The Australian Society for Operations Research

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Allison Higgins
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1 h Asalian Sociy fo Opaions sach ASO Bllin Vol 33 ss 4 Pags 4-48 A Coninos viw nvnoy Mol fo Dioaing s wih Sochasic Dan an Pic Discon on Backos Manisha Pal an Sjan Chana Dpan of Saisics Univsiy of Calca nia Eail: anishapal@gail.co Dpan of Saisics Halia Gov. Collg nia Asac h pap sis a coninos viw invnoy ol fo ioaing is whn an is ano an only a facion of h n an is acklogg. a i is ass o consan an h invnoy anag offs a pic iscon on ackos o p csos o wai. Kywos: Coninos viw invnoy ol; ponnially ioaing is; sochasic an; fi la i; pic iscon on ackos.. nocion n classical invnoy ols wih shoags i is gnally ass ha h n an is ih coplly los o coplly acklogg. Howv i is qi possil ha whil so csos lav ohs a willing o wai ill flfiln of hi an. n so siaions h invnoy anag ay off a iscon on ackos an/o cion in waiing i o p csos o wai. Oyang al. 999 consi cion in la i an oing cos in a coninos viw ol wih paial ackoing. Dan an Hsian si invnoy ols wih acko iscon an ano la i. Chang al. 4 iscss a isiion f poc fo i invnoy ol wih acko iscon an vaial la i. al. 7 si an invnoy policy involving ack-o iscons an vaial la i an Uhayaka an Pavai 8 consi a coninos viw ol wih only fis wo ons of h la i an known an oain h opi acko pic iscon an o qaniy in ha siaion. Pal an Chana si a inisic invnoy ol wih pissil lay in payn an pic iscon on ackos. S also Chng an Hang 998 vino al. 993 Ki al. 99. h Asalian Sociy fo Opaions sach 4

2 4 ASO Bllin Vol 33 ss 4 Pags 4-48 n his pap w consi a coninos viw invnoy ol fo ioaing is whn an is ano an a pic iscon is off o csos who a willing o wai fo ackos. a i is ass o fi an h an a of ioaion is also consan. h pap is oganiz as follows. Scion givs h noaions s an h asspions a in h sy. Scion 3 folas an analyzs h ol. n scion 4 a snsiiviy analysis is cai o hogh an apl. Finally concling aks a a in scion 5.. Noaions an Mol Asspions h invnoy ol consi is a coninos viw ol. h following noaions an asspions hav n s h.. Noaions = invnoy lvl a i poin = facion of h an acko ing sock o = pp on on acko aio θ= a of ioaion = fi la i X = ano an ing la i = pc lngh of a plnishn cycl = h o qaniy a ach o poin = sock high which iggs off an o = pc i akn fo sock o co own o K = oing cos p o p = pchas cos p ni h = holing cos p ni p ni i s = fi shoag cos p ni sho p ni i s = los sals cos p ni an los = pic iscon on ni acko off = aginal pofi p ni. Asspions h asspions govning h ol a as follows:. h ol consis only on i in invnoy.

3 4 ASO Bllin Vol 33 ss 4 Pags h sock on han ioas a a consan a. h is no pai o placn of ioa is. 3. Dan is ano wih an an a. 4. Shoags a allow an a facion of n ans ing sock-o is acklogg. 5. a i is fi. 6. Dan ing la i follows an ponnial isiion wih paa. 7. Only on o can osaning a any poin of i. 8. Ding h sock-o pio h acko facion is icly popoional o h pic iscon off y h invnoy anag on ni acko. hs wh. 3. Mol Folaion an Analysis h invnoy policy is o plac an o fo is whnv h sock high cos own o. W ass h planning pio o of infini lngh so ha h ol is si ov a plnishn cycl. A plnishn cycl fins h i inval wn wo consciv spplis of h o qaniy. no h invnoy lvl a i poin in h plnishn cycl. s consi h siaion wh h sock on han a h n of h la i is ga han o qal o zo. hn sinc plion of sock occs owing o an an ioaion h following iffnial qaions will fin ansiions in invnoy ing h la i: / wih onay coniion =. H - = h la i. Hnc. say. Sinc w s hav hs h iffnial qaions fining ansiions in invnoy ing a plnishn inval will as follows: i fo /.

