A min-max Approach for Improving the Accuracy of Effort Estimation of COCOMO

Transcription

1 Internatonal Journal of Soft Computng and Engneerng (IJSCE) ISSN: 3-307, Volume-, Issue-3, July 0 A mn-max Approah for Improvng the Auray of Effort Estmaton of COCOMO H. S. Hota, Ramesh Pratap Sngh Abstrat hs study proposed to extend the Construtve Cost Model (COCOMO) by norporatng the onept of mn-max approah to estmaton. Formal effort estmaton models le Construtve Cost Model (COCOMO) are lmted by ther nablty to manage unertantes and mpreson surroundng software proets early n the development lfe yle. A mn-max approah s suggested to retfy data unertantes and modelng errors. he proposed method of mn-max s used to mprove the auray of effort estmaton of COCOMO and ts result have been ompared wth the gradent desent, robust fuzzy lusterng, -mean lusterng methods of estmaton. It has been observed that the proposed method have lowest and steady state absolute estmate error AE() and mean absolute estmate error MAE() for dfferent value of (tme seres) and dfferent step-sze s. Index erms Sugeno fuzzy nferene system, mn-max method, Construtve Cost Model (COCOMO), effort estmaton, Absolute Estmate Error, Mean Absolute Estmate Error. I. INRODUCION he preson and relablty of the effort estmaton s very mportant for software ndustry beause both overestmates and underestmates of the software effort are harmful to software ompanes. As hghlghted by MConnell n [], several surveys have found that about two-thrds of all proets substantally overrun ther estmates. In realty estmatng software development effort remans a omplex problem attratng onsderable researh attenton. It s very mportant to fnd novel method to handle vague and unertan nformaton for mprovng the auray of suh estmates. A mn-max approah [] based ost estmaton model s more approprate when vague and mprese nformaton s to be aounted for. he best nown tehnque usng LOC (Lnes of Code) s the COCOMO (Construtve COst MOdel), developed by Boehm[3]. hs model performs estmaton usng LOC, and other fators suh as development envronment, produt attrbutes and hardware lmtatons. hese fators provde one or more adustment fators whh adust the estmate of the effort needed. In COCOMO's ase, there are fourteen suh fators derved by Boehm. But Construtve Cost Model (COCOMO) s lmted by ther nablty to manage unertantes and mpreson surroundng software proets early n the development lfe yle [4]. Assumptons mae estmates more aurate. Fuzzy log-based ost estmaton Manusrpt reeved June, 0. Ramesh Pratap Sngh, Department of Computer Sene, D. P. Vpra College, Blaspur, Inda, M: , (e-mal: snghramesh30@gaml.om). H. S. Hota, Department of CS & I, Guru Ghasdas Vshwavdyalaya (A entral unversty), Blaspur, Inda, M:945658, (e-mal: hota_har@redffmal.om). models are more approprate when vague, mprese and unertan nformaton s to be aounted [5] for suggested the use of fuzzy sets n mprovng the auray of effort estmaton through fuzzy sets. But the amount of unertanty nvolve n the estmaton proess, the mn-max method s propose here to use n effort estmaton of software development. hs study for mprovng the auray of effort estmaton of COCOMO model assumng as physal proess y=f(x,.,x), s onerned wth the fuzzy parttonng of n-dmensonal nput spae nto K dfferent lusters then estmatng the proess behavor yˆ f ( xˆ ) for a gven nput xˆ ( xˆ xˆ,... n) X and then fuzzy approxmaton of the proess wth unertan nput-output data x ) x y( usng Sugeno type fuzzy (, ) y,... nferene system. he sze of the proet n COCOMO s represented by fxed numeral values. In mn-max based ost estmaton models, ths sze s represented wth fuzzy partton spae. So, the transton from one partton spae to an adaent partton spae s abrupt rather than gradual. II. RELAED LIERAURE REVIEW Estmaton auray s largely affeted by modelng auray [6]. Fndng good models for software estmaton s very rtal for software engneerng n bddng and plannng. In the reent years many software estmaton models have been developed [7, 8, 9, 0]. Gray and MaDonell ompared funton pont analyss regresson tehnques, feed forward neural networ and fuzzy log n software effort estmaton []. her results showed that fuzzy log model aheved good performane, beng out performed