The Man-hour Estimation Models & Its Comparison of Interim Products Assembly for Shipbuilding

Transcription

1 Internatonal Journal of Operatons Research Internatonal Journal of Operatons Research Vol., No., 9 4 (005) The Man-hour Estmaton Models & Its Comparson of Interm Products Assembly for Shpbuldng Bn Lu and Zu-hua Jang Department of Industral Engneerng, Shangha Jao tong Unversty, Shangha 00030, Chna Receved August 004; Revsed September 004; Accepted January 005 Abstract The Man-hour of products s the basc data for ratonal producton plan. To estmate the man-hour of shp s nterm products both relably and accurately, three models for man-hour estmaton are proposed n ths paper. They are Smple Lnear Regresson Model and Multple Lnear Regresson Model, as well as Artfcal Neural Network Model. To evaluate the performance of the models, mathematcal methods were used to express the relablty and accuracy of each model. The relablty of the model was represented both by the consstency of the error center wth 0 and the consstency of the error dstrbuton wth normal dstrbuton. The accuracy of the model was represented by the Resdual Sum of Squares. It s verfed that the Model of Neural Network can elct more accurate and relable results than others va analyss and comparson. Keywords Man-hour, Shpbuldng, Neural network, Lnear regresson model. INTRODUCTION The man-hour quotaton of products s the basc data for ratonal producton plannng, cost accountng, performance evaluaton and vtal n achevng hgh productvty. The assembly man-hour quotaton accounts for about 30% to 50% of the total of shpbuldng man-hour quotaton. Conventonally, man-hour estmaton for assemblng a new nterm product was smply calculated as a functon of weght, followng ths method, the smple lnear regresson model was developed. In that model, the weght of product s regarded as the only valuable of man-hour. But the assembly man-hour depends not only on the weght of product but also many other factors (Bunch, 999). Thus, the multple lnear regresson model was developed (Chou, 00). Compared wth smple lnear regresson model, the latter s more advanced n precson and applcaton scope (Salenm, 997). Prevous studes have been lmted to smplfed lnear models, whch are nadequate for applcaton n current shpyard practces. In addton, the relatons between man-hours and other ndependent varables are very complex, and are regarded as a system of none-lnear and hgh order. Thus, t s almost mpossble to be descrbed under lnear theory framework. Ths s due to the deep gap between the postulated condton and practcal stuaton. Only a new model s establshed under the theory of nonlnear can ths problem be solved. The objectves of the research are to nvestgate the prevalng factors for man-hour of assembly by emprcal analyss and to construct man-hour estmaton model. Further more, applcaton of these models wll provde more accurate man-hour estmaton and allow decsons to be made more rapdly and scentfcally, thus enhancng the compettveness and proftablty of the shpyard corporatons. The remanng three sectons are devoted to the llustraton of the objectves of the research: the lnear regresson model was ntroduced n secton, n secton 3, a man-hour model usng neural network was proposed; The quanttatve and qualtatve comparson of the results of the three models s contaned n secton 4, whle the last secton s devoted to the conclusons.. LINEAR REGRESSION MODEL Conventonally, lnear regresson model s one of the most wdely used statstcal technques to descrbe and predct the relatonshp between man-hour and factors assocated to the man-hour. The lnear regresson models nclude smple lnear regresson model and multple lnear regresson models. Datum n Table show the product characterstcs and man-hour of 8 samples, whch were randomly collected on spot. Table. The Characterstc & Man-hour Data of Samples No. M (Kg) S (m ) L (m) Man-hour T Correspondng author s emal: lubn@sjtu.edu.cn 83-73X Copyrght 005 ORSTW

2 Lu and Jang: The Man-hour Estmaton Models & Its Comparson of Interm Products Assembly for Shpbuldng IJOR Vol., No., 9 4 (005) 0 Here, M s the weght of product ; S s projectve area of product ; L s the length of weldng lne of product ; T s assembly man-hour of product.. Smple lnear regresson model Smple lnear regresson model s descrbed as follow, T = β + β M + ε 0 T s man-hour for assembly the product, (man-hour), M s the weght of product (Kg;), β 0 and β are regresson coeffcents; ε s error term. Table. the Result of Smple Lnear Regresson β 0 β S e S A S T R F S e s the value of resdual sum of squares; S A s the value of sum of squares about regresson: S T s the value of devaton sum of squares; R s the value of fttngs, denotng the