Generalized solutions for the joint replenishment problem with correction factor

Transcription

1 Geerzed soutos for the ot repeshet proe wth correcto fctor Astrct Erc Porrs, Roert Deer Ecooetrc Isttute, erge Isttute, Ersus Uversty Rotterd, P.O. Box 73, 3 DR Rotterd, he etherds Ecooetrc Isttute Report EI 5-9 I ths pper we gve copete yss of the ot repeshet proe JRP uder costt deds d cotuous te. We preset souto ethod for the JRP whe correcto s de for epty repeshets, d we test the souto procedures wth re dt. We show tht the soutos oted dffer fro the stdrd JRP whe o correcto s de the cost fucto. We further show tht the JRP wth correcto outperfors depedet orderg. Addto uerc experets re preseted. Keywords: vetory, ot repeshet, correcto fctor. Itroducto Durg the st two decdes uch tteto hs ee gve the terture to the deterstc ot repeshet proe JRP cotuous te. I ths proe t s ssued tht or orderg cost s chrged t sc cyce te, d tht the orderg cyce of ech te s soe teger utpe of, whch s ced, pocy. Furtherore, t s geery ssued tht or set-up cost s chrged for ech te cuded sge order. I ths pper we wt to gve coprehesve yss of the JRP. Frst of we preset copete theory provg soe cs de erer ppers; secod we cosder the cuso the oectve fucto of correcto fctor for epty repeshets, d thrd we e coprso wth depedet orderg. Athough y heurstcs d exct ethods hve ee proposed to sove the JRP effcety, o pper so fr corportes the correcto fctor the yss. I ths pper we show tht the cuso of the correcto fctor the cost fucto y yed very dfferet soutos ters of the sc cyce te d the vues of the. We show y experetto tht ths s ofte the cse for rge vues of the or set-up costs d oderte or set-up cost. hs y hve rge pct o the quty of the souto fro peetto pot of vew, especy whe usg the deterstc JRP s pproxto to the stochstc cse. oreover, or theoretc shortcog of the stdrd JRP s tht t does ot dcte whether depedet orderg souto wth EOQ pped to the dvdu ot szes s etter. We show tht the cuso of the correcto fctor the cost fucto of the JRP yeds souto tht wys outperfors depedet orderg. uerc resuts support ths theoretc fdg. Correspodg uthor: porrsuse@few.eur.

2 I ths pper we preset ethod sr to v Es [], ut odfed to e sute for geer teger poces GI wth correcto fctor the oectve fucto. We show tht the oectve fucto wth correcto fctor s st pecewse covex wth respect to ut dscotuous. Addtoy, we susttte the c tht the ethods provded the terture to sove the JRP re gorths. he ethod preseted ths pper s sr to the oe gve Porrs d Deer [], who studed the JRP uder u order quttes for the ot szes of dvdu tes cuded the repeshet order. he set-up of the pper s s foows: I the ext secto we gve the defto of the proe d the reevt terture revew. I secto 3 we preset the gorths to sove the JRP usg GI poces wth d wthout correcto fctor. I secto 4 uerc experets re preseted d further theoretc resuts re dscussed. he f cocusos re cuded secto 5. For crty of the des preseted, we cude the theoretc dets of the ethods proposed the ppedces t the ed of the pper.. Proe defto Cosder the proe of orderg tes tht c e oty repeshed gst or set up cost S. he ded D for te s ssued costt d ow. o corders re owed. I the se foruto oe sees terv d vector of s whch ze the tot hodg d orderg costs, gve y: S C, s h D where s d h re the or set-up cost d ut hodg cost of te. he foruto for the JRP wthout correcto fctor s gve eow: JRP stdrd foruto P { C, >, tegers for,,..., } ote tht souto, y hve epty repeshets, e.g.,3, for whch the or set-up cost s st chrged. herefore, correcto fctor shoud e cuded the oectve fucto, s foows: C c S, s h D he fctor the prevous equto s the frcto of o-epty repeshets per yer. Whe such correcto for epty repeshets s de the oectve fucto, we deote ths y proe P c. he foowg foru c e derved usg the prcpe of cuso d excuso for the evuto of Dgpur []:

