Grey forecast model for torpedo development cost based on neural network

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1 Idia Joural of Geo Marie Scieces Vol. 46 (08), August 07, pp Grey forecast model for torpedo developmet cost based o eural etwork Qigwei Liag *, Miqua Zhao & Megi Wag College of Marie Sciece ad Techology, Northwester Polytechical Uiversity, Xi a Shaaxi, 7007, P.R.Chia No Uit of the PLA, Qihuagdao, Hebei, 06600, P.R.Chia *[ liagqigwei@wpu.edu.c] Received 5 Jauary 05 ; revised 30 October 06 Cumulative curve of developmet cost for torpedo shows similar to shape S. The grey Verhulst model of developmet cost for torpedo ca be established by collectig the developmet cost historical data for several years. The fitted values ad the predictive value ca be gotte. To take full advatage of iformatio cotais i the historical data, a mappig betwee historical data ad the predictive value usig BP eural etworks ca be foud through BP Neural Networks. Example shows that this method has good predictio accuracy. [Keywords: Developmet cost; BP eural etworks; Grey Verhulst model; Torpedo] Itroductio The LCC of torpedo meas all expeses paid i the life cycle, icludig demostratio cost, developmet cost, productio cost, operatio ad maiteace cost. The developmet cost is a importat part of LCC. It oly accouts for 8% of total cost, but determies 95% of the total cost. Substatial icrease i cosumptio fuds with the progress of life cycle, however, the impact capacity of LCC is gettig less ad less. This shows that studyig developmet cost has great sigificace for reducig LCC. May methods is used i modelig the developmet cost of products. Referece preseted a evolutioary learig process, called DEP(MGA), usig a modified geetic algorithm(mga) to desig the dilatio-erosio perceptio(dep) model of developmet cost. Referece 3 proposed a o-parametric approach that itegrates a eural etwork method with cluster aalysis to estimate developmet cost. Referece 4 used wavelet eural etworks to set up developmet cost model. Referece 5 ewly itroduced the Cost Correctio Factor(CCF) ad Low Cost Small Satellite (LCSS) adjustmet factor as additioal parameters for developmet cost estimatio. Referece 6 developed the Lagrage relaxatio decompositio(lrd) method with heuristics to solve the developmet cost problems. Referece 7 proposed a rule-based approach for estimatig software developmet i the requiremets aalysis phase cost. Referece 8 proposed a method combied kowledge miig by rough set ad eural etwork. Referece 9 researched developmet cost estimatio for airbore fire cotrol radar based o effectiveess idexes. Referece 0 predicted missile developmet cost based o eural etwork. The author of this paper proposed grey Verhulst model of developmet cost i Referece. Grey model has several advatages, such as relevat factors for forecast is t eed take ito accout, the radomess of time series is reduced, ad wealthy iformatio from small sample sequeces ca be gotte. But it has several drawbacks too, such as predictio accuracy is low, ad error accuracy ca ot be cotrolled. Because of powerful processig capabilities of oliear problems, eural etwork has a strog advatage i data fittig ad fuctio approximatio. But it eeds a large amout of computatio, ad eeds a large umber of sample data i traiig. I this paper, full advatage of the complemetarities of the two theories &4 is take, iformatio of developmet cost data is mied as much as possible.

