Technical Appendix for Inventory Management for an Assembly System with Product or Component Returns, DeCroix and Zipkin, Management Science 2005.

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1 Techc Appedx fo Iveoy geme fo Assemy Sysem wh Poduc o Compoe eus ecox d Zp geme Scece 2005 Lemm µ µ s c Poof If J d µ > µ he ˆ 0 µ µ µ µ µ µ µ µ Sm gumes essh he esu f µ ˆ > µ > µ > µ o K ˆ If J he so ˆ If K he c If J he ˆ so s s If K he s The poof of Poposo ees o few em esus we se d pove hese fs Lemm A If B he fo y µ 0 K Aso so µ µ fesy mpes h Poof of Lemm A Fs suppose P d cosde seve cses Cse I hs cse J If ˆ ˆ µ ˆ

2 µ µ µ µ µ µ µ µ µ µ µ µ µ µ If sed µ < ˆ he sme gume hods excep h ech uppe m o epced y µ ˆ Cse 2 K d J I hs cse ˆ Cse 3 K µ s µ The gume Cse hods fe doppg ems Ag he gume Cse hods fe doppg ems The defo of echeo veoy mpes h d fesy mpes h Ths esshes he esu fo P By choosg P such h A we c show h µ µ fo y epeedy ppyg he gume ove Ths exeds he esu o B µ 0 K y Poposo A Wh he css of o-cpoy poces he opm pocy hs he foowg popees: If > he If he 2 Poof of Poposo A The poof dps he poof of Lemm osg 989 o ou seg wh eus Suppose s opm pocy d voes -2 fo some em some peod e > d > Lemm A mpes h he e eoshp co hod f B so ssume B Cosuc ee pocy wh esug ses s foows: mx d 2

3 q s q q s m fo } { A q Ths pocy s f e d ose h hodg coss ude e scy ess h ude sh show h co Cse s Thee e fou su-cses: ˆ ˆ ˆ Suppose ˆ The ˆ Now mx 0 < whee s es s esy o dem We fo - whch guees h de J Choose em A such h P coss ude he wo poces e equ s d s d mx 0 m Now so mx 0 mx 0 m 0 mx mx 0 whch mpes mx 0 0 mx Now Theefoe oe of hese eoshps mus hod: < By cosdeg hee posses we see h o 3

4 0 mx 0 mx 0 mx Exedg hs sme e of gume hough of A u yeds 0 mx 3 Sm gumes essh 3 fo he o he hee sucses ˆ ˆ d ˆ Now pc m A such h m P Cosde he quy m m m mx 0 4 Suppose > If m he Sce suppose sed h > The 4 mpes h > whch u mpes h m m so If we c show h 5 he we woud hve m m whee he s equy foows fom Lemm A Ths woud yed m so m ˆ ˆ ˆ > > d ˆ ˆ > ˆ To show 5 we cosde hee su-cses: c Noe h we do o hve o cosde cses wh ˆ < Ude he o-cpoy ˆ pocy esco d 2 uomcy hod hose cses Su-cse I hs cse Su-cse I hs cse 0 sce mx ˆ ˆ If > he ˆ 0 ˆ 4

5 If ˆ he ˆ ˆ 0 Su-cse c s sm o su-cse d s omed Suppose ow h Comg he fc h 0 show fo m cses hough c ove wh yeds m m m m 0 whee he s equy foows fom Lemm A Comg hs wh 4 yeds m m whch mpes m Cse 2 K so If K he fo P d A osg's gumes mmedey yed 3 If ddo K osg's gumes so dec y essh Suppose sed h J cse of Ag choose m A such h m P We woud e o show h 5 hods The s coveed y he o-cpoy pocy esco so ssume h ˆ ˆ > The 5 hods sce ˆ 0 The gumes Cse fo oh > d m he mpy h m Cse 3 J K I hs cse ˆ so ˆ 0 Ag he gumes Cse fo > d m mpy h m 5