effects of ndependent varable on assembly man-hour; F s a statstc value for denotng the relablty of the model. As shown n Table, t can be seen that the sgnfcance level of lnear relatonshp between man-hour and the weght of products s very hgh, F s equal to.7. Ths ndcates that weght and man-hour do have postve correlaton. Thus, t can be concluded that the conventonal method of man-hour estmaton s ratonal n some degree. But on the other hand, the ftness value (R ) s small than 0.44, means that only about 40 percent varance of assembly man-hour s caused by weght of product, whle about 60 percent varance result n other factors omtted. Then, to mprove the correctness of man-hour model, t s reasonable to desgn a new model coverng more factors concerned man-hour. As a result, a multple lnear regresson model comes forward.. Multple lnear regresson model Multple lnear regresson model s as follow: T = β + β M + β S+ β L + ε 0 T s assembly man-hour of product, man-hour; M s the weght of product, Kg; S s projectve area of product, m; L s the length of weldng lne of product, m; β 0, β, β and β 4 are regresson coeffcents; ε s a random error term. Three parameters were selected as the ndependent varables for assembly man-hour estmaton. They are weght, length of weldng lne and projecton area of product. The results of Multple Lnear Regresson Model are showed n Table 3. S e s resdual sum of squares of error term: S A s sum of squares about regresson of error term; S T s devaton sum of squares of error term; R s ftted value, whch denotng the effects of ndependent varable on assembly man-hour; F s a statstc value for denotng the relablty of the model. t(), t(), t(3) are statstcs, whch denotng the ndependency level among ndependent varables. Table 3. The Result of Multple Lnear Regresson Model β 0 β β β 3 S e S A S T R F t() t () t (3) As shown n Table 3, the value of R reached 0.90, whch s far hgher than that n Table. It means that about 90 percent varance of man-hour s caused by the three factors aforementoned. Compared wth smple lnear regresson model, the descrpton capablty of multple lnear regresson model s more comprehensve. The rato of man-hour varance, caused by omtted factors, reduced from about 50 percent to 0 percent. As shown n Table 3, the values of t(), t() and t (3) are small. It means that the ndependency among ndependent varables s not sgnfcant. In other word, some nonlnear relatonshps exst among them and t s more complex and can be explaned by ths model. In ths case, to estmate assembly man-hour more accurately, t s necessary to develop a new model based on other theores and technques. 3. ARTIFICIAL NEURAL NETWORK MODEL There s a complex multple nonlnear relatonshp between assembly parameters and ts man-hour, whch s rather dffcult to deal wth n the theory of lnear regresson. A new theory or technology s needed to solve ths problem. In ths paper, artfcal neural network (ANN) wth a back propagaton algorthm s proposed for the generaton of man-hour, and t s verfed wth some examples n solvng the non-lnear hgh order problems. 3. BP (Back Propagaton) neural network ANN s bologcally nspred and s confrmed as an effectve algorthm. The technque deduces desred parameters by relatng learned functons to knowledge processng and pattern recognton. The fundamentals and practces of ANN are found n many books, such as, (Freeman, 99). The BP Network s chosen to map assembly man-hour and ts parameters of products, and then generate the man-hour of each. Fgure shows a conceptual dagram for an applcaton of artfcal neural network. As show n Fgure, the process of man-hour estmaton usng artfcal neural network s consttuted of three steps. In the frst step, man-hour data are collected on spot, and put nto man-hour database; n the second step, the neural network wll be traned usng the data of parameters and man-hour of samples whch was collected

3 Lu and Jang: The Man-hour Estmaton Models & Its Comparson of Interm Products Assembly for Shpbuldng IJOR Vol., No., 9 4 (005) n the frst step; n the thrd step, the assembly man-hour of new products wll be generated usng the network whch was traned n step. Cases of product Man-hour samplng 3. Man-hour generaton The precson of the model s very hgh after the tranng of 0,000 teratons, and sum of squares (MSE) about error term s close to zero wth a value of A scattergram of neural network predcted man-hour vs. lne of man-hour collected on spot s shown n Fgure 4. The graph shows that the neural network s predctons s close to the actual man-hour. Data-Base Tranng Neural Network Data-base Paramete r Input Neural Network Fnshed Tranng Man-hour Outpur Fgure. A conceptual dagram for applcaton of neural network A three