output of each fuzzy rule s a numercal value. By analyzng the rules of Table, we notce that the output of rule R s postve (7.44), whereas that of rule R 5 s negatve (-30.). These two values are the synapse weghts of the hdden unts representng the clusters C and C 5 to the output unt. Consequently, the natural /04/$ IEEE. 4

5 nterpretaton that we can gve to the outputs of the 59 fuzzy rules s that they can be consdered as partal contrbutons to the total effort. They have the same role as the effort multplers n the COCOMO 8 model. They can ncrease (postve value) or decrease (negatve value) the total development effort. The only dfference between them s that the outputs of the fuzzy rules are postve or negatve. Ths s because the cost functon of the FRBS uses the summaton operator, whereas n the COCOMO 8 model, due to ts cost functon that uses the multplcaton operator, the effort multplers are greater than (ncrease the effort) or less than (decrease the effort). Up to now, we have been suggestng a natural nterpretaton of the premses and outputs of the obtaned fuzzy rules. It stll remans to consder the meanng of each fuzzy rule taken as a whole, n other words, the nterpretaton of the mplcaton contaned n each fuzzy rule. Accordng to our understandng of the premses and the outputs, each fuzzy rule R can be wrtten as follows: R : f x (P) s approxmately equal to c ( A ) and x (P)s approxmately equal to c ( A ) and M x M (P)s approxmately equal to c ( A ) then effort (P)= β cluster. Thus, each fuzzy rule can be nterpreted as follows: where x (P) are the values of the cost drvers of the new proect P and c( A ) are those of proect C, that s, the center of the th cluster. If we reduce the wdth σ of the hdden neurons, the values β, derved from the Delta learnng rule, converge to the real efforts of C s, where C s are the centers of clusters. Indeed, n ths case, receptve felds (hdden neurons) do not overlap and consequently each new proect s classfed nto only one. Table : Three examples of the obtaned fuzzy rules P s smlar to C mples that effort(p) s smlar to effort(c ) Ths affrmaton s the bass of cost estmaton models based on reasonng by analogy [,0]. The man advantage of usng reasonng by analogy to estmate software development effort s that we can understand ts process and easly explan t to users. Ths s because humans are famlar wth ths knd of reasonng, as they use t every day of ther lves 7. Concluson and Future work In ths paper, we have studed one of the most mportant lmtatons of neural networks, whch s the dffculty of understandng why a neural network makes a partcular decson. Our study s ntended for applcaton to the cost estmaton feld, and the neural network that we have used to predct the software development effort s the Radal Bass Functon network. We used the entre COCOMO 8 dataset to tran and test the RBF. We have found that the accuracy of the RBF depends essentally on the parameters of the mddle layer, especally the number of hdden neurons and the values of the wdths σ. The number of hdden neurons s determned by one of the exstng clusterng algorthms; n partcular, we have adopted the APC-III algorthm because t s a one-pass clusterng technque and therefore t makes the estmaton process faster than f we had used an elaborate but tme-consumng clusterng algorthm. In addton to the number of hdden unts, the accuracy of the RBF depends on the wdths σ. We have shown that choosng one value closer to the radus R 0 for all the σ mproves the accuracy of the RBF. If : R R 4 R 5 DATA s approxmately equal to 7.74 and s approxmately equal to and s approxmately equal to 8.47 and VIRTm s approxmately equal to 6.47 and s approxmately equal to.48 and s approxmately equal to 0.40 and TIME s approxmately equal to and s approxmately equal to and s approxmately equal to and STORE s approxmately equal to 94.4 and s approxmately equal to 7.38 and s approxmately equal to and VIRTma s approxmately equal to and s approxmately equal to and s approxmately equal to 4.99 and TUR s approxmately equal to 3.50and s approxmately equal to.049 and s approxmately equal to.536 and ACAP s approxmately equal to and s approxmately equal to and s approxmately equal to and AEXP s approxmately equal to and s approxmately equal to 6.44 and s approxmately equal to and PCAP s approxmately equal to 8.05 and s approxmately equal to and s approxmately equal to and VEXP s approxmately equal to 3.3 and s approxmately equal to and s approxmately equal to 3.95 and LEXP s approxmately equal to.93 and s approxmately equal to and s approxmately equal to and SCED s approxmately equal to.338 and s approxmately equal to 90.3 and s approxmately equal to and KDSI s approxmately equal to 3.0 and s approxmately equal to 5.75 and s approxmately equal to 5.3 and Then Y=7.44 Y=03.37 Y=-30. After evaluatng the accuracy of the RBF, we appled the Jang and Sun method to extract the f-then fuzzy rules from the most accurate RBF archtecture. These fuzzy rules express the nformaton encoded n the archtecture of the network. The nterpretaton of each fuzzy rule s determned by analyzng ts premse, ts output and ts mplcaton. Our case study shows that we can explan the meanng of the output and the propostons composng the premse of each fuzzy rule. Indeed, each proposton of a premse can be expressed /04/$ IEEE. 5