3 c,..., { } {,..., }: L c, c,, c,...,, {,..., },, {,..., } where c,..., deotes the est coo utpe of the tegers,...,. ote tht f for soe,. I other cses t s ore dffcut to copute. I Appedx A we preset gorth for the evuto of. Both forutos P d P c re o-er xed-teger progrg proes, whch re dffcut to sove especy for rge uer of tes. Goy [5] d other uthors rgue tht the cost proveet ged y the cuso of the correcto fctor s oy of few percetge pots, d hece t shoud e eft out. However, Goy does ot cosder y posse effect tht the correcto fctor y hve o the vues of d. Wde et. [] showed tht the souto of rexto of the JRP c e used s ower oud for the souto of the JRP wth correcto fctor. Jcso et. [6] d Roudy [] proposed the use of the so-ced power-of-two Po poces, y ettg p, p p: teger. he forer showed tht Po pocy for the JRP produces 94%-effectve soutos wth respect to the vue of the oectve fucto whe usg GI poces. Fg-Chu d g-jog [3] provded go zto procedure for the JRP usg Po pocy, d they showed tht the Po souto cots t est oe of the s equ to oe. Such souto hs ssocted correcto fctor of oe. Cosequety s poted out y Goy, ts oectve vue w e ether equ or sghty worse th the oectve vue of GI pocy souto wth correcto fctor. I the ext secto we gve the souto ethods for P d P c. 3. Souto ethods 3.. Souto ethod for proe P he fucto C, s ot oty covex wth respect to d. However, for fxed vector the fucto C s covex, wth gve y: s S,..., h D Susttutg c C, we get the C for fxed : s C,..., S hd 3 3

4 ow cosder the foowg equvet foruto of proe P s suggested y Wde et. []: S P C s.t. > z where the fuctos z re gve y: s z hd : tegers for,,. 4 Wde et. [] showed tht for fxed the vue of s gve y:, 5 where s. hd Let - d correspodg - - e gve. ow et I e the terv [, - ssocted to -. ext oserve tht for [, the rguets 5 crese s. he vector w chge whe oe or ore of ts eeets creses y oe ut ust eow. herefore, c e ccuted fro: x { } 6 where s for,,. 7 h D he eeets of the vector ust eow, sy, re gve y: for J for J where J s the set of eeets of for whch the xu 6 s tted. he prevous yss ows us to e prtto o the set I, of vues usg equto 6, wth vectors gve y 5. ote tht we do ot eed fu euerto o, sce we oy cosder the vectors tht ze the tot cost for gve. herefore, f we c estsh ower d upper ouds o, sy 4

5 ow d upp, we oy eed to evute fte uer of tervs. We c ot the oc of C wth foru 3 sde ech terv of such prtto, d copute the est souto og tervs. he ove procedure ws frst proposed y Goy [5]. However, he dd ot expcty show tht the ty of gve y 5 deteres the uer of vectors tht shoud e cosdered etwee ow d upp. herefore, hs procedure oe eeds to euerte vectors strtg,,. I the ethod we propose ths pper, y usg 5 we c drecty strt the serchg procedure the vector ssocted to upp. Aother ptf of Goy s ethod, s poted out y v Es [], s tht the ower oud tht he used coud oy gurtee soutos for strct cycc poces, where the sest. ow otce tht for gve, the vue of gve y, sy, does ot ecessry eogs to the terv [, where the vector - zes C. However, the over souto for C hs ssocted, sy, equ to soe see Fgure. herefore, we eed to evute C oy the tervs for whch [,. We forze ths resut the foowg theore. heore. Let e the vector of vues tht ze the fucto C, og posse vues s gve y equto 5. Let [, e the terv ssocted wth. he,,..., [,. Proof. See ppedx B. Fgure. Schetc represetto of the serchg procedure for the JRP. 5

6 he ove resut ws ot provded prevous ppers to show tht Goy s gorth d odfed versos of t cudg the oe preseted ths pper provde deed the souto for the JRP. oreover, ths resut y ot hod for exteded versos of the JRP, e.g. whe costrt s posed o the ot szes for dvdu tes [] or whe the correcto fctor s cuded the oectve fucto. For these cses the y e o the oudry of gve terv the prtto of vues. Bouds o Before gvg the gorth to sove the JRP, we eed to estsh ouds o. I order to overcoe the proe ssocted wth Goy s ower oud, V Es [] proposed the foowg ower oud to esure souto for GI poces: S / RC VE ow f where RC f s the tot cost ssocted wth fese souto for the JRP. VE Athough ow c e proved tertvey y sertg the st equto the est C foud so fr the gorth, for hgh vues of the or set-up cost the resutg ower oud c e very s. V Es uses the se upper oud s the oe proposed y Goy, gve y: VE upp,..., Vswth [] preseted tertve ethod to oted tghter ouds o for the JRP. Strtg wth the v Es ower oud, Vswth uses tertvey forus d 5 to ot oc souto of C, sy ow. He shows VE tht the fucto C s ootocy decresg etwee ow d ow. A sr procedure s used to fd upper oud o, sy upp. he Vswth ouds re pproprte for GI poces d therefore we c use the our ethod. Wde et. [] uses rexto of proe P, sy R, for whch the souto of C R s foud, sy, R. he fese souto for the JRP s oted usg R d foru 5. Fy, y deterg the tersecto etwee the eve e correspodg to the fese C d the C R curve, ower d upper oud o, sy d, re oted usg secto. hs procedure W ow W upp c yed tghter ouds o wth respect to the oes Vswth [] for uer of proe cofgurtos, ey for oderte or set-up costs d retvey hgh or set-up costs. oreover, the t Wde ower oud c e further proved y repetg the secto procedure usg the est vue of C foud so fr the gorth, wheever C<CR. otce tht gve the t Wde ouds, tghter ouds o c e oted y the Vswth procedure descred ove for uerc coprso o the perforce of these procedures see Porrs d Deer [9]. Bsed o ths yss, we proposed the foowg gorth. 6