n terms of auray only by neural networ model wth onsderably more nput varables []. he frst realzaton of the fuzzness of several aspets of COCOMO was that of [3]. hey observed that an aurate estmate of delvered soure nstruton [4] annot be made before startng a proet and t s unreasonable to assgn a determnate number for t. he data drven onstruton of fuzzy models [5] has beome an mportant top of researh wth a wde range of real-world applatons. A ommon prate s to hoose the type of fuzzy model.e. Sugeno type [6], that s lnear n onsequent parameters and therefore standard least-squares lnear tehnques an be apply for ther estmaton. he onstrants n the estmaton of anteedent parameters may arse n order to preserve the nterpretablty of the fuzzy model [7]. he dervatve based estmatons methods le gradent-desent offer the advantage of fast onvergene n omparson to the dervatve-free methods le genet algorthms [8], but they tend to onverge to a loal mnma. 74

2 A mn-max Approah for Improvng the Auray of Effort Estmaton of COCOMO However, f the assumptons are not met due to modelng errors, they perform poorly. hs motvates the study of robust methods for fuzzy lusterng, estmaton and dentfaton of unnown physal proess. In ths researh paper t s proeted to unfed mn-max approah to fuzzy lusterng, estmaton and dentfaton wth unertan data suh that worst ase effet of regresson vetor unertanty, model output unertanty, and modelng errors on estmaton performane s mnmzed wthout mang any assumpton and requrng a pror nowledge of unertantes. he Sugeno-type fuzzy nferene systems are onsdered approprate models, sne they deally ombne smplty wth good analytal propertes [9]. Moreover the data-drven onstruton of Sugeno fuzzy systems allows qualtatve nsght nto relatonshps [0]. herefore we onsder the dentfaton of a Sugeno type fuzzy model wth unertan data that parttons the nput spae nto dfferent lusters and approxmate the nput-output mappngs n an optmal manner. III. FORMULAION OF HE PROBLEM Let us onsder a Sugeno-type fuzzy nferene system.e. Fs:X Y, mappng n-dmensonal nput spae (X=X x X x X 3 x.. x X n ) to one dmensonal real lne, onsstng of K dfferent followng rules: If x belongs to a luster wth Centre C HEN y=α If x belongs to a luster wth Centre C HEN y=α.. If x belongs to a luster wth Centre C HEN y=α Where xr n s n-dmensonal nput vetor, C R n s the entre of th luster, and the values α, α.. α are real numbers. In COCOMO effort s expressed as Person Month (PM). It determnes the effort requred for a proet based on software proet s sze n Klo Soure Lne of Code (KSLOC) as well as other ost drvers nown as Sale Fators (SF) and Effort-Multplers (EM) as shown below: s SF M sze.0 PM= n EM () Where, M s a multpler onstant and the set of Sale Fators (SF) and Effort Multplers (EM) are defned the model. Here n=5 EM and s=5 SF. he standards numer values of the ost drvers are gven n able 3 (see Appendx). Let x be a multvarable membershp funton :X [0,] that represents the degree of membershp of nput vetor x X to the th luster (n plae of fuzzy nterval values). Now the dfferent rules [] an be aggregated as F s x= x x he optonal shape of membershp funton () x ensure by the method of fuzzy C-means (FCM) as below xx m x. x Mnmum x = (3) an Where m> s the fuzzfer and. denotes the Euldean norm. he soluton of prevously onstraned optmzaton problem s gven as: = x, C,..., C x 0 m x (4) for x X \{ } for for x { } x,,...,,,..., \ { } How ever, ths lusterng rteron suffers more than a few drawbas le: () In presene of unertanty n the data, t may be dffult sometmes to expltly express how to reonstrut the data from a luster soluton. () he membershp funtons for the lusters are non onvex and for generaton of fuzzy sets, the membershp funtons should be onvex. o overome the above drawbas another lusterng rteron s adopted assumng that there s a nose luster outsde eah data luster. Followng lusterng rteron s hosen: J x x x x x x (5),,..., log xx where the seond term n the obetve funton s ntended as a nose luster. he term x log x x may be nterpreted as the degree to whh x does not belong to the th luster and thus the membershp of x to the nose luster. he rteron to set equal to the dstane of nearest luster enter from s: = mn o mnmze x J J. of equatons: x,,..., 0 (6) (7) hs result n the followng expresson for optmal membershp funton x,,..., exp x (a) Membershp funtons for eq n (4) before retfaton (8) 75