2 6 INDIAN J. MAR. SCI., VOL. 46, NO. 08, AUGUST 07 Materials ad Methods Grey Verhulst predictio model 5 The raw data series is as follows: X x, x,, x () Accumulative geeratio operatio series (amed - AGO series) ca be gotte: X x, x,, x () k 0,,,, ; x k x i k i From the -AGO series X, the MEAN series is: Z z, z,, z (3) The, ( ) z k 0.5 x k x k, k,,, 0 X az b Z is the grey Verhulst model. I this formula, ab, are model parameters. a is called developig coefficiet, the value reflects the 0 growth rate of series X. b is called grey iput. LSE parameters vector of grey differetial equatio is met: T T T aˆ a b B B B Y (4) z z 0 x 0 z 3 z 3 x 3 B, Y= 0 x z z The time respose series of grey Verhulst model is: xˆ k 0 xˆ 0 x. ax 0 ak bx 0 a bx 0 e (5) Iverse accumulated geeratig operatio series (amed -IAGO series) of x i ca be gotte: 0 xˆ k xˆ k xˆ k k,,, The grey Verhulst model series is: ˆ X xˆ, x,, x (6) Ad the predictive value of poit is 0 xˆ. Grey Verhulst predictio model Accuracy Test Usually, posterior differece method is used i grey model accuracy testig 5. Accuracy is determied by the mea square error ratio (MSE ratio) ad small error probability together. The basic method is as follows: x is raw data series, ˆx is model series, is residual error series, 0 k xˆ 0 k x 0 k (7) The MSE of the raw data series is S, ad the MSE of the residual error series is S, the 0 0 S x k x, S k - - k k k 0 0 x x k, k k The mea square error ratio is defied as S C (8) S Small error probability is defied as 0 g p P k S (9) g is the total umber of 0 k S. Idex C is the smaller, the batter. Small C meas S is big ad S is small. Big S idicates that the 0 0 MSE of x is big, amely, the swig rag of x is 0 big, ad the rule of x is ot obvious. Small S 0 idicates that the MSE of x is small, amely, the 0 0 swig rag of x is small, ad discrete degree of x is small. Small C idicates that although the rule of 0 x is ot obvious, the differeces of fitted values ad fact values are ot discrete. Idex p is the bigger, the batter. Big p meas more poit which differece of residual error ad the average residual error less tha S. The grade of predictive precisio for grey model is show i table 6. BP Neural Networks Neural etwork 7,8,9&0 has the ability of self learig, self adaptig ad oliear processig. It obscures the kowledge gettig from self learig ito the etwork structure. Its way dealig with complex oliear system has the essetial differeces from traditioal methods ad has its ow predomiace. As a typical learig algorithm of artificial eural etworks, BP eural etwork is multi-layer etwork usig oliear differetiable fuctio with weight traiig. It cotais the essetial part of the eural etworks. Its structure is simple ad fictile.

3 LIANG et al.: GREY FORECAST MODEL FOR TORPEDO DEVELOPMENT COST BASED ON NEURAL NETWORK 63 Table Grade of predictive precisio for grey model Grade of predictive precisio p C grade (good) p 0.95 C 0.35 grade (qualificatio) 0.95 p C grade (grudgig qualificatio) 0.80 p C grade (disqualificatio ) p 0.70 C 0.65 Results ad Discussio Grey Forecast Model for Torpedo Developmet Cost Based o Neural Network From study, it is foud that if the develop course follows a ormal procedure (that is, blue prit demostratio, elemetary desig, detailed, trialmaufacture, experimet, validate, fialize the desig.), the developmet course accordig to the pla o the whole, ad there is o big gusty alteratio, the developmet cost for torpedo eeds few at the begiig, ad as the time goes by, it grows ad reach a piacle, the drops. The cumulative developmet cost curve of torpedo shows a growth tred, icreased slowly at the begiig, ad the grows rapidly. Fially, it teds to a limit slowly. So the developmet cost of torpedo preset a growth situatio similar to the growth of plat: rise slowly at the begiig, the icrease rapidly, go to limit at last. The figure is similar to shape S. As show i fig.. y Fig. The curve of cumulative developmet cost for torpedo Grey Verhulst model is fit for characterizatio the tred of saturated S shape, so it is selected to set up the model of developmet cost for torpedo. Although grey Verhulst model has the superiority of less iformatio ad simple, BP eural etwork has the characteristic of effective simulatio. To take full advatage of the iformatio cotaied i the origial data, combiatio model of grey Verhulst eural etwork is