7 Agorth to sove P Step. Itzto Evute Wde ouds W ow d Vswth tertve procedure. Set W upp usg equto 5. Set C, d. Evute for,, usg foru 7. W upp, d prove the usg Step. For detere usg 6 d set J { : x { }}. For,..., evute usg equto. Set: C { C, C [,..., }, f otherwse Ot the eeets of the ew vector ccordg to d set f J. Otherwse Step. If ow SOP wth C, C d. Otherwse set d GOO step. ED of the gorth. he ove gorth s sr to the oe proposed y v Es [], though peeted sghty dfferet wy d wth tghter ouds o the sc cyce te. otce tht ech roud of the gorth we chec whether the es sde the terv for whch the ssocted vector zes C. If ot, o evuto of C s doe, whch y sve soe coputto te, especy for rge uer of tes.. Coputto copexty of the proposed gorth A ddto resut of our gorth coes fro the use of foru 5, whch ws ot prevousy corported gorths to sove the JRP. Usg 5 d otg tht the s chge step szes of oe, we c evute the xu uer of tervs eeded to ot the souto. hus, gve ower d upper ouds o we provde the foowg foru: xu # of steps : ow upp 7

8 For fxed ow d upp ths uer creses ery the uer of tes,. hs hs ee uotced the terture, s ost ppers gve o expct expresso for the -vues, e equto 5. ext ssue tht the t st of - vues s sorted efore eterg Step of the gorth. Sce the tes chge ther vues oe y oe t ech step of the gorth wth oy oe -vue updted ech roud, t foows tht the uer of coputto steps of the gorth s O og uder costt upper d ower ouds. I the reder of the copexty yss, we eed to set ouds o the s d hd vues. hs coes fro prctc reso, sce we ssue tht rety there s wys effort ssocted wth the hdg or recevg of te. Sry, tes re ssued to cuse hodg costs whe ept o stoc. hus, for s [s, s x ] d h D [hd, hd x ] we dstgush the foowg cses: S fxed. Frst otce tht ow,ve s proporto to /, sce the tot cost C dds up postve ters s d h D, pus costt ter S. For the Wde ouds, sce we te the tersecto of rexto of P wth the C curve, t foows y W W sr resog tht ow d upp re proporto to / for copete yss W see Porrs d Deer [9]. herefore, fro 5 we hve tht d W ow upp re proporto to. Fro ths t foows tht the uer of steps the gorth s proporto to og. It c so e show tht the copexty to ot the souto R of the rexto s O og, sce dervtves of CR eed to e sorted the procedure []. herefore the copexty of the over gorth s O og uder Wde ouds. S creses ut /S s ouded. I ths cse we hve tht d re ouded s creses. herefore W ow W upp the uer of steps the gorth creses ery. It foows tht the gorth copexty s O og. For S, the uer of steps of the gorth creses ore th the prevous cses, however t s ot such terestg cse sce prctc ower oud o c e used. oreover, for s vues of S the JRP s ess reevt, d depedet orderg for the tes shoud e pped. 3.. Souto ethod for proe P c ow we cosder the JRP whe correcto fctor s cuded the cost fucto. As efore we cosder the foowg tertve foruto of the fucto C c,: C c S z where the fucto z re defed the se wy s for C.

9 he proe P c s geery ore dffcut to sove th proe P, sce the cuso of the correcto fctor es the fucto C c dscotuous. Sr to C, the fucto C c s ot oty covex wth respect to d. As efore, for fxed vector the fucto C c s covex, wth gve y:,..., s S h D 9 Susttutg 9 c C c, we get the C c for fxed : C c,..., s S hd oreover, the cuso of requres the evuto of equto every step of the serch gorth. I ddto to tht, uerc experets preseted the ext secto suggest tht the fucto C c teds to fuctute roud cert vue s goes to zero, rther th gog to fty s the cse of C see Porrs d Deer [] for deted descrpto. herefore, the trdto ower ouds o preseted the terture re ot vd yore, d copetey ew yss s ecessry. O the other hd, sce f t est oe equs oe, proe P c d proe P re the se for rge vues of, d therefore the trdto upper ouds o preseted the terture [], [] to sove P re st vd, s og s. Lower oud o for proe P c We w derve ower oud o for proe P c sr wy s Porrs d Deer []. Here we provde the resuts d the reder s referred to Appedx B for dets d proofs. As tht pper, we use the foowg proposto for our yss: Proposto. Gve products wth ssocted vector, the foowg hods: c, Proposto w e used to estsh upper d ower ts o the fucto C c s goes to zero. We gve frst the foowg defto. Defto. Let for,,. For two tes, wth / Q, where Q,, deotes the set of rto uers, et, e the sest tegers for whch the foowg equty hods: 9

