Iraioal Joural of Opraios Rsarh Iraioal Joural of Opraios Rsarh Vol., No., 9 (5) Dad Forasig Usig Baysia Expri wih No-hoogous Poisso Pross Modl Hug-Ju Wag,, Ch-Fu Chi,*, ad Chig-Fag Liu Dpar of Idusrial Egirig ad Egirig Maag, Naioal Tsig Hua Uivrsiy, Hsihu 3, Taiwa. R.O.C. Miisry of Eoois, Taipi, Taiwa. R.O.C Absra This sudy prss a ovl ahaial odl usig Baysia odl for dad forasig wih o-hoogous Poisso pross odl. This sudy ais o osru a frawork o iiiz h ovrproduio ad udrproduio oss by usig h i-dpd uraiy of auulaiv dad urv. Spifi odls wr drivd as h fudaals of his approah. Furhror, his sudy also proposd a hod o valua dad forasig usig Baysia xpri wih o-hoogous Poisso pross odl. Kywords Dad forasig, Poisso pross, Baysia, Mahaial odl, Disio aalysis. INTRODUCTION Sussful prfora of rvu aag syss havily rlis o forasig ad opiizaio (Rajopadhy al., 999). Basd o h hisorial dad daa, rsarhrs hav applid i sris or ohr saisial aalysis hods for dad foras. For xapl, Hol-Wirs xpoial soohig odl for opial forasig is applid for shor-r forass for sris of sals daa or lvls of dad for goods (Sgura ad Vrhr, ). Rahr ha usig sigl forasig hod, Wi ad Wi (995) proposd auo-rgrssio, xpoial soohig, ad ooris for forasig ouris dad. Wih aggrga slak orol or ulisag produio orol, h asr plaig produr wih h varia of produio ad ivory lvls a avoid h ursraid growh of ivory ad h uorollabl osupio of apaiy (Barzzaghi ad Vrgai, 995, Hirakawa, 996). Alraivly, his sudy ais o osru a frawork o iiiz h ovrproduio ad udrproduio oss by usig h i-dpd uraiy of auulaiv dad urv i whih so propris of h Poisso pross ar irodud ad h rlaio bw h Poisso pross ad Bays hory is idifid. Du o h sohasi hararisis of h fuur apaiy ds by ipu ad oupu pross (Laior ad Bakr, 997), h Poisso pross ad Bays hory (Cilar, 975) ar adopd hri. This approah is diffr fro h Baysia aalysis of h Muh odl ad ixd Markov wih la lass odl.g., Urba al. (996), Goulias (999). Baus his ahaial odl is drivd fro h pas sals xpri, h hology of pr-ark forasig of w produ ay o b suiabl by lakig of h hisorial daa (Gaviri al., 998). Furhror, his sudy also proposd a hod o valua dad forasig usig Baysia xpri wih o-hoogous Poisso pross odl. Ths idis of valuaio ar ssial i rvu aag. Th rs of his papr is orgaizd as follows. Sio sablishs h horial foudaio ad dsribs h proposd ahaial odls. Sio 3 irodus h valuaio pross ha osidrs ovrproduio ad udrproduio oss o assss h auulaiv dad urv ad is uraiy. Coludig rarks ar fially ad i sio 4, iludig h ris ad liiaios of h proposd produr.. MATHEMATICAL MODEL Th followig riology ad oaios ar grally usd i his sudy. (, ): h v ra for ah v ours i priod. : h avrag of rado v ra. h (): h auulaiv dad fuio of i. ( ) : osa of v ra. k( ): volu of produ k a i. : i priod or i irval. P (, (, )): a Poisso pross prss h probabiliy of produ s dad a giv i ad v ra (, ). f(, (,)) : a odiioal probabiliy fuio of gaa disribuio for i irval giv ad (, ). Hri, h Poisso pross is adopd for dad forasig. Espially, h likag bw dad forasig ad ivory aag a b applid i a odl wih odsd ad opoudd Poisso ixd ovr i (Boyla ad Johso, 996). I h gral odl of Poisso pross, rsarhrs usually us h * Corrspodig auhor s ail: fhi@x.hu.du.w 83-73X Copyrigh 5 ORSTW