6 as a statement lke x s approxmately equal to v where the lngustc value approxmately equal to s represented by a fuzzy set wth a Gaussan membershp functon. After provdng a natural nterpretaton of the premse and output of each fuzzy rule, we have suggested an explanaton of the entre fuzzy rule. Each fuzzy rule can be nterpreted as an analogy affrmaton smlar software proects have smlar costs, whch s easy for users to understand. 8. Bblography [] J. M. Bentez, J.L. Castro and I. Requena. Are Artfcal eural etworks Black Boxes? IEEE Transactons on eural etworks, vol. 8. no. 5, September 997, pp [] B.W. Boehm. Software Engneerng Economcs. Prentce- Hall, 98. [3] B.W. Boehm et al. Cost Models for Future Software Lfe Cycle Processes: COCOMO.0. Annals of Software Engneerng on Software Process and Product Measurement, Amsterdam, 995. [4] L. Brand, T. Langley and I. Weczorek. Usng the European Space Agency data set: A replcated assessment and comparson of common software cost modelng. In Proc th IEEE Internatonal Conference on Software Engneerng, Lmerck, Ireland, 000, pp [5] J. J. Buckley, Y. Hayash and E. Czogala. On the Equvalence of eural ets and Fuzzy Expert Systems. Fuzzy Sets and Systems, no. 53, 993, pp [6] C. J. Burgess and M. Lefley. Can Genetc Programmng Improve Software Effort Estmaton? Informaton and Software Technology, vol. 43, 00, pp [7] S. Hayken. eural etworks: A Gude to Intellgent Systems. Addson-Wesley, 994. [8] J. Hertz, A. Krogh and R. G. Palmer. Introducton to the Theory of eural Computaton. Addsson-Wesley, 99. [9] Y-S Hwang and S-Y Bang. An Effcent Method to Construct a Radal Bass Functon etwork Classfer. eural etworks, vol. 0, no. 8, 997, pp [0] A. Idr, and A. Abran, A Fuzzy Logc Based Measure for Software Proect Smlarty: Valdaton and Possble Improvements. 7 th Internatonal Symposum on Software Metrcs, IEEE Computer Socety, 4-6 Aprl, England, 00, pp [] A. Idr, T. M. Khoshgoftaar and A. Abran. Can eural etworks be easly Interpreted n Software Cost Estmaton. FUZZ-IEEE, Hawa, 00, pp [] A. Idr., A. Abran and T. M. Khoshgoftaar. Estmatng Software Proect Effort by Analogy based on Lngustc values. 8 th IEEE Internatonal Software Metrcs Symposum, 4-7 Aprl, Ottawa, Canada, 00, pp [3] A. Idr, T. M. Khoshgoftaar and A. Abran. Investgatng Soft Computng n Case-based Reasonng for Software Cost Estmaton. Internatonal Journal of Engneerng Intellgent Systems, vol. 0, no. 3, 00, pp [4] J. S. Jang and C. T. Sun. Functonal equvalence between radal bass functon networks and fuzzy nference systems. IEEE Trans. on eural etworks, vol. 4, 99, pp [5] J. Moody and C. Darken. Fast Learnng n etworks of Locally-Tuned Processng Unts. eural Computaton, vol., 989, pp [6] I. Myrtvet and E. Stensrud, A Controlled Experment to Assess the Benefts of Estmatng wth Analogy and Regresson Models. IEEE Transactons on Software Engneerng, 5, 4, July/August, 999, pp [7] M. egnevtsky. Artfcal Intellgence: A Gude to Intellgent Systems. Addson-Wesley, 00. [8] H-X. L and C. L. P. Chen. The Equvalence between Fuzzy Logc Systems and Feedforward eural etworks. IEEE Tran. on eural etworks, vol., no., 000, pp [9] J. Park and I. W. Sandberg. Approxmaton and Radal Bass Functon etworks. eural Computaton, vol. 5, 993, pp [0] M. Shepperd and C. Schofeld. Estmatng Software Proect Effort Usng Analoges. Transactons on Software Engneerng, vol. 3, no., 997, pp [] S. Shepperd and G. Kadoda, Usng smulaton to evaluate predcton systems. 7 th Internatonal Symposum on Software Metrcs, IEEE Computer Socety, 4-6 Aprl, UK, 00, pp [] K. Srnvasan, and D. Fsher Machne Learnng Approaches to Estmatng Software Development Effort. IEEE Transactons on Software Engneerng, vol., no.. February, 995, pp [3] A. R. Verkatachalam. Software Cost Estmaton usng Artfcal eural etworks. Internatonal Jont Conference on eural etworks, ogoya, IEEE 993. [4] S. Vcnanza, and M.J. Pretolla, Case-Based Reasonng n Software Effort Estmaton. Proceedngs of the th Int. Conf. on Informaton Systems, 990. [5] G. Wttg and G. Fnne. Estmatng Software Development Effort wth Connectonst Models. Informaton and Software Technology, vol. 39, 997, pp [6] L.A. Zadeh. Fuzzy Set. Informaton and Control, vol. 8, 965, pp [7] H. J. Zmmermann. Fuzzy Set Theory and ts Applcatons. nd ed, Boston, MA, Kluwe, /04/$ IEEE. 6