Transitional Probability Model for a Serial Phases in Production

Transcription

1 Jurnal Karya Asli Lorkan Ahli Matmatik Vol. 3 No. 2 (2010) pag Jurnal Karya Asli Lorkan Ahli Matmatik Transitional Probability Modl for a Srial Phass in Production Adam Baharum School of Mathmatical Scincs Univrsiti Sains Malaysia Pulau Pinang, Malaysia Tl: , Fax: , adam@cs.usm.my Abstract : In this papr, a transitional modl is proposd for analyzing data mrging from longitudinal study in a production facility. At diffrnt tim points, w obsrvd th procss of a systm in trms of attaining crtain targts within a plannd tim for a sris of componnts. If som or all componnts ar dlayd, th ovrall tim will b longr than th plannd tim. Th proposd modl shows th transition probability for compltion of a job within th stipulatd priod for ach componnt as wll as th ovrall probability of a sris of jobs within th spcifid priod. Th transition modls for ach componnt ar xprssd as a function of tim and a dcision modl is proposd on th basis of attainmnts of th dsird targts. Kywords: transitional modl. 1. Introduction It has bn a challng to th rsarchrs for a long tim to modl a sris of machin componnts in ordr to rlat th undrlying covariats and th duration of prvious componnts with th compltion of jobs in ach componnt. It is vry usful to know how th covariats and prvious Bnjaafar and ElHafsi (2006) usd a Markov dcision procss and proposd optimal policis for th optimal machin and invntory control of an assmbl-to-ordr systm with m componnts, on-nd machin and n customr classs. Thy proposd a control policy spcifying whn to produc ach componnt. Gyr, Wassnhov and Atasu (2007) showd an conomic modl for limitd componnt durability and finit machin lif cycls. In a rcnt rsarch work, Hopp, Iravani and Yun (2007) pointd out that in many systms, particularly srvic and profssional work, judgmnt is frquntly ndd to dtrmin how much tim to allocat to a spcifid task. This papr highlights a rlatd issu how th prvious componnt in a sris affcts th subsqunt componnt. 2. Th Modl with Singl Mod of Failur Th modl proposd hr is basd on th rsarch works conductd by Bonny (1987), Islam (1994), Diggl, Hagrty, Liang and Zgr (2002), Islam t al (2004), Islam and Chowdhury (2006, 2007). Islam (1994) dvlopd transitional modl for continuous failur tims and Bonny (1987), Islam t al. (2004) and Islam and Chowdhury (2006) showd th conditional modls for a finit stat spac of first or highr ordrs. This papr applis th framwork of ths works to dvlop a modl for a sris of componnts. Lt us dfin th hazard function for occurrnc of vnt as a function of covariat vctor Z: t; X lim P t T t t / T t; X t / t (2.1) t 0 whr X(t) dnots th valu of th rgrssion vctor of rprsnting p covariats ( X ( t) [ X1( t), X2( t),..., X p ( t)]) at tim t. Thn th hazard modl can b xprssd as 2010 Jurnal Karya Asli Lorkan Ahli Matmatik Publishd by Pustaka Aman Prss Sdn. Bhd.

2 Jurnal KALAM Vol. 3, No. 2, Pag () 0 t (2.2) ( t; X ) ( ) Xt whr X(t) is a vctor of covariat valus, possibly dpndnt on tim, thought to influnc th failur rat, is vctor of cofficints and th is unspcifid arbitrary function (s Cox, 1972). 0 ( t ) Now, if w considr a sris of vnts taking plac at tims T1, T2 and T 3, thn th modls for diffrnt componnts of failur can b dfind as follows: X( t1 ) X( t2) X( t3) 3 ( t ; X ) ( t ) ( t ; X ) ( t ) ( t ; X ) ( t ) (2.3) W can furthr gnraliz th abov modls to tak account of th rlationship with prvious durations as shown blow: X( t1 ) X ( t2 ) 2 21t ( t ; X ) ( t ) ( t ; X t ) ( t ) X ( t3 ) 3 31t1 ( 32t t 2 3; X3 t1, t2 ) 0 ( t3) (2.4) Th liklihood function is n Xi1( ti1) 1 Xi2 ( ti2 ) 2 21ti1 Xi3( ti2 ) 3 31ti1 32t i2 L(, ) i1 1 Xl ( t ) 1 l 1 2 Xl ( t ) 2 l 2 1tl 1 3 Xl ( t ) 2 l 2 31tl 1 32tl 2 l1r ( ti ) l2r( ti ) l3r( ti ) Now w can diffrntiat th log liklihood function with rspct to th paramtrs to obtain th stimats as follows: (2.5) ln L ln L 0 0 (2.6) whr and ar paramtrs corrsponding to covariats and prvious durations rspctivly. W can mploy th liklihood ratio tst and th Wald tst for tsting th significanc of th modl and th individual paramtrs of a modl. 3. Th Modl with Multipl Mods of Failur Th problm of multipl mods of failur has bn studid by Farwll (1979), Kay (1984), Islam (1994), Jiang and Murthy (2003), Islam t al. (2004). This papr applis th framwork of ths works to dvlop a a modl for a sris of componnts with multipl mods of failurs at ach phas. Lt us dfin th caus spcific hazard function at k-th phas (k=1,2,,k) and jth failur (j=1,2,,j) for occurrnc of vnt as a function of covariat vctor X: 50