10 ,, For ese of otto, the seque we drop the super dex,. heore. Gve products wth deds D,,D, d or set-up costs s,,s. If / Q,, the the foowg hods: f C c S, : s h D d supc c S s h D ote: Atertvey, the ters e repced y. sde the secod suto of f, c heore c e exteded for the cse where / R\ Q, the set of rrto uers. As Porrs d Deer [] we gve the foowg theore: heore 3. Gve products wth deds D,,D, d or set-up costs s,,s. If / R\ Q,, the the foowg hods: C c S s h D Athough for ost re ppctos the rtos / re ofte tructed to rto uers, for whch heore s the oe of prctc terest, ote tht for rtrry uers, the vues of d re ey to e rge. I tht cse, the gp etwee the two ts heore s ey to e s. he yss used to estsh heore, so yeds ower oud o. Frst we gve the foowg e. Le. If / Q,, the there exsts te vue foowg hods: I the proof of e see Appedx B, we fd tht c ow, s.t. for y c ow s oted fro: c ow the

11 { } c ow,, : where, s te vue for whch the foowg hods e 4 of Appedx B: x c, <, Rer. he ove procedure to evute ower oud o c yed very s vues e.g. ess th hour, whch y ot e usefu for prctc purposes. I such cse, prctc ower oud o the sc cyce te shoud e estshed, sy oe dy or oe hour, whch s resoe ssupto for ost vetory trcg systes. ote tht s ust utper the deterstc JRP s the repeshet terv for ech te s gvg y. However, f we use the deterstc souto of the JRP s pproxto to the stochstc cse, y we e cosdered s re revew te for the vetory syste d the vue gude o the orderg te of te. oreover, for the deterstc JRP, ctuy represets the precso of the orderg terv for the tes. herefore, usg ower oud of oe dy o, es tht the tes re ordered wth precso of dy. Gve the prevous resuts, we ow forute the foowg gorth. Agorth to sove P c Repce the fucto C y C c the gorth to sove P preseted secto 3. c d use the ew ower oud ow, or prctc ower oud o. Use foru 9 for the evuto of step of the gorth. As we c see the uerc experets preseted secto 3, the cuso of the correcto fctor es the fucto C c dscotuous t the pots where the vector chges. I ths cse heore does ot ppy d we hve to chec the extree pots of the tervs s we. herefore, the foowg foru shoud e used ech roud of the gorth for the evuto of C : C { C, C,..., c c c { C, C,, C, } } f [, ] otherwse 3.3. Idepedet orderg We ow cosder the geer cse of depedet orderg of tes, where ech te pys the or set-up cost S ddto to ts or set up cost s, ut c e schedued t ow te, s foows: S s P IO hd, s.t. >

12 he souto of P IO s gve y: S s,, hd Susttutg c to the tot cost fucto yeds the cost for depedet orderg: C IO S s h D We ow woud e to prove tht the souto of proe P c wys outperfors the depedet orderg souto gve y, whe ths does ot eed to e the cse for proe P. Let e gve y equto d et ε >. By cotuty of the ter h D t foows tht there exsts δ >,,,, such tht for stsfyg < δ we hve: hd hd < ε ext reze tht f we choose the sc cyce te P c s eough we c fd tegers,,,, s.t. < < δ. Hece, usg equty : S s h D S < S s s h D h D ε otce tht the coto,,,, s fese souto to P c d hece c C s ess th the frst ter of the prevous equty. Sce the prevous c ccuto c e doe for ε>, we hve proved tht C CIO. A sr resut c e oted for orderg oy suset of tes depedety. We forze ths resut the foowg theore. heore 4. For the stdrd foruto of the JRP the foowg hods: C C. c IO As sde resut, otce tht fro heore 3 t foows tht the t s of C c gves the tot cost of the syste ssug tht the tes re ordered ccordg to ther EOQ evuted wth s oe d tht ech te pys ddto set-up cost S every repeshet. ht s, ech te s repeshed every