layer network model was developed, followng the theory (Hecht-Nelsen, 989) that any L functon transformaton from [0, ], n to Rm can be mplemented to any desred degree of accuracy wth a three layer back propagaton neural network. The nput layer s conssted of three neurons correspondng to the three nput parameters, whle the output layer s conssted of a sngle neuron producng the desred man-hour estmate. The number of neurons n the hdden layer was found by tral and error snce the number of hdden unts per layer s more of an art than a scence (Karayanns, 99). A sample of 8 dfferent products was used. The rate of tranng rangng from 0.0 to.0 and the teraton s 0,000. The followng Fgure s the network structure. Fgure. The Structure of Neural Network. The process of tranng va ths model s showed n Fgure 3. In ths fgure, axs Y s error of the model, whle axs X s teraton of tranng. As the teraton becomes larger and larger, the error converged quckly. Iteraton Fgure 3. Convergent Cure of error. Fgure 4. Target/Output VS Pattern Number. 4. COMPARISON OF VARIOUS MODELS The ams of the establshment of a model nclude as follow () the descrpton and explanaton of a relatonshp among varables by estmatng the model parameters, and () the predcton of the dependent varable values gven the levels of ndependent varables. In ths secton, the correctness and precseness of smple regresson model, multple regresson model and neural network model are compared. And the technque for the comparson s manly based on error theory, statstcs theory combned wth characterstc of assembly process. The data and results of each model are analyzed, to fnd out the statstcal laws, and then analyze the dstrbuton shape of data. Then the performance of each model wll be compared based on the statstcal values. Table 4. The descrpton of Model Codes Model Code The descrbe of the Model Code SLR Smple Lnear Regresson Model (SMR) between the man-hour and product weght. MLR Multple Lnear Regresson Model of L, S, M and T NN Neural Network Model (NN) of L, S, M and T 4. Resdual sum of squares S e The value of resdual sum of squares of term error (Se) reflects the degree of precson of a model, the bgger the value, the bgger the mprecson of model, and the lower the degree of precson. On the contrary, the smaller the value s, the hgher the degree of precson. The value of Se of dfferent models s shown n the followng table. Table 5. The Resdual Sum of Squares (RSS) of Models Model Code SLR MLR NN Resdual sum of square S e As shown n Table 5, the value of Se of smple lnear

4 Lu and Jang: The Man-hour Estmaton Models & Its Comparson of Interm Products Assembly for Shpbuldng IJOR Vol., No., 9 4 (005) regresson model s hghest, and that of multple lnear regressons model s n medal, the Se of the NN s the smallest. Ths proved that the error of NN model s lowest, and the degree of precson s hghest. And the error of smple lnear regresson model s the bggest; the degree of precson s the lowest. 4. Dstrbuton shape of errors As mentoned above, the method-usng Se to evaluate the performance of a model s dstnct. But t couldn t gve the dstrbuton shape of error term of a model. We know, dstrbuton shape of error term s an mportant characterstc for model performance. So t s not suffcent to evaluate models only based on the statstcs of Se. And t s necessary to evaluate the dstrbuton shape of error term. Fgure 5 shows the dstrbuton shape of error terms of dfferent models. Here we can see, error dstrbuton of dfferent models has sgnfcant dfference. Wth the mprovement of the model, () rato of sample wth smaller error s ascendng, whle the rato of the bgger ones s descendng; () the error range becomes narrow; (3) the error center s closer to zero; and (4) dstrbuton shape s closer to normal dstrbuton. Is these dfference contngent? Accordng to the theory of error, any results of estmate can be expressed by: y = y + ε y s the valued of estmated man-hour of the sample ; y s the value of measured man-hour of the sample ; ε s the value of error term; s the code number. Errors ε consst of systematc error and chance error. In general stuaton, systematc error s greatly bgger than chance error, and t determnes the correctness of estmated results, whle the chance error reflects the degree of estmaton precson. The narrower the fluctuatng error range, the more precse the estmated results. However, systematc error s always hdden n measure results, and t s dffcult to make out. Thus, the systematc error s more dangerous than chance error. Generally, the systematc error s mostly caused by the mperfect relatonshp, the technques and methods for model establshments. Whle on the contrary, a chance error s caused manly by unconcerned, dstnct, ruleless factors, n general, ther nfluence on the estmated value s slght and dffcult to predct. Ths showed that: (a) chance error results from multple factors; (b) these factors are ndependent and unrelated to each other; (c) every fact alone has a slght effect on the fnal error. In another words, one factor has no sgnfcant effect, f there s, ts effects are equal and small. These are agreed wth Lyapunov condton. In such a case, the error term should present ts correspondng statstc law----normal dstrbuton law. As analyzed above, the correctness of a model depends on whether the model has systematc error n the model. And the precson of a model depends on the value of chance error. A model should not have any systematc error. So the systematc error must be elmnated from the model, or be elmnated enough to be neglected. Only n ths way, a model s correct. Under ths condton, the errors of a model are mostly comprsed of chance error. Then, the error of a model should represent characterstc of chance error normal dstrbuton wth mean of zero. 4.3 Normal dstrbuton test of errors Error Dstrbuton of SLR Error Dstrbuton of MLR Error Dstrbuton of NN Fgure 5. The Error Dstrbuton of Models. As mentoned above, estmated results of the model are correct only on the condton that there s no systematc error, f there s, t s small enough to be neglected. In that case, the errors of model are manly conssted of chance error, whch s normally dstrbuton wth mean of zero. In another words, there wll be no systematc error n the model f the error of the model obeys normal dstrbuton wth mean of zero. Hence, the correctness of the model can be test through the dstrbuton shape of the error test of the model. If the error s normally dstrbuton wth mean of zero, the model should be correct. The test of normalty dstrbuton of the three models wll be carred out, and the method of x ft test was stated by a mathematcan, K. Porson. The processes are:. Dvde the error term nto k nconsstent smaller ranges accordng the value of them. A = [ a0, a ), A = [ a, a ), Ak = [ ak, a k]. Calculate the observe value of computed as: x k = = ( n ˆ np ) npˆ x, whch s

5 Lu and Jang: The Man-hour Estmaton Models & Its Comparson of Interm Products Assembly for Shpbuldng IJOR Vol., No., 9 4 (005) 3 n s the sample sze; n s the frequency that sub-samples whch belong to range A. p ˆ s the estmator probablty that parent populaton whch belong to rang A ; p ˆ s the probablty of sub-sample whch belong to A. m s the number of parameters n probablty functon of samples. (For example, there are two parameters n normal dstrbuton functon: expected mean and varance). 3. Gven a certan confdence coeffcent, for example, α = 95%, and Degree of freedom, v = k m, search t n the table for x dstrbuton. Then get the crtcal value x α. Compare the observed value of x wth the crtcal value of x α. If the observed value s smaller than the crtcal value, then the error term obeys normal dstrbuton. To test whether the error term of the three models (SLR, MLR and NN) mentoned above s normal dstrbuton, frst, we dvde the orgnal data of each error term nto 8 ranges, gven a confdence coeffcent α =95%; then calculate correspondng observe value of x. In the end, from the table of x -dstrbuton, we get the value of x α (8 ). Compare the observe value x wth correspondng crtcal value x α (5), a concluson s made that whether the dstrbuton of each error term agree wth normal dstrbuton. Table 6. The Test of Normal Dstrbuton of Models SLR MLR NN Observed Value x Crtcal Value x Confdence Coeffcent α Normalty Test, Pass or not? NO PASS PASS As shown n Table 6, the error term of smple lnear regresson model ddn t pass the test of normalty. Whle the error terms of multple lnear regresson model and artfcal neural network model passed. That means that model of SLR has systematc error, whle the other two may not. To ensure models, MLR and NN, do not has systematc error, t s necessary to make a consstency check, yet. 4.4 Consstency test between mean error wth zero Accordng to the analyss above, that the dstrbuton shape of error term of a model obeys normal dstrbuton s a necessary condton rather than suffcent condton. To ensure there s no systematc error, f there s, ts systematc error s small enough to be neglected n the model, t s necessary to make a consstency check. In another words, on the condton that the errors dstrbuton shape obeys normal dstrbuton, t s necessary to test whether the error center (mean) s close to zero or not. Ths test s also called consstency check between error mean and zero. Here, the method s test. It s to say, the object of ths test s to confrm the consstency between error mean and zero on condton of