4 43 ASO Bllin Vol 33 ss 4 Pags 4-48 ii Fo / / wh is h i akn fo sock on han o has ing h la i whn. n i h onay coniion givs an h onay coniion givs ln ln. n ii fo h onay coniions w g. Sinc h laionship wn an is oain as ln. h pc invnoy ing a cycl is hn givn y

5 44 ASO Bllin Vol 33 ss 4 Pags 4-48 E = f f [ { }{ } { Ei ln - Ei }] wh Ei. h n of nis ioaing p cycl is hfo E. Sinc shoag occs ov h pio [ ] whn h la i an cs an nos h acko aio h n of ackos p cycl is givn y S [ f h pc n of los sals ing a cycl is { ln Ei Ei }] hs h oal pc cos p ni pc lngh of a plnishn cycl is givn y C [ K h p E ss s ] ln

6 45 ASO Bllin Vol 33 ss 4 Pags 4-48 [ ] [ [ { ln[ ] C k s S s h E P E s k Ei }] [ ln { }{ } s Ei h P { }]] ln[ ] Ei Ei N Fo givn h opial vals of an which iniiz C s saisfy C an C which giv C P h.. }}] ] [ { { } } { { [ C Ei P h s Ei s Fo. an. an a oain as non-lina fncions of. Hnc C can pss as a fncion of alon iniizing which w oain h opi vals of an hnc of an.

7 46 ASO Bllin Vol 33 ss 4 Pags Snsiiviy Analysis Sinc algaic acailiy is alos ipossil w ap o g an ia of h havio of h opial wih chang in h ol paas hogh nical apls. al : Showing h opial vals of an cosponing pc cos p ni i fo iffn vals of whn K=4 =35 p=3 h=7 s=5 s =5 = =5 =.5 =.5 C al : Showing h opial vals of an cosponing pc cos p ni i fo iffn vals of h whn K=4 =35 p=3 s=5 s =5 θ=. = =5 =.5 =.5 h C al 3: Showing h opial vals of an cosponing pc cos p ni i fo iffn vals of whn K=4 =35 p=3 h=7 s=5 s =5 θ=. = =5 =.5 C

8 47 ASO Bllin Vol 33 ss 4 Pags 4-48 al 4: Showing h opial vals of an cosponing pc cos p ni i fo iffn vals of s whn K=4 =35 p=3 h=7 s =5 θ=. = =5 =.5 =.5 s C al 5: Showing h opial vals of an cosponing pc cos p ni i fo iffn vals of s whn K=4 =35 p=3 h=7 s=5 θ=. = =5 =.5 =.5 s C h following osvaions a a fo als 5: a an a casing fncions of an h whil incass as /h incass. an a casing fncions of. c As s incass an incas cass. an a non-casing fncions of s. 5. Conclsion h pap sis a coninos viw invnoy ol fo ioaing is whn an follows an ponnial isiion an la i is fi. n o o ncoag csos o wai fo flfilln of hi an whn sock on han is zo h invnoy anag offs a iscon on ackos. h opial o qaniy o lvl an iscon on ackos hav n oain so as o iniiz h oal pc cos of h invnoy anag. h na of chang in h opial vals of h cision vaials wih chang in h ol paas has n invsiga hogh nical apls.

9 48 ASO Bllin Vol 33 ss 4 Pags 4-48 fncs. Chang B.. Oyang.Y. Chang K.W.: A no on pioic viw invnoy ol wih conollal sp cos an la i. Cops & Opaions sach Chng K.J. Hang C.K.: Econoic anfacing qaniy ol involving la i an sp cos cion invsn as cision vaials. nnaional Jonal of Opaions an aniaiv Managn Dan an Hsian: nvnoy ols wih acko iscons an vaial la i. nnaional Jonal of Sys Scinc Ki K.. Hayya J.C. an Hong J.D.: Sp cos cion in conoic pocion qaniy ol. Dcision Scincs W.C. W J.W. i C..: Opial invnoy policy involving ack-o iscons an vaial la i an. nnaional Jonal of Avanc Manfacing chnology Oyang. Chng C. Chang H.: a i an oing cos cions in coninos viw invnoy syss wih paial ackos. Jonal of h Opaional sach Sociy Pal M. an Chana S. : A inisic invnoy ol wih pissil lay in payn an pic iscon on ackos. o appa in Opsach. 8. vino J. Hly B.J Fiich W.: A ahaical ol fo h conoic jsificaion of sp i cion. nnaional Jonal of Pocion sach Uhayaka. Pavahi P.: nvnoy Mols wih i of Backos involving cil a i an Sp Cos. Opsach

Partial Fraction Expansion

Partial Fraction Expansion Paial Facion Expanion Whn ying o find h inv Laplac anfom o inv z anfom i i hlpfl o b abl o bak a complicad aio of wo polynomial ino fom ha a on h Laplac Tanfom o z anfom abl. W will illa h ing Laplac anfom.