6 Poposo A2 Suppose he ssemy sysem ss og-u ce he sysem expeeces oges-ed-me ecovey d fo ˆ ˆ N The y pocy ssfyg he eques Poposo A w eep he sysem og-u ce Poof of Poposo A2 The poof s vo o he poof of Theoem osg 989 exeded o ou seg wh eus Assume he sysem s og-u ce me e µ µ fo µ 0 K fo K N Comg Lemm wh og-u ce yeds Sce he ssfy he eques of Poposo A hs mpes h he poof we cosde hee cses Cse Assume h J To fsh whch mpes J due o oges-ed-me ecovey The so so epeg hs gume esshes h s s fo s Now cosde peod µ fo 0 µ d cosde hee su-cses Su-cse µ ˆ ˆ I hs cse µ µ µ µ µ µ µ µ µ µ µ Su-cse µ The sme eoshps s su-cse hod fe epcg he > ˆ ˆ uppe m o ech wh ˆ µ d epcg he f equy wh " " Su-cse c > µ Sme s su-cse ˆ ˆ 6

7 µ µ We hve show h fo µ By epeg he gumes sg wh 2 ec sed of we c show h µ µ s s fo s µ fo µ 0 K Ths u mpes h peods s µ µ s s fo µ 0 K Cse 2 K d J d Cse 3 K d K c e hded usg sm gumes Poposo Suppose ssemy sysem ss og-u ce he sysem expeeces oges-ed -me ecovey d fo N The ude he esc o o ˆ ˆ o-cpoy poces he ssemy sysem s equve o sees sysem wh eus sge L he sme cos coeffce p H echeo hodg coss α h d ed mes L Poof The cos fuco 8 c e ewe s m E α N L L α p H cos whch s he cos fuco of sees sysem wh he modfed cos pmees d ed mes The cos em fo he se es sysem queso s so gve y 7 usg he modfed cos pmees d ed mes L L α α h Sce he sysem ss og-u ce d sys hee ude y opm pocy Lemm d og-u ce mpy h fo d d y P Bsed o og-u ce s so posse o show h [ ] m m P > so Poposo A mpes h y opm pocy w ssfy Theefoe 6 c e epced y Wh hs susuo he Assemy Poem hs he fom of he specfed sees sysem 7

8 Cooy Suppose ssemy sysem ss og-u ce d ecoves he ed poduc o of s mmede pedecessos The he sysem s equve o sees sysem wh eus sge he cse of ed-em ecovey o sge 2 f he mmede pedecessos e ecoveed Poof If he ed poduc s ecoveed he J { K N} ˆ 0 fo d 0 fo Theefoe he o-cpoy pocy e sco ppes o oe of he ems ˆ so c e dopped The esu he foows fom Poposo If mmede p edecessos of he ed em e ecoveed he J { 2 K N} 2 ˆ fo d ˆ fo Fo K he se J s empy so g he o- cpoy pocy esco ppes o oe of he ems d c e dopped Fo 2 so he sme hods The esu he foows fom Poposo ˆ ˆ Poposo 2 If P {2 K N} he he opm hodg/code cos fo he sysem whou eus s owe oud fo he opm hodg/code cos of sysem wh eus d y ecovey pe ssfyg J { 2 K N} Poof ec h ou oecve fuco 8 cosss of hodg/code coss pus ppee coss whe em s s o s successo d hodg coss o em po o s v sge Sce he e wo coss e cos wh espec o he odeg pocy choosg pocy o mmze 8 s equve o mmzg hodg/code coss Now cosde modfed sysem whee he decso me es he c u ume of ech compoe h w e ecoveed peods oe ed me fom ow I ddo he decso me hs oe-me opo of ccepg o eecg dv ce some o of hese fuue ecove es Oe fese pocy woud e o ccep fuue ecovees d goe he fomo u ohewse ode opmy Ths woud epce he opm pocy fo he og sysem wh eus so he opm cos hs ew sysem s owe oud fo he og sysem cos Now oce h he ccep/eec decso fo ecoveed compoe s mde excy he sme me s he odeg decso fo h compoe So wh espec o hodg/code coss he sysem does o dsgush ewee ccepg fuue ecoveed compoe d odeg ew oe wh mes s he comed ccep/ode quy 8

9 Fuhemoe y ccep/ode quy c e ed y eecg ecovees d smpy odeg he desed quy Ths s equve o mgg he sysem wh o eus so he opm pocy fo h sysem s so oe opm pocy fo he ew sysem 9

Generalisation on the Zeros of a Family of Complex Polynomials

Generalisation on the Zeros of a Family of Complex Polynomials Ieol Joul of hemcs esech. ISSN 976-584 Volume 6 Numbe 4. 93-97 Ieol esech Publco House h://www.house.com Geelso o he Zeos of Fmly of Comlex Polyomls Aee sgh Neh d S.K.Shu Deme of hemcs Lgys Uvesy Fdbd-