a certan confdence coeffcent, for example, α =95%. The formulaton of calculatng the Statstc t: ε 0 t = n S n ε s error mean; n s sample sze; S n s sample varance. Table 7. The Consstency Check of Models Error Center MLR NN confdence coeffcent α 95% 95% Observed Value t <0. <0.0 Crtcal Value t Consstency check, pass or not? PASS PASS Table 7 shows the consstency check results of error mean wth zero of dfferent models. As shown n the table, the observed values of MLR Model and NN are greatly less than the crtcal value,.708. Ths result suggests that both error means of the two models have passed the consstency check. Further more, t gves proof that the systematc errors of the two models are too lttle to be neglected, or even there s no systematc error. 4.5 Comparson of precson The above analyss ndcates that the error term of the models (MLR, NN) s normally dstrbuted wth mean of zero. It means that there s systematc error n the model or t s lttle enough to be neglected. That s to say, the estmated results are correct and relable. Hence, an nterestng queston arse, whch, MLR model or NN model, would be a choce under such crcumstance. It s known that the value of Resdual sum of squares (S e ) of error term reflects the precson of a model. The smaller the value of S e s, the more precse the model. But when the values of S e of two models are dfferent, are ther precsons sgnfcant dfferent? Can we say the precson of a model wth smaller S e s greatly more sgnfcant than the other one wth bgger S e? Ths problem can be solved by mathematcal and statstcal methods. And how to make a concluson whether the value of S e of two models s sgnfcant? In other words, s the model wth a smaller S e more greatly precse than another under a certan confdence coeffcent, for example, α = 95%r? Statstc F can use to check that. the formulaton of calculatng the Statstc F: * S S n n F = = * S n S n n n n n S s sample varance of error term ε from model ; S s sample varance of error term ε from model ;

6 Lu and Jang: The Man-hour Estmaton Models & Its Comparson of Interm Products Assembly for Shpbuldng IJOR Vol., No., 9 4 (005) 4 n s sample sze of model ; n s smple sze of model. As shown n Table 8, n ths case, the crtcal value s.93, whle the observed value of statstc F s 38.5 >>.93. The two values of Se of the two models are sgnfcant dfference wth confdence coeffcent 95%. Also n ths way, t can be assumed true that the S e value of Multple Lnear Regresson Model s bgger than that of Artfcal Neural Net Work Model. That s to say, the precson of ANN model s greatly hgher than that of MLR model. network, Internatonal Jont Conference on Neural Networks, Washngton, DC. 7. Cavalera, Sergo, Maccarroneb, Paolo, and Pntoa, Roberto. (004). Parametrc vs. neural network models for the estmaton of producton costs: A case study n the automotve ndustry. Internatonal Journal of Producton Economcs, 9: Younger, M.S. (985). Handbook for Lnear Regresson, The Unversty of Tennessee Press, Knoxvlle. Table 8. The Results of F check MLR NN Confdence Coeffcent α 95% 95% Sample varance S n Observed value of F 38.5 Crtcal Value of F Is there sgnfcant dfference? Yes 5. CONCLUSIONS To estmate the assembly man-hour of vary products n shpbuldng both correctly and precsely, three models are proposed n ths artcle, namely, smple lnear regresson model, multple lnear regresson model, and artfcal neural network model. And the descrptve capablty s compared. The man fndng n ths paper maybe summarzed as follows: (a) Whle estmatng the assembly man-hour of the ntermedate products, Systematc errors of Lnear Regresson model and Artfcal Neural Net Work Model are so lttle that they can be neglected. (b) The precson of Artfcal Neural Network Model s better than Multple Lnear Regresson Model. Comparng, the descrptve capablty of Artfcal Neural Network Model s far better than the other two models, and the assembly man-hour estmated usng t wll be more correct and precse. REFERENCES. Bunch, Howard (990). Study of the causes of man-hour varance of naval shpyard work standards. Journal of Shp Producton, 6(4): 68.. Chou C-C. and Chang P-L. (00). Modelng and Analyss of Labor Cost Estmaton for Shpbuldng: The Case of Chna Shpbuldng Corporaton. Journal of Shp Producton, 7(): 9-96(5). 3. Freeman, J.A. and Skapura, K.M. (99). Neural Networks Algorthms, Applcaton and Programmng Technques, Addson-Wsedy, Calforna. 4. Salenm, M.D. (997). Multple Lnear Regresson Analyss for Work Measurement of Indrect Labor. The Journal of Industral Engneerng, Karayanns, N. (99). Theory of Back Propagaton Neural Network, Internatonal Jont Conference on Neural Networks, Baltmore, MD. 6. Hecht-Nelsen, R. (989). Theory of back Propagaton neural