13 s / h D uts of te d pys ddto u cost of S s / h D. / ote however tht ths resut dffers fro C, whch yeds fte costs. 4. uerc experets I ths secto we w show y uerc experetto tht the soutos of proe P d proe P c gve y the ove gorths c e rdcy dfferet. We ppy the gorths to the foowg dt te fro re cse [7] tht cse dd ot provde s s, sce the or set-up costs where repced y u order qutty for the ot sze of ech te cuded the order: tes S 95 euros h.35 euros/utyer for e shows the ded rtes for the tes, where vues for ded set were te fro the cse d vues for ded set were rdoy geerted fro [5, 5]. For the ove dt we perfored set of experets for vues of the or set-up costs rgg fro 5 to 5, euros, s show es -3. hs choce of vues of s ows us to yze the effect o the souto of the JRP uder two codtos: whe there s cetve to cude the tes every repeshet opportuty ow vues of s d whe orderg the tes utpe te of the sc cyce te es ore ecooc sese for rge vues of s. he verge te etwee repeshets vg s so reported for the souto, where vg /. e. Deds per yer, D Ite Set Set,34 7,96,76 5,3 3 6,796 3,36 4,4 45,376 5,6 43,449 6,4,46 7 5,4 35,7 5,4 9,567 e. Soutos for ded set Soutos wth correcto fctor Soutos wthout correcto fctor s:,, C c c c vg C vg %Dff. C wees wees 5 9,747,,, 9,746.7,,,. 5,3,,,,3.7,,,. 5 4,364 6,,, 6 4, ,,, 6. 35,97,,,,969.53,,,. 7 5,7 7,,, 7 5,7.43 7,,, 7.,5 9,953 3,,, 3 9, ,,, 3.,5 35,5 3,,, 3 35,5.36 3,,, 3., 4,33 44,,, 44 4, ,,, 44. 3, 4,46 5,,,3,,3,, ,93. 53,,,,,,, , 56,,,,3,,3,, ,.4 54,,,,,,, , 6,37 3,,,3,,3,, , ,,,,,,, 6.43, 7,465 44,,,3,,3,, , ,,,3,,3,, 67.4, 3,6 6,,,3,,3,, , ,,,3,,3,, 94. 4, 73,79 9,9,9,,,,, ,999.3,,,3,,3,, 3.7 5, 94, 4 9,9,9,,,,, , ,,,3,,3,, 4.7 3

14 e 3. Soutos for ded set Soutos wth correcto fctor Soutos wthout correcto fctor s:,, C c c c vg C vg %Dff. C wees wees 5,5 9,,, 9,5. 9,,, ,,,, 3,.5,,,. 5 6,35 3,,,,,,, 3 6,35.4 3,,,,,,, 3. 35,439 6,,,,,,, 6, ,,,,,,, 6. 7,,,,,,,,,.34,,,,,,,.,5 33,445 4,,,,,,, 4 33, ,,,,,,, 4.,5 39,7,3,,,,,, 39,7.37,3,,,,,,., 44,774 9,3,,,,,, 9 44, ,3,,,,,, 9. 3, 54,4 3,6,3,,,3,, , ,3,,,,,, , 6,953 3,6,4,,,3,, ,6. 4,3,,,,,, 4.6 5, 6,9 3,6,4,,,3,, , ,3,,,,,, 45.4, 96,39 3 3,6,4,,,3,, , ,6,4,,,3,, , 35, ,6,4,,,3,, , ,6,4,,,3,, , 9,9 6 3,6,4,,,3,, , ,6,4,,,3,, , 3, ,6,4,,,3,, , ,6,4,,,3,,4 4. I Fg. we show pot of the fuctos C d C c for ded set d s 4, for tes. As oe c see fro the pot, the fucto C s soother th C c, sce the tter exhts dscotutes for ech terv [, wth ssocted costt vector -. Furtherore, we c see tht the fucto C c does ot go to fty s pproches to zero, s stted heore. Fro the resuts preseted es -3 sever cocusos re drw. Frst ote tht for s vues of s s,, the soutos for P d P c re excty the se. hs s due to the fct tht for these vues of s the s re gve y for soe, wth the correspodg correcto fctor equ to oe. Fro equto 6 t foows tht s the s s decrese, so does the te whch the vector chges ts st coordtes fro to step of the gorths. I other words, for s s s the vue of the fucto C or C c s ey to e the rego whch t est oe eeet of equs oe. Fgure. Pot of C c d C For rge vues of s s 3, the vues of the oectve fucto for proes P d P c dffer o ore th.54% proes soved for oth ded sets. evertheess, for ost of the cses, the d dffer sgfcty. I proes soved, the oted y usg foruto 4