Wag, Chi, ad Liu: Dad Forasig Usig Baysia Expri wih No-hoogous Poisso Pross Modl IJOR Vol., No., 9 (5) osa v ra as h parar of (Hogg ad Tais, 983). So gaps xis bw h Poisso pross ad Bays hory i rs of h v ra updad. Tha as h ould b variabl isad of osa parar. Thor : If a spifi v i a sys ours as a Poisso xpri, h drivd Baysia odl o h v ra will hav h liklihood fuio as a Poisso disribuio. Tha is, (, ) d (, ) d P (, (, ))!! Proof: s Wag ad Chi (). ( ) h( τ ) I Thor, h Poisso pross ad Bays hory ar rlad i rs of h radoss of v ra. Th iuiio of his ovrsio os fro h odl i Drikig War Copay of Liburg (WML) who hags h osa produio flow io opiizaio of h quaiaiv orol Bakr al. (998). Fro h abov hor, w a ifr ha h posrior disribuio of v ra is gaa disribuio. For h radoss of v ra, w assu hr is a dad urv wih i dpd fuio h () ha affs h a of ha v ra. Th o-hoogous Poisso pross odl a b applid o dal wih h radoss of a v ra a obi h Poisso pross ad Bays hory o solv h probl of i dpdy o h v ra. Thor : If a spifi v i a sys ours as a Poisso xpri wih h i dpd v ra, h drivd Baysia odl will hav h liklihood fuio as a Poisso disribuio. Tha is, h d! ( τ) τ f(,, (, )) Proof: s appdix. H, Thor liks h o-hoogous Poisso pross ad h Bays hory. Th, h dad urv h () is h auulaiv ad h v ra (, ) is qual o wih h radoss of o whih i quaio (9). Fro h propris of Poisso pross, w kow ha h probabiliy disribuio of h rado variabl i, () (9) rprsig h ubr of produs dad i a giv i irval dod by. Thor 3: If ourr spifi vs (), (),, () i a sys or opo our as is Poisso xpri rspivly o h sa v ra, h drivd Baysia odl wih axiu liklihood siaor (Cilar, 975) will saisfy E [ ] k k h ( τ ) d τ Proof: s appdix. () Thor 3 spifis h rlaioship bw h radoss v ra ad h auulaiv dad h () is of rlva or. Thor 4: If ourr spifi vs (), (),, () i a sys our as is Poisso xpri rspivly o h sa v ra (, ), h drivd Baysia odl wih axiu liklihood siaor will saisfy + E[, ] (3) h( τ ) + ad Var(, ) (5) h( τ ) Proof: s appdix 3. Baus h auulaiv dad urv is a o-drasig fuio as i gos by, h largr dad rquir iplis a sallr uraiy of v ra. Thor 5: Miod abou avrag i as of hr is a saddl poi i warig produiviy sragy wih irasig h () If f ''(), h() h () Proof: s appdix 4. 3. EVALUATION PROCESS (7) Th abov hors ad propris i h o-hoogous Poisso pross a b applid o dvlop a valuaio pross of dad forasig as
Wag, Chi, ad Liu: Dad Forasig Usig Baysia Expri wih No-hoogous Poisso Pross Modl IJOR Vol., No., 9 (5) 3 show i Figur. I his valuaio pross, o oll dad urv basd o hisorial daa of siilar produs is h ai prossig of daa aalysis i h proposd frawork (Lrpalagsui ad Cha, 998). As illusrad i Figur, h valuaio pross osiss of six sps. Firsly, h hisorial dad daa of a spifi produ or siilar produs wih (), (),, () a so rai i ar olld. Followig h aur of a lupy dad (Barzzaghi al., 999), if his produ is a w o, so siilar produs a b usd o rpla h avrag dad daa (), (),, (). If h dad daa a o b olld, h auulaiv sals daa a also b usd o subsiu h avrag dad daa, hough h udrproduio os (Fishr ad Raa, 996) ay b udrsiad. Sodly, h ubiasd iiu varia siaor is usd as h avrag dad daa. Thr ar so spifi i pois for ollig h avrag dad daa. Th ubiasd iiu varia siaor uss h a valu of hs daa as h prdiio of h auulaiv urv a h sld i pois. Tha is, h d () k k is a poi siaor of h avrag dad a i. Thirdly, h ubi spli hiqu i urial aalysis (Burd al., 985) is ployd o sooh h avrag dad urv. Si oly h daa a so spifi i pois ar drivd i h sod sp, h oiuous dad ra h () a b drivd, whih is asir o driv h avrag dad urv by igraio. Fourhly, Thor 4 is ployd o obai h a ad varia valu of v ra for asurig h dad uraiy. I pariular, + Var(, ) h( τ ) iplis a larg uraiy a h bgiig of sals, if hr is a idl i passd. Th produ sragy sigifialy affs h profi ouo a h bgiig of sals ad hus aks h produ sals or urai of h bgiig ha so sals priods lar afr so sals priod. Fifhly, Thor is ployd o valua h radoss of v ra afr so sals priods. Aordig o quaio (7), h varia of h radoss i v ra is ovrg wih a iras of h auulaiv dad. Fially, rdud os of dad uraiy hology (Fishr ad Raa, 996) a b applid o iiiz h ovrproduio ad udrproduio oss. Wh sals ra of h () is irasig ad bfor rahig h saddl poi of h d, w a xpad our produiviy by h avrag dad urv ad is radoss o driv h probabiliis of ovrproduio ad udrproduio. Th xpd valu of ovrproduio os a b drivd fro h produ of disou os i ah produ ad is ovrprodud probabiliy. Siilariy, h xpd valu of udrproduio os a b drivd fro h produ of shorag os i ah produ ad is udrproduiv probabiliy. Morovr, h produio pla ad shdul a b valuad by h oal ovrproduio os ad udrproduio os durig h sals priods. Thr is a sigifia diffr bw his odl ad origi-dsiaio (OD) dad prdiio (Caus al., 997). Rahr ha usig h i sli i OD dad aris, his odl provids a igral siaio a ay i. Idd, h produ of h () is siilar o h siplifid forula i xpoial soohig odls (Wir, 96, Sal ad Jaqus, 999). Th fdbak loop of valuaio pross is h rsuls rfi ad validaio i ha frawork. k k C h d ( τ ) τ h( τ ) Figur. Evaluaio pross of dad forasig. 4. CONCLUDING REMARKS This sudy drivs ahaial odls i dad forasig ad proposs a orrspodig pross for valuaig dad foras. Th proposd odl a provid usful iforaio suh as variaio of produ dad a diffr is. Wih h uraiy of auulaiv dad urv big siad, his hod a b usd o iiiz h ovrproduio ad
Wag, Chi, ad Liu: Dad Forasig Usig Baysia Expri wih No-hoogous Poisso Pross Modl IJOR Vol., No., 9 (5) 4 udrproduio oss. Thrfor, h proposd odl a b usd o valua h produio pla ad shdul basd o h oal produio oss iludig ovrproduio ad udrproduio oss. Th rsuls for dad forasig drivd i his approah a b igrad wih rvu aag o axiiz h rvu i ligh of h fixd disou os, shorag os, ad sohasi avrag dad urv durig a sals priod. Furhr sudy is dd o us pirial daa for validaig h praial viabiliy of h proposd odl. ACKNOWLEDGEMENTS This rsarh is sposord by Naioal Si Couil, Taiwa, R.O.C. (NSC 93-3-E-7-8). APPENDIX Cosidr ha hr is a fuioal rado variabl (, ) i a Poisso pross suh ha (, ) wih () is a rado v ra ad h() is a fuio of i. Hr, w us h avrag of rado v ra o rprs h parar (). Th, for ah v ours i priod, w a g (, τ ) d τ ( ) (, τ) h( τ ) P (, (, ))!! () I addiio, l f(, (,)) b a odiioal probabiliy fuio of gaa disribuio for i irval giv ad (, ), whr i is a oiuous variabl Thus, (, τ ) d τ ( ) (, τ ) h( τ ) f(, (,))!! () O o had, P (, (, )) k f(,, (, )) f( k,, (, )) f(,, (, )) P(, (, )) f( k,, (, )) f(,, (, ))! ( ) f k k k (,, (, )) (3) O h ohr had, f(, (,)) f(,, (, )) f(,, (, )) d f(,, (, )) f(, (, )) f(,, (, )) d by (), ( ) (,, (, )) (,, ) f f d! (4)
Wag, Chi, ad Liu: Dad Forasig Usig Baysia Expri wih No-hoogous Poisso Pross Modl IJOR Vol., No., 9 (5) 5 Fro quaio (4), w driv: f(,, (, ))! + ( ) h( τ ) f(,, (, )) d h +! ( ) f(,, (, )) d + h! + + + ( ) h( τ ) f(,, (, )) d + + ( ) f(,, (, )) d! h() h () + f(,, (,)) h( τ ) (5) Th, l h( τ ) l [ f(,, (, )) ] [ l ] (6) Thus, f(,, (, )) ( ) h( τ ) (7) Copar h rsul of quaio (7) ad quaio (3), w obai: ( ) (,, (, )) (,, (, )) f f k k! k! ( ) h( τ ) f( k,, ( k, )) (8) k So, For h as,, h osrai of quaio (8) is saisfid.