3 (b) Membershp funtons for eq n (8) after retfaton Fgure (a & b) shows the membershp funtons before (equaton 4) and after (equaton 8) the optmzaton and retfyng the nose lusters respetvely for three dfferent lusters wth entres at 0, 50 and 80.hus t an be seen (fg (b)) that the shape of membershp funtons beng onvex an be used to generate fuzzy sets. he output of Sugeno-type fuzzy nferene system for ths shape of membershp funton (fg (b)) s gven by: x F s =, x,..., x exp x exp = then F s x= x, Internatonal Journal of Soft Computng and Engneerng (IJSCE) ISSN: 3-307, Volume-, Issue-3, July 0 (9) x exp...(0) x exp,..., () Let us ntrodue the notatons, R.... n R,... and x, x, R... he equaton () an be wrtten as Fs x x, () Let us onsder the fuzzy approxmaton [] of software effort estmaton proess desrbed by tme seres equaton as gven below: y f x x v (3) µ where Input Varables x s unertanty present n nput vetor x v s the orrespondng unertanty n and y. Let, be some true value of fuzzy model parameters, whh approxmate the software effort estmaton proess then y x x v (4), or y x v, (5) Where x, x, s the unertanty n regresson vetor due to nose y f x n x x. For software effort estmaton proess, a unfed approah to robust fuzzy lusterng, estmaton and dentfaton s to estmate reursvely the parameters,, at, say measurements y x, v th tme ndex, wth the x, n presene of unertantes, wthout mang any assumpton and requrng pror nowledge of upper bounds, statsts and dstrbuton of data unertantes and modelng errors, suh that () For a nown x X x, f x mnmum (6) () For all x X x, f x where, lm, mnmum (7) IV. MIN-MAX APPROACH O SOFWARE EFFOR ESIMAION Now, we present a mn-max approah to the robust estmaton of luster enters vetor and fuzzy onsequents. he approah onssts of solvng a loal mn-max estmaton problem where the worse-ase effet of data unertantes and modelng error on estmaton performane s mnmzed. he software effort estmaton proess an be modeled as: y x, v (8) where v nludes not only the data unertantes, but also the error resultng from a dfferene between and ~. We defne the followng error measures for our problem. denotes the dfferene between and estmates.e. ~ (9) and e a denotes the pror estmaton error.e. x ~ (0) e a, o measures the estmaton performane, we defne nstantaneous absolute estmaton error (AE) at tme nterval as: AE 00 f x x 00 where the ponts,..., 00 x, () are unform dstrbuted n the -dmensonal nput spae, If the smulatons run from = to =, then the mean absolute estmaton error (MAE) an be defned as: MAE AE () We run over smulaton for =0000 for a far omparson, the same step-sze has been hosen for Gradent-desent, Robust fuzzy lusterng, K-means lusterng 76

4 A mn-max Approah for Improvng the Auray of Effort Estmaton of COCOMO and mn-max methods. Sne the dfferent methods may have ther best performane at dfferent step-sze, therefore we perform smulatons at dfferent values of step-sze rangng from S=0.0 to S=0.. able : Desrbes the dentfed value base usng mn-max approah at =0000 Rule s R R R3 R4 R5 R6 R7 R8 R9 Anteedent If x belongs to the luster havng entre 0.5,0.04 then y s equal to If x belongs to the luster havng entre 0.4,0.676 then y s equal to If x belongs to the luster havng entre 0.54,.5 then y s equal to If x belongs to the luster havng entre.55, 0.67 then y s equal to If x belongs to the luster havng entre 0.4,0.83 then y s equal to If x belongs to the luster havng entre.5,.5 then y s equal to If x belongs to the luster havng entre.5, 0.37 then y s equal to If x belongs to the luster havng entre.5,.04 then y s equal to If x belongs to the luster havng entre.5,.5 then y s equal to Conse quent AE(0000)) and mean absolute estmaton error (.e. MAE) of all four approahes at dfferent values of step szes S. A remarable good performane of proposed mn-max approah n omparson to others an be easly seen n Fg and Fg 3. hs verfes the robustness propertes of mn-max estmaton sheme n presene of data unertantes and modelng errors. he better performane of mn-max estmaton s due to the fat that no only lnear parameters (onsequents), but