15 P c s ower or equ th the oe correspodg to proe P. he cuso of the correcto fctor prevets the to rech hgh vues, d the vector tes o vues tht ow ore frequet repeshet occsos. For stce cosder the soutos for ded set wth s 4,. he revew te for proe P c s c wees, wth verge revew te of 4 wees, wheres for proe P these vues equ d 3 wees, correspodgy. We c oserve ths ehvor grphcy the pots of Fg.. If we use the deterstc JRP s pproxto for the stochstc cse, whe the or set-up costs re hgh, usg foruto P c the syste gets ore opportutes to e revewed. hs y hve gret pct o the perforce of the syste. I ddto to the prevous experets, we crred out rge set of experets for 5,, 5 d tes usg the re cse s se ut wth expded rges for the ded d hodg costs. Accordgy, for ech proe sze we cosdered 7 dfferet vues of the or set up cost S, 5, 75, 5, 5, d 5. herefore we cosdered dfferet proe stces. For ech of the we soved proes usg oth gorths pus depedet orderg, wth deds rdoy geerted fro [5, 5], hodg costs rdoy geerted fro [.,] d or set-up costs rdoy geerted fro [5, 5]. hus, dfferet proes were soved wth ech gorth. We preset the uerc resuts e 4, where the verge vues re reported over the ded reztos. o e fr the coprso, we peeted ddto step the gorth for proe P where the fucto C s corrected wth the correcto fctor ssocted wth the vector P corr. Both vues of C re reported. For the depedet orderg souto wth tot cost C IO, the ot sze for ech te s evuted usg foru together wth D. Vues of ow,prct etwee. d. yers were used the gorth for proe P c. he verge uer of tervs evuted d the verge CPU te secods s reported for ech gorth. he percetge dfferece etwee C vues s ccuted fro: C % Dff. C corr C C c % A sr foru ws used for %Dff. C IO, %Dff. d %Dff. vg. Fro the uerc resuts preseted e 4 we derve the foowg cocusos:. As the or set-up cost S decreses, so decreses d the svgs wth respect to depedet orderg ecoe ser. Evetuy t does ot py off yore to ppy the ot repeshet pocy d therefore ppyg EOQ suffces. ote however tht for soe proe stces, eve for ow S, portt svgs wth respect to depedet orderg c st e cheved. E.g. for tes d S the rge 5~75, svgs of 5.6~.% re cheved w.r.t depedet orderg. For these proe stces the percetge dfferece verge repeshet te c e s hgh s 6.% for S 5. ote tht the souto of P c wys outperfors the depedet orderg souto, whe ths s ot wys the cse for P see the resut for 5 tes d S. 5

16 e 4. Coprso of gorths for the stdrd JRP Av. o. of tervs Averge C %Dff. %Dff. Averge %Dff. Averge vg %Dff. Averge ow Averge upp Av. CPU o. of Agorth for dep. Agorth for C C IO Agorth Agorth vg Agorth Agorth te sec. tes S P P c order. P P corr P c Pcorr - P c P P c P P c P P c P P c P P c ,,,6,9.%.% % % ,6 3,9 3,9 3,75.%.5%.35..% % ,36 4, 4, 4,59.5% 7.% % % ,97 4,774 4,774 4,774.%.7% % % ,79 6,59 6,59 6,59.% 7.%.9.9.%.9.9.% , ,,7,7,7.% 35.3%.5.5.%.5.5.% , ,674 3,79 3,79 3,79.% 47.5% % % ,965,964,95,6.9%.% % % ,6 6,5 6,77 6,79.6% 3.3% % % ,933 7,6 7,6 7,4.7% 9.% %.33..7% ,59 7, 7, 7,.% 5.% % % ,694 3,43 3,43 3,43.% 3.% % % , ,7 3,6 3,6 3,6.% 4.9%...%...% , ,9 47,43 47,43 47,43.% 5.%.3.3.%.3.3.% ,976 37,95 37,935 37,56.97%.% % % ,47 39,994 39,93 39,74.4% 4.%..6 3.%.9.7.% ,37 4,66 4,66 4,57.%.% % % ,95 4,377 4,377 4,377.% 7.%.7.7.%.7.7.% ,947 43,9 43,9 43,9.% 35.4%.6.6.%.6.6.% , ,34 46,96 46,96 46,96.% 46.3% % % , ,9 64,4 64,4 64,4.% 63.% % % ,57 5,9 5,96 5,.95%.% % % ,764 5,746 5,634 5,73.77% 5.6% % % ,645 53,53 53,6 5,5.%.% % % , 54,69 54,69 54,.%.% %.5.5.4% ,39 57,45 57,45 57,45.% 36.% % % , ,543 6, 6, 6,.% 47.4%.7.7.%.7.7.% , ,43 7,35 7,35 7,35.% 66.3%.5.5.%.5.5.% A te uts re yers. For oderte vues of the or set-up cost S 5, oth gorths yed the se souto for 5, d 5 tes. However for tes the soutos re dfferet, wth the gorth for P c chevg ower verge repeshet te dfferece of.4%. For ths proe stce the svgs w.r.t. depedet orderg re of %. 3. For rge vues of the or set-up cost S > 5 oth gorths yed the se souto proes soved. However, the effect of cresg S ecoes ess portt s the uer of tes creses, s c e see for oderte vues of S. 4. Athough we oy preset e 4 sury forto for the experets,.e. verge vues over rdo ded reztos, we pr-checed the stteets the soutos of oth gorths for dvdu proe stces. he c stteets, d C < C were wys cofred. c c vg vg 5. Athough the coputto te ssocted wth the souto of P c s uch hgher th tht of P, we eeve tht for the cses dscussed the prevous prgrph the gorth wth correcto fctor s reevt d c yed etter soutos, especy whe we use the deterstc JRP s pproxto stochstc evroets. oreover, the JRP s reted to tctc ger decsos, s the JRP s soved oy oce over cert perod of te oths up to yer. I ths respect the hgh dfferece coputto te etwee oth gorths ecoes ess reevt. he ove uerc oservtos c e geerzed the foowg eprc oservtos. Eprc oservto. he geer shpe of foows decresg ptter s decreses, s c e see the pot of Fg. 3. Expto. Fro equto d the prcpe of cuso d excuso, we c estsh upper oud o y repcg the est coo utpe wth the utpcto of the tegers, s foows: IO 6