established. The method is as follows: First, grey Verhulst model accordig to origial data series () is established, ad the model series (6) ( ˆX 0 gotte. ) ad residual error series (7) ( 0 k t )ca be Residuals may be positive or egative, therefore, all the residuals are made positive first: plus the absolute value of the miimum egative ad plus 0 for each data. Use e k, k,,, to distiguish e k k mi k, k,,, Regroupig series () ad, s cosecutive years of practical value is iput, ad the modified residual of grey model of ext year is output, the all data ca be divided ito t groups( s + t - = ). Matrix ca be accessed: x x x s e s x x 3 x s e s () x t x t x e x t x t x e 0 e is ukow. Neural etwork model is established usig the matrix metioed above. The iput layer is s, amely the s row of the matrix. Ad output layer is, amely the s th row of the matrix. For etwork traiig, frotal t - lie of data is used, ad a series of weights ad threshold correspodig to each ode ca be gotte. To avoid fallig ito local optimal solutio, LM algorithm is used traiig BP etwork i this paper. The etwork cosists of iput layer, hidde layer ad output layer. The taget Sigmoid fuctio is used as activatio fuctio betwee iput layer ad hidde layer, ad liear fuctio is used betwee hidde layer ad output layer. The rage of Sigmoid fuctio is [0, ]. At the same time, the Sigmoid fuctio varies very slowly, which is ot good for the draw of character. I this paper, the iput sample is ormalized to [0., 0.9] for improvig the speed of etwork covergece. X x ( k) is the raw series, ad is Suppose the sample capacity, the ormalize method is as follows: * x ( k) x x ( k), ( k,, L, ;) () S

4 64 INDIAN J. MAR. SCI., VOL. 46, NO. 08, AUGUST 07 x * ( k) is the ormalized data (defie X * x * ( k ) x ( k) x ). ( ) ; x x k k k S. With this ormalizatio, the mea value of the variable is 0, ad the MSE of the variable is, ad the * ifluece of dimesio is elimiated. But X is ot i the rage of [0., 0.9] defiitely. The followig trasform ca reach this aim: * * 0.8 x ( k) mi x ( k) # k x ( k) 0. (3) max * ( ) mi * x k x ( k ) k i After completig the traiig, k lie of matrix (9) is used to predict, the predictive value of residual 0 e at time + ca be gotte. 0 e ca be coverted ito ; e mi i, i,,, (4) 0 Predictive value of residual is compesated i the predictive value of grey Verhulst model. Ultimately, the combiatio predictio is 0 xˆ ( ) xˆ ( ). Example The developmet costs of a torpedo year after year are show as table. It s assumed that the developmet costs of 0 th year is ukow. I order to predict the developmet costs of 0 th year, a model usig the data of former 9 years is established, ad it s compared with the actual costs. 0 X 55, 58, 90,5, 5, 63, 45, 0,38,08 (5 ) a ad b ca be gotte from equatio (4): a , b From equatio (5), the time respose fuctio ca be gotte: xˆ k e k From the time respose fuctio, the fitted value series of grey Verhulst model is: X ˆ (55, 0, 8, 3.6, 503.7, 747.3, 004, 6.8, 389.8, 494.9) 0 Xˆ (55, 46.0, 80, 30.6, 9., 43.6, 56.7,.8, 63., 05.) (6) ˆ 0 X () ca be predicted. The predicted value of ˆ 0 X () is 557., the the predicted value of ˆ 0 X () is 6.. From equatio (7), grey Verhulst residual series ca be gotte: 0 (0,.0, 9.96,.43, 9.43,.69, 0.80, 5.08,.9) Substituted ito the accuracy formula (8) ad (9), it ca be gotte that the accuracy of model is. I order to obtai more accurate model, BP eural etwork is used to improve model. 0 Accordig to fuctio, ca be coverted 0 to positive series e : 0 e (7) 3.9,.9, 3.95,.48,, 4.48, 35.59, 44.7, 48.99, 0.98 Accordig to matrix (), data (5) ad (7) is grouped. s is 4, the it ca be divided ito 7 lie of data( t is 7). The matrix ca be gotte: e () The BP etwork ca be structured by MATLAB cotrol box. After multiple test, BP eural etwork of 4 3 is used, taget Sigmoid fuctio betwee iput layer ad hidde layer is selected, ad liear fuctio betwee hidde layer ad output layer is selected. 6 lie of data is used as a sample for traiig etwork, the maximum learig times is set 000, learig rate is set 0.0, the error sum of squares as learig objectives is set 0.000, iput values are ormalized to [0., 0.9](equatio (3)), the iitial value of etwork coectio weights is set a stochastic value i [-, ]. Table. Developmet Cost Uit:0 000 RMB, (first fiscal year) Year Developmet Cost