also the non-lnear parameters (luster entre s) attempt to math the output of the fuzzy model to unnown proess output n an optmal manner. Experments were done by tang orgnal data from COCOMO dataset [3]. he software development efforts obtaned when usng COCOMO [3, 4] and other membershp funtons observed. After analyzng the results attaned by means of applyng COCOMO model usng Gradent-desent, Robust fuzzy lusterng, K-means lusterng and mn-max methods, t s observed that the effort estmaton of the proposed model s gvng more prese results than the other models. he effort estmated by means of fuzzyfyng step-sze usng bell shaped membershp funton s yeldng better estmates, whh s very nearer to the atual effort. herefore, usng fuzzy sets (membershp funtons) sze of a software proet an be spefed by dstrbuton of ts possble values, by means of whh we an evaluate the assoated mpreson resdng wthn the fnal results of ost estmaton. V. SIMULAION OF PROBLEM Now we provde smulaton studes to ompare gradent-desent, robust fuzzy lusterng and K-means lusterng wth the proposed mn-max approah, for ths purpose, onsder a software effort estmaton proess for two nputs as: 0 x 0.06 x 0.06 y f x, x (3) x x where x and x 0.6,.6 0.6,.6 he fuzzy model dvdes the two dmensonal nput spae nto nne lusters wth ntal guess about luster enters as (equally spaed) ,..., 9 (4) he ntal guess 0 s taen equal to a zero vetor. he nonlnear system was smulated by hoosng x and x from a unform dstrbuton on the nterval (-0.6,.6). he unertan nput-output dentfaton data s generated by the sequene: x x, y y (5) x x where x, x and y are random entres, hosen from a normal dstrbuton wth mean zero and varane 0.0. Fg and Fg3 summarzes the smulaton results by omparng the fnal value of absolute estmaton error (.e. Fg. : Comparson of AE() for dfferent methods for dfferent values of step-sze S Fg 3: Comparson of MAE() for dfferent methods for dfferent values of step-sze S. VI. DISCUSSION From the above, t s observed that by fuzzyfyng the some step-sze of the proet dataset usng belshaped urved after retfaton, t an be proved that the resultng AE() and 77

5 MAE() for effort estmaton of COCOMO model for the proposed mn-max method s found better n performane. hs llustrates that by fuzzyfyng step sze usng belshaped membershp funton after ts orreton, the auray of effort estmaton an be mproved by usng proposed mn-max method and estmated effort an be very lose to the atual effort. hs onept open a new oneptual dmenson to the models of software ost estmaton. he COCOMO-8 onssts of 63 proets dataset. he quantfaton of COCOMO multplers of the dataset s a tas that must be done by an expert based on hs experene. herefore, there s an unertanty lyng n the dentfaton data (unertan measurements of parameters and omments of expert). VII. CONCLUSION hs study has outlned a new mn-max approah to the fuzzy lusterng, estmaton and dentfaton wth unertan data. he proposed approah mnmzes the worst-ase effet of data unertantes and modelng errors on estmaton performane of effort estmaton of COCOMO model, wthout mang and statstal assumpton and requrng a pror nowledge of unertantes. Smulaton studes have been provded to show the better performane of proposed method n omparson to the standard tehnques applyng on effort estmaton of COCOMO model. he performane of proposed method s found to be better n omparson to the standard tehnques. APPENDIX COCOMO COS DRIVERS Cost Drvers Range Desrpton RELY Requred Software Relablty DAA Database Sze CPLX Produt Complexty RUSE Developed for Reusablty DOCU Doumentaton Math to Lfe-Cyle Needs IME Exeuton me Constrant SOR Man Storage Constrant PVOL Platform Volatlty ACAP Analyst Capablty PCAP Programmer Capablty PCON Personnel Contnuty APEX.