Wag, Chi, ad Liu: Dad Forasig Usig Baysia Expri wih No-hoogous Poisso Pross Modl IJOR Vol., No., 9 (5) h d ( τ) τ f(,, (, ))! 6 (9) H, f( (, ), ) ( ) ( ) h( τ ) h( τ ) ( τ ) τ h d d () APPENDIX Suppos hr ar ourr Poisso pross, h vs our a (), (),, () durig i priod wih h sa v ra (, ). Fro Thor, w a g f( (, ), ) ( ) ( ) h( τ ) h( τ ) d k k h τ h τ k h τ h τ f( ( k, ) k, ) k k k k h( τ ) f( (, ), ) k k k d d k k ( ) ( ) k k ( ) k d ' k k k ( ) ' k h( τ ) k ( τ) τ ( τ) τ k k k h( τ ) d h( τ ) h d h d k ( ) d k For h purpos of opial,
Wag, Chi, ad Liu: Dad Forasig Usig Baysia Expri wih No-hoogous Poisso Pross Modl IJOR Vol., No., 9 (5) 7 k f( ( k, ) k, ) ' ' Th, k () k H, C k k h ( τ ) d τ () Thrfor, E [ ] E [ ] E [ ] k ( ) () k APPENDIX 3 Suppos hr ar ourr Poisso prosss, (), (),, () ar vs ourr durig i priod wih (). Fro quaio (9), h sa v ra f( (, ), ) ( ) ( ) ( ) h( τ ) h( τ ) h( τ ) d + ( ) h( τ ) ( ) ( ) d E[ (, ), ] f( (, ) d, ) d ( + )! +! (3) + ( ) ( ) h( τ ) d E[ (, ), ] f( (, ) d, ) d ( ) ( + )! ( + )( + )! h( τ ) (4)
Wag, Chi, ad Liu: Dad Forasig Usig Baysia Expri wih No-hoogous Poisso Pross Modl IJOR Vol., No., 9 (5) 8 Thus, Var( (, ), ) E[ (, ), ] E[ (, ), ] ( + )( + ) + + h τ (5) APPENDIX 4 L f() h ( τ ) d τ ' h ( τ) d τ h () f () + h d h h f " + ( τ) τ () ' () 3 If If f ', h (6) ' f ''(), h() h () (7) Tha as if w build a oior wih h d h warig of saddl poi o war h drasig ra of h() (i.. h'( ) < ) i h ar fuur. ( τ) τ ad h irasig ra of h() (i.. h'( ) > ), hr is a REFERENCES. Babok, M.W., Lu, X., ad Noro, J. (999). Ti sris forasig of quarrly railroad grai arloadig. Trasporaio Rsarh, Par E: 43-57.. Bakr, M., Vrb, A.J.P., ad va Shag, K.M. (998). Th bfis of dad forasig ad odlig. War Qualiy Iraioal, 5-6: -. 3. Barzzaghi, E., ad Vrgai, R. (995). Maagig dad uraiy hrough ordr ovrplaig. Iraioal Joural of Produio Eoois, 4: 7-. 4. Barzzaghi, E., Vrgai, R., ad Zori, G. (999). A siulaio frawork for forasig urai lupy dad. Iraioal Joural of Produio Eoois, 59: 499-5. 5. Boyla, J.E., ad Johso, F.R. (996). Varia laws for ivory aag. Iraioal Joural of Produio Eoois, 45: 343-35. 6. Burd, R.L., ad Fairs, J.D. (985). Nurial Aalysis. PWS Publishrs, 3 rd Ediio. 7. Caus, R., Caarlla, G.E., ad Iaudi, D. (997). Ral-i siaio ad prdiio of origi-dsiaio aris pr i sli. Iraioal Joural of Forasig, 3: 3-9. 8. Cha, C.K., Kigsa, B.G., ad Wog, H. (999). Th valu of obiig forass i ivory aag-a as sudy i bakig. Europa Joural of Opraioal Rsarh, 7: 99-. 