17 UB L, {,..., },, {,..., }... Fro equto 5 t foows tht s decreses, the upper oud o decreses, d therefore the geer shpe of w e decresg. Fg. 3. Pot of ote: It shoud e poted out tht ofte the vues the souto re oted whe the vue of oserves drop. hs resut s ot surprsg sce ths w hppe whe the vues of the s ow etter coordto etwee the orders for the dfferet tes. I ths cse, soe of the vues w e ether equ to ech other or utpe of ech other. For these vues of the s, w e ser th whe o coordto s oserved. Eprc oservto. Let d c e the sc cyce tes for c proes P d P c, wth correspodg vectors d. he the foowg hs ee oserved the uerc experets: c c If the d c c c C C. For <, cses c we oserved tht d c c c C < C. oreover, f the c <. ote: We fed to fd for proof of ths fdg. Oe c prove tht the dervtve wth respect to of C c s rger th tht of C, pyg tht y u of C hs u of C c eft of t. oreover, f were ootoc, the the resut coud e show. However, there re soe cses where t s ot. Rer. For rge vues of the or set-up costs s, t foows fro equto 5 tht the s re ey to e rge. herefore, y the resut of the prevous c oservtos t foows tht ths cse ofte <. hs s oserved the uerc resuts preseted es -3. he equvet resut s foud e 4 for s vues of S. hs c so e see grphcy the pots of Fg.. 7

18 I ddto to the ove resuts, we vestgted the ehvor of the syste whe S s very s coprso wth s for,,. I order to g soe theoretc sght ths respect, we et S d oserve tht proe P or equvety P c ecoes: P s s C { } φ s.t. > s where φ hd for,,. It s ot dffcut to verfy tht φ s strcty covex, wth u tted :. O the other hd, Appedx B we show tht. otce tht the ser s, the coser s towrds. Accordgy, for S we expect s we. hs cocdes wth the oservto tht for very ow or set-up cost, the souto s ot to use ot repeshet t. I other words, t s to chec the syste cotuous fsho, d to order ech te depedety every uts of te. hs theoretc resut s ustrted the uerc expe show e 5 t c so e oserved the resuts correspodg to S e 4. As we c see e 5, s S goes to zero, the vector of repeshet tes defed y,,, teds to ts t gve y:,,..., d the oectve fucto goes to ts t gve y: s φ hd e 5. Ded set wth vryg S d fxed s, S C c wees vg 44,6.3 3,3,,,,,, , ,6,3,,,3,, ,.7 4 3,6,4,,,3,, , ,6,4,,,3,, ,6..5,3,,,,6,,4.53 4, ,7,44,4,4,34,7, , ,9,56,3,3,43,34, , ,3,7,3,39,54,43,3.35 5

19 5. Cocusos I ths pper we preseted copete yss for the JRP, y showg tht the ethods foud the terture to sove the JRP provded deed soutos. Furtherore, we provded effcet souto ethod to sove the JRP whe correcto s de the cost fucto. We showed tht though the cost proveet whe usg the correcto for epty repeshets s oy of few percetge pots, the quty of the souto ters of d s hgher. Prtcury ths proves to e the cse for rge vues of the or set-up costs d oderte or set-up costs. We further showed tht the souto wth correcto fctor outperfors the souto gve y ppyg depedet orderg usg EOQ s. hs s ot the cse for the foruto of the proe wthout correcto fctor, whch proves fory tht ths s prtcur cse of the ode wth correcto fctor. Refereces [] J.S. Dgpur, Foruto of ut te sge supper vetory proe, Jour of the Operto Reserch Socety [].J.G. v Es, A ote o the ot repeshet proe uder costt ded, Jour of the Operto Reserch Socety [3] L. Fg-Chu, Y. g-jog, A go u serch gorth for the ot repeshet proe uder power-of-two pocy, Coputers & Opertos Reserch [4] R.Y.K. Fug, X., A ew ethod for ot repeshet proes, Jour of the Operto Reserch Socety [5] S.K. Goy, Deterto of u pcgg frequecy of tes oty repeshed, geet Scece [6] P. Jcso, W. xwe, J. ucstdt, he ot repeshet proe wth powers-of-two restrcto, IIE rsctos [7] E. Porrs, R. Deer, Cotrog vetores suppy ch: A cse study, Iterto Jour of Producto Ecoocs [] E. Porrs, R. Deer, A effcet souto ethod for the ot repeshet proe wth u order quttes, Report Seres Ecooetrc Isttute, Ersus Uversty Rotterd, EI [9] E. Porrs, R. Deer, ew ouds for the ot repeshet proe: tghter, ut ot wys etter, Report Seres Ecooetrc Isttute, Ersus Uversty Rotterd, EI 5-5. [] R. Roudy, Roudg off to powers of two cotuous rextos of cpctted ot szg proes, geet Scece [] S. Vswth, A ew gorth for the Jot Repeshet Proe, Jour of the Operto Reserch Socety [] R.E. Wde, J.B.G. Fre, R. Deer, A effcet souto ethod for the ot repeshet proe, Europe Jour of Operto Reserch