-0.8 Applatons Experene PLEX Platform Experene LEX Language and ool Experene OOL Use of Software ools SIE Mult ste Development SCED Requred Development Shedule REFERENCES [] Steve MConnell. Rapd development: tamng wld software shedules. Mrosoft Press, 996. [] Moht Kumar et al, A Mn-Max Approah to fuzzy lusterng Estmaton and Identfaton, IEEE ransaton on fuzzy system, vol. 4, No., PP 48-6, 006. [3] B.W. Boehm, Software Engneerng Eonoms, Englewood Clffs, NJ, Prente Hall, 98. [4] Iman Attarzadeh, Sew Ho Ow, Improvng the Auray of Software Cost Estmaton Model Based on a New Fuzzy Log Model, World Appled Sene, Vol. 8(), pp , 00. Internatonal Journal of Soft Computng and Engneerng (IJSCE) ISSN: 3-307, Volume-, Issue-3, July 0 [5] C.S. Reddy and KVSVN Rao, Improvng the auray of effort estmaton through fuzzy set representaton of sze, Journal of Computer Sene, Vol 5, pp , 009. [6] M. Jorgensen, B. Faugl,. Grushe, Charatersts of software engneers wth optmst predton, Journal of Systems and Software, Vol. 80, Issue. 9, 007, pp [7] C.L. Martn, J.L. Pasquer, M.C. Yanez,.A. Guterrez, Software Development Effort Estmaton Usng Fuzzy Log: A Case Study, IEEE Proeedngs of the Sxth Mexan Internatonal Conferene on Computer Sene (ENC 05), 005, pp [8] M.. Su,.C.Lng, K.K.Phang, C.S.Lew, P.Y.Man, Enhaned Software Development Effort and Cost Estmaton Usng Fuzzy Log Model, Malaysan Journal of Computer Sene, Vol. 0, No., 007, pp [9] A. Heat, Comparson of artfal neural networ and regresson models for estmatng software development effort, Informaton and Software ehnology, Vol. 44, Issue 5, 00, pp [0] X. Huang, Danny Ho, J. Ren, L.F. Capretz, Improvng the COCOMO model usng a neuro-fuzzy approah, Appled Soft Computng, Vol.7, Issue, 007, pp [] A. R. Gray, S. G. MaDonell, Applatons of Fuzzy Log to Software Metr Models for Development Effort Estmaton, Fuzzy Informaton Proessng Soety 997 NAFIPS 97, Annual Meetng of the North Ameran, 4 September 997, pp [] P. hrft, Fuzzy log synthess wth genet algorthms, n pro., 4 th Int. Conf. on Genet Algorthms, pp , 99. [3] Fe Z, Lu X (99) f-cocomo: fuzzy onstrutve ost model n software engneerng. In: IEEE nternatonal onferene on fuzzy systems, pp [4] D. Smon, Desgn and rule base reduton of a fuzzy flter for the estmaton of Motor urrents, Int. Jou. of Approx. Reason, Vol. 5, pp 45-67, 000. [5] J. A. Roubos and M. Setnes, Compat fuzzy models through omplexty reduton and evolutonary optmzaton, n pro. IEEE Int. Conf. fuzzy systems, San Antono, X, pp , 000. [6] J. S. R. Jang, ANFIS: Adaptve-networ-based fuzzy nferene systems IEEE rans. Syst., Man, Cybernets, Vol. 3, No. 3, pp , 993. [7] M. Setnes et al, Rule-based modelng preson and transpareny, IEEE rans. Syst. Man. Cybern. C. Apple. Rev., Vol. 8, No., pp 65-69, 998. [8] J. S. R. Jang, C.. Sun and E. Mzutan, Neuro-fuzzy and soft omputng: a omputatonal approah to learnng and mahne ntellgene, Upper Saddle Rver, NJ: Prente-Hall 997. [9] M. Kumar et al, Robust Adaptve Fuzzy Identfaton of me-varyng Proesses wth Unertan Data. Handlng Unertantes n the Physal Ftness Fuzzy Approxmaton wth Real World Medal Data: An Applaton, Fuzzy Optmzaton and Deson mang, Vol., No.3, pp 43-59, 003. [0] J. C. Bezde, Pattern Reognton wth fuzzy obetve funton algorthms, New Yor: Plenum, 98. [] L. X. Wang and J. M. Mendel, Generatng fuzzy rules by learnng from examples, IEEE rans. Syst., Man, Cybernets, Vol., No. 6, pp 44-47, 99. [] D. Nan and R. Kruse, A Neuro-fuzzy approah to obtan nterpretable fuzzy systems for funton approxmaton, n pro. IEEE Int. Conf. fuzzy systems 998 (FUZZ-IEEE 98), Anhorage, AK, pp 06-, May 998. [3] Menzes,., 005. Promse software engneerng repostory. [4] B. Boehm, E. Horowtz, R. Madahy, D. Refer, B. K. Clar, B. Steee, A. W. Brown, S. Chulan, and C. Abts, Software Cost Estmaton wth Coomo II. Prente Hall, 000. H.S. Hota: reeved MCA degree from the Bho Open Unversty, Bhopal (M.P.), Inda n 00, and the Ph.D. degree n Computer Sene from the Guru Ghasdas Unversty, Blaspur (Chhattsgarh), Inda n 006. Presently I am worng as Assstant Professor, Department of Computer Sene and Informaton eh., Guru Ghasdas Unversty, Blaspur (Chhattsgarh), Inda. My researh nterests nlude Software Engneerng and Soft Computng tools. 78

6 A mn-max Approah for Improvng the Auray of Effort Estmaton of COCOMO Ramesh Pratap Sngh: reeved M.S. degree n Informaton ehnology from the Guru Ghasdas Unversty, Blaspur (Chhattsgarh), Inda n 00, the M. Phl degree n Computer Sene from Alagappa Unversty, Karaud (.N.), Inda n 009. Presently I am worng as Assstant Professor & Head, Department of Computer Sene, D. P. Vpra College, Blaspur (Chhattsgarh), Inda. My researh nterests nlude Cost and Effort estmaton n Software Engneerng and Soft Computng tools. 79