3 Adam Baharun j k j, k, t ; X lim P t T t t / T, J j, K k t ; X t / t (3.1) t 0 whr X(t) dnots th valu of th rgrssion vctor of rprsnting p covariats ( X ( t) [ X1( t), X2( t),..., X p ( t)]) at tim t. Thn th hazard modl can b xprssd as X() t jk j, k ( t; X ) 0jk ( t) (3.2) whr X(t) is a vctor of covariat valus, possibly dpndnt on tim, thought to influnc th failur rat, is vctor of cofficints and th is unspcifid arbitrary function (s Cox, 1972).. jk 0 jk () t Now, if w considr a sris of vnts taking plac at tims T1, T2 and T 3, whr K=3, thn th modls for diffrnt componnts of failur can b dfind as follows: X( t1) j1 j1( t1; X1) 0 j1( t ) X( t2) j2 j2( t2; X2) 0 j2( t2) X( t3) j3 j3( t3; X3) 0 j3( t3) (3.3) W can furthr gnraliz th abov modls to tak account of th rlationship with prvious durations as shown blow: X( t1) j1 j1( t1; X1) 0 j1( t1) X ( t2 ) j2 j21t1 j2( t2; X2 t1) 0 j2( t2) X ( t3) j3 j31t1 j32t2 j3( t3; X3 t1, t2) 0 j3( t3) (3.4) Th liklihood function is 3 J n jk Xi1( ti1) j1 Xi2 ( ti2 ) j2 j21ti1 L(, ) k 1 j1 i1 j1xl ( t ) 1 l 1 j2 Xl ( t ) 2 l 2 j21tl 1 l R( t ) l R( t ) 3 X ( t ) t t i2 i2 j3 j31 i1 j32 i2 X ( t ) t t l R( t ) i 3 l2 l2 j31 l1 j32 l2 1 i 2 i (3.5) Now w can diffrntiat th log liklihood function with rspct to th paramtrs to obtain th stimats as follows: 51

4 Jurnal KALAM Vol. 3, No. 2, Pag ln L jk ln L jk 0 0 (3.6) and j=1,2,,j; k=1,2,3, whr ar paramtrs corrsponding to covariats and prvious durations rspctivly. W can mploy th liklihood ratio tst and th Wald tst for tsting th significanc of th modl and th individual paramtrs of a modl. 4. Application W hav considrd a st of hypothtical data for application of th modl prsntd in sction 2. Th data is comprisd of thr componnts, A, B and C. In componnt A th dpndnt variabl is duration to complt th procss A (T1). In addition, w hav considrd two typs of machins (X1=1 for on typ and 0 for th othr typ). Similarly, in th scond componnt, w hav considrd th duration to complt th procss B (T2). In this cas, w hav thr catgoris of machins undr considration (X2=1 for th first typ, X2=0, othrwis, X3=1 for th scond typ of th machin, X3=0, othrwis; and th third typ of machin is th rfrnc catgory). Finally, th third componnt is comprisd of th compltion tim for th componnt C (T3), thr catgoris of machins (X4=1 for th first typ of machin, X4=0, othrwis; X5=1 for th scond typ of machin, X5=0, othrwis, third typ is th rfrnc catgory). If w fit th modls, thn w obtain th rsults prsntd in Tabls 1, 2 and 3. Tabl 1 displays th rsults for compltion of componnts A, B and C and w hav considrd th covariats X1, T1 and T2. It is vidnt that X1 dos not show any significant association with compltion of A, B and C procsss. Duration to complt A dos shows statistically significant association with th compltion of B or C but a longr duration to complt A rducs th tim of compltion of B and incrass th duration of C. Tabl 2 mploys all th variabls w hav discussd abov. In Tabl 2, th rsults indicat that X1, T1 and T2 show associations similar to that of Tabl 1. In addition, X2 and X4 display ngativ associations with durations of compltion of componnts B and C rspctivly. It is notworthy that th modl for combind tim is not significant. This is indicativ of th fact that although thr ar significant impacts of th slctd variabls on th compltion of ach componnt, th long trm impact disappars du to prssur to complt th ovrall task in tim. In othr words, th duration in short trm componnt spcific tasks ar influncd by th duration in prvious componnts but in th long run th ovrall task is compltd in tim du to subsqunt masurs at th subsqunt componnt. Tabl 1. Estimats of th Conditional Modls for Componnts A, B and C of a Procss Variabls 1. T1 2. X1 3. T2 Estimats (standard rror) Componnt A Componnt B Componnt C *** *** ( ) ( ) ( ) ( ) ( ) *** ( ) *** Significant at 1% lvl ** Significant at 5% lvl * Significant at 10% lvl 52