20 Appedx A I ths ppedx we provde gorth for the evuto of the correcto fctor. otce tht the out of wor for the evuto of creses expoety wth the uer of tes. evertheess, we show tht y cses the sze of the vector c e reduced d therefore so the uer of ters. Frst oserve tht for gve vector wth soe of ts eeets eg equ or utpes of ech other, t s esy to verfy the foowg: foru w gve the se uerc vue for f we ppy t to reduced vector, sy ew, wth ts eeets extrcted fro the org d stsfyg: / for reted-prs,. Sce ew couts the frcto of effectve repeshets, t s cer tht t w so cude the repeshets whch products wth eeet of te pce. I such cse, we c reduce cosdery the out of wor eeded to evute. Gve vector,,, the foowg gorth s used to evute the vue of usg foru. Agorth for the evuto of Step. If for y,,, the. SOP. Step. Re-rrge the eeets of s.t. L d defe the set K {,,, }. Set R K. Set D dk d. Step 3. For to D do f / the R R \ { } ese R R ext Set K R d D dr. Step 4. If D GOO Step 5. Ese set d R K. GOO Step 3. Step 5. Appy foru to the ew vector ew wth eeets gve y K. Appedx B Proof of heore Frst ote tht fro t foows tht,, s ootoe decresg. ow et e the dcet ocy vector to for > d suppose tht >. By the covexty of C t foows tht C s decresg [, whch pes tht the u of C s foud. It foows tht C s cresg for >. Ag y the covexty of C ths pes tht < <, whch s cotrdcto y the ootocty of. herefore, < d the u of C s to the eft of Proceed sr wy to show tht, pyg..

21 Proof of proposto Frst ote tht the equty proposto s equvet to: c } {,...,,, ext otce tht sce the frcto of repeshets of te per yer s / the RHS of the ove equty hods. ow rese tht through the prcpe of cuso-excuso the uer of o-epty repeshets due to te s rger th the uer of repeshets of te us the ot repeshets of prs of products cudg te. Hece, the LHS of the equty hods. ote: If t est oe of the, the d C c, cocdes wth C,. We use proposto to estsh ower oud o for proe P c d the ss for heore d Le. Frst we hve: utpyg oth sdes of the ove equty y d tg the t s goes to zero yeds: I the foowg yss, we w see tht the ehvour of the secod ter the LHS of s s very uch detered y the ture of the rtos /. Athough for prctc purposes these rtos c e cosdered s rto uers, we foud terestg ehvour of the product c, for whe the rtos re regrded s rrto uers, s s the cse whe deds re cotuous vres, rther th dscrete see Porrs d Deer []. herefore, we cosder oth cses our yss. Accordgy, we frst cosder the cse for whch the s re rto uers, d we proceed sr wy s Porrs d Deer []. Let R \Q deote the set of rrto uers, where R s the set of re uers d Q the set of rto uers. We frst try to costruct susequece of gog to zero, sy,,, s.t. d

22 where, re gve tegers wth gcd, d Z. Hece, c, Such susequece of shoud stsfy the foowg syste for,,, < < Or equvety: d < < Fro the prevous syste t foows tht we c fd such sequece of s f d oy f d Lettg we ot: d

23 3 he ove equtes yed: hs pes tht the oy, for whch d hve fte uer of soutos s gve y. Let us c these vues,,,. For spcty of otto, the seque we drop the super dex,. ext ote tht for,,, we hve such sequece for whch:, c Oserve tht we c so seect,,, Usg ths vue d equto we hve: c, ow cosder rtrry, gve wth gcd, d te for whch d for soe teger >. ote tht stsfes the foowg syste: < 3 < Syste 3 pes tht

24 4 d Rewrte the ove syste s foows: Sovg the ove syste for yeds:, Usg the root the ove syste of equtes yeds tht, s ozero f, sry s the resut foud Porrs d Deer [7]. Let { }, }, :, {, < <. ow cosder, < d et w..o.g., for soe,, wth gcd,, d fro 3 we hve tht:, x. he, c, > otherwse f 4

25 5 Sce, < we cot hve y 4 oth < d <. Hece, ether or. I oth cses. Fro the prevous yss we hve estshed the foowg e: Le 4. If Q,, the the foowg hods: c, for y,. Proof of Le By e 4 d equto the resut foows. Le 5. If Q,, the the foowg hods