9. Chi, C.F., Ch, S. ad Li, Y. (). Usig baysia work for faul loaio o disribuio fdr of lrial powr dlivry syss. IEEE Trasaios o Powr Dlivry, 7(3): 785-793.. Cilar E. (975). Iroduio o Sohasi Prosss. Pri-Hall I.. Daio, S.J., ad Lair, J.A. (997). Modlig hologial hag i rgy dad forasig: A gral approah. Thologial Forasig ad Soial Chag, 55: 49-63.. Faulkr, B., ad Valrio, P. (995). A igraiv approah o ouris dad forasig. Touris Maag, 6(): 9-37. 3. Fishr, M., ad Raa, A. (996). Rduig h os of dad uraiy hrough aura rspos o arly sals. Opraio Rsarh, 44(): 87-99. 4. Gaviri, S., Bollapragada, S., ad Moro, T.E. (998). Priodi rviw sohasi ivory probl wih forasig updas: Wors-as bouds for h yopi soluio. Europa Joural of Opraioal Rsarh, : 38-39. 5. Goulias, K.G. (999). Logiudial aalysis of aiviy ad ravl par dyais usig gralizd ixd Markov la lass odl. Trasporaio Rsarh, Par B, 33: 535-557. 6. Hirakawa, Y. (996). Prfora of a ulisag hybrid push/pull produio orol syss. Iraioal Joural of Produio Eoois, 44: 9-35. 7. Hogg, R.V., ad Tais, E.A. (983). Probabiliy ad Saisial Ifr. Maillia Publishig Co., I., d Ediio.
Wag, Chi, ad Liu: Dad Forasig Usig Baysia Expri wih No-hoogous Poisso Pross Modl IJOR Vol., No., 9 (5) 9 8. Jio, J.F. (99). Th rlaiv ipora of aggrga ad sor-spifi shoks a xplaiig aggrga ad soral fluuaios. Eoois Lrs, 39: 38-385. 9. Laior, P.K., ad Bakr, J.R. (997). Dad siaio wih failur ad apaiy osrais: A appliaio o prisos. Europa Joural of Opraioal Rsarh, : 48-43.. Lrpalagsui, N., ad Cha, C.W. (998). A arhiural frawork for h osruio of hybrid illig forasig syss: Appliaio for lriiy dad prdiio. Egirig Appliaio of Arifiial Illig, : 549-565.. Moo, M.A., Mzr, J.T., ad Thoas, D.E. Jr. (). Cusor dad plaig a lu hologis. Idusrial Markig Maag, 9: 9-6.. Rajopadhy, M., Ghalia, M.B., ad Wag, P.P. (999). Forasig urai hol roo dad. Prodig of h Aria Corol Cofr, Ju. 3. Sal, M.B., ad Jaqus, J.F. (999). Coribuio of aggrga ad soral shoks o h dyais of ivoris: A pirial sudy wih frh ad Aria daa. Iraioal Joural of Produio Eoois, 59: 33-4. 4. Sgura, J.V. ad Vrhr, E. (). A spradsh odlig approah o h Hol-Wirs opial forasig. Europa Joural of Opraioal Rsarh, 3: 375-388. 5. Slywozky, A.J., Chriss, C.M., Tdlow, R.S., ad Carr, N. G. (). Th fuur of or. Harvard Busiss Rviw, Jauary-Fbruary. 6. Urba, G.L., Wibrg, B.D., ad Hausr, J.R. (996). Prark forasig of rally w produs. Joural of Markig, 47-6. 7. Wag, H. ad Chi, C.F. (). A proposd baysia ifr frawork ad h propry of h liklihood fuio. Joural of Maag ad Syss, 7(3): 35-36. 8. Wir, P.R. (96). Forasig sals by xpoially wighd ovig avrags. Maag Si, 6: 34-34. 9. Wi, S.F., ad Wi, C.A. (995). Forasig ouris dad: A rviw of pirial rsarh. Iraioal Joural of Forasig, : 447-475.