5 Adam Baharun Tabl 2. Estimats of th Conditional Modls for Componnts A, B and C of a Procss with Additional Slctd Covariats Variabls 4. T1 5. X1 6. X2 7. X3 T2 8. X4 9. X5 Estimats (standard rror) Componnt A Componnt B Componnt C *** *** ( ) ( ) ( ) ( ) * ( ) ( ) ( ) ( ) ( ) *** ( ) ** ( ) ( ) *** Significant at 1% lvl ** Significant at 5% lvl * Significant at 10% lvl Tabl 3. Estimats of th Unconditional Modl for th Combind Procss Variabls Estimats (standard rror) 4. Conclusion 10. X1 11. X2 12. X3 13. X4 14. X ( ) ( ) ( ) ( ) * ( ) *** Significant at 1% lvl ** Significant at 5% lvl * Significant at 10% lvl A modl is proposd in this papr to analyz th tim of compltion in a sris of componnts of a procss both for singl and multipl mods of failur. Th major objctiv of this papr is to link th prvious procsss with th subsqunt compltion of job in th nxt componnt. Th modl can provid usful insights in idntifying th rol of various factors on th prformanc of compltion of th ovrall procss. W hav mployd conditional modls for ach componnt and th duration of compltion a particular componnt is linkd with covariats and duration of compltion of prvious componnts. 53

6 Jurnal KALAM Vol. 3, No. 2, Pag Rfrncs [1] Bnjaafar, S and ElHafsi, M. (2006). Production and Invntory Control of a Singl Product Assmbl-to-Ordr Systm with Multipl Customr Classs. Managmnt Scinc, 52 (12): [2] Bonny, G. E. (1987). Logistic Rgrssion for Dpndnt Binary Obsrvations. Biomtrics, 43, [3] Cox, D. R. (1972). Rgrssion Modls and Lif Tabls (with discussion). Journal of th Royal Statistical Socity B, 34: [4] Diggl, P. J., Hagrty, P. J., Liang, K. Y., Zgr, S. L. (2002). Analysis of Longitudinal Data (Scond Edition). Oxford Univrsity Prss, Oxford. [5] Gyr, R., Van Wassnhov, L. N. and Atasu A. (2007). Th Economics of Rmanufacturing Undr Limitd Componnt Durability and Finit Machin Lif Cycls. Managmnt Scinc, 53(1), [6] Hopp, W. J., Iravani, S. M. R. and Yun, G. Y. (2007). Oprations Systm with Dirctionality Task Compltion. Managmnt Scinc, 53(1), [7] Islam, M. A. (1994). Multistat Survival Modls for Transitions and Rvrs Transitions: An Application to Contracptiv Us Data. Journal of th Royal Statistical Socity, 157: [8] Islam, M. A., Chowdhury, R. I., Chakroborty, N. and Bari, W. (2004). A Multistag Modl for Matrnal Morbidity during Antnatal, Dlivry and Postpartum Priods. Statistics in Mdicin, 23(1): [9] Islam, M. A. and Chowdhury, R. I. (2006). A Highr-ordr Markov Modl for Analyzing Covariat Dpndnc. Applid Mathmatical Modlling 30, [10] Islam, M. A. and Chowdhury, R. I. (2007). First and Highr Ordr Transition Modls with Covariat Dpndnc. In Progrss in Applid Mathmatical Modling, F. Yang (d), pp