5 LIANG et al.: GREY FORECAST MODEL FOR TORPEDO DEVELOPMENT COST BASED ON NEURAL NETWORK 65 Year Actual value Table 3. Compariso Table of th Actual ad its Predicted Value Grey Verhulst Model Combiatio Model Predicted Value Relative Error( %) Predicted Value Relative Error( %) Fig. The effect of traiig The effect of traiig is show as fig.. Network coverges i 000 Epochs or so whe simulated by Matlab, ad the expected error ca be met. Through sectio 7 sets of data, the residual of th 0 year e is 0.36, ad the actual residual of th 0 year is through equatio (4). The the combied predictive value of th year is accordig to equatio (7). From table 3, it ca be draw that the accuracy of grey Verhulst eural etwork model is better tha grey Verhulst model. Re-modelig by all raw data of table, the the predicted value of th year is Coclusio Torpedo developmet cycle is ot too log, which determies that the sample of aual developmet cost is less. Grey theory suitable for predictio of small sample, but its accuracy is lower. I the paper, the complemetarily of the two theories is take full advatage of, ad iformatio value of data is mied as much as possible. Grey Verhulst eural etwork successfully predicted the developmet cost for torpedo. Example shows that the model has higher accuracy. Ackowledgemet This work is supported by Chia Scholarship Coucil (CSC NO ), for which the authors are grateful. Refereces Wolter J. Fabrycky ad Bejami S, Blachard. Life-cycle cost ad ecoomic aalysis. Pretice-Hall, (99). Ricardo de A. Araújo, Adriao L.I. Oliveira, Sergio Soares, Silvio Meira, A evolutioary morphological approach for software developmet cost estimatio. Neural Networks, 3: Aita Lee, Chu Hug Cheg, Jaydeep Balakrisha. Software developmet cost estimatio: Itegratig eural etwork with cluster aalysis. Iformatio & Maagemet, (998) 34: K. Viay Kumar, V. Ravi, Mahil Carr, N. Raj Kira, Software developmet cost estimatio usig wavelet eural etworks. The Joural of Systems ad Software, (008) 8: Youg K. Chag, Hograe Kim, Ji S. Kag, Developmet of reliability-corrected cost model for Small Earth Observatio satellites. Acta Astroautica, (03) 88: Wu ZQ, C.K. Kwog, Tag JF, J.W.K. Cha, Itegrated model for software compoet selectio with simultaeous cosideratio of implemetatio ad verificatio. Computers & Operatios Research, 39: Soohwag Choi, Sooyog Park, Vijaya Sugumara, A rule-based approach for estimatig software developmet cost usig fuctio poit ad goal ad sceario based requiremets. Expert Systems with Applicatios, 39: Wag F, Wu XY, MA Da-wei, Cost forecastig of rockers weapos developmet based o kowledge miig. Systems Egieerig ad Electroics (007) 9(): Zhag L, Zhag XB, Zhag TM, Research o developmet cost estimatio for airbore fire cotrol radar based o effectiveess idexes. Fire Cotrol & Commad Cotrol 37(5): Liu GL, Tag XB, Liu YL, Predictio for missile developmet cost based o eural etwork. Tactical Missile Techology, (003) (): Liag QW, Sog BW, Jia Y, Grey Verhulst model of developmet cost for torpedo. Joural of System Simulatio (005) 7(): Shi WR, Wag YX, Tag J, Fa M, Water quality parameter forecast based o grey eural etwork modelig. Computer Applicatios, (009) 6: Zhag L, Wag L, Performace Assessmet Based o BP Neural Networks i Kowledge based Compaies. Proceedigs of the 7th World Cogress o Itelliget Cotrol ad Automatio, (008): Zhag X, Li SP, Dyamic data sequece forecastig base o combied model of grey eural etwork. Electroic Measuremet Techology, (009) 30: Deg JL, Grey predict ad Grey decisio-makig, Wuha: Huazhog Uiversity of Sciece ad Techology Publishig House, (00).

6 66 INDIAN J. MAR. SCI., VOL. 46, NO. 08, AUGUST 07 6 Xu XX, SUu Hua-you, Zhag CP, Wag W, Applicatio of optimized o-equidistace GM(,) model for predictig foudatio settlemet i the base pit viciity. Mathematics i Practice ad Theory (04) 44(): Wag HS, Wag YN, Wag YC, Cost estimatio of plastic ijectio moldig parts through itegratio of PSO ad BP eural etwork. Expert Systems with Applicatios, (03) 40: Wu QM, Wei M, A mathematical expressio for air ESD curret waveform usig BP eural etwork. Joural of Electrostatics, (03) 7: Zhuo L, Zhag J, Dog P, Zhao YD, Peg B, A SA GA BP eural etwork-based color correctio algorithm for TCM togue images. Neurocomputig, (04) 34: Zheg DZ, Peg P, Fa SC, A research of dyamic compesatio of coriolis mass flowmeter based o BP eural etworks. Istrumets ad Experimetal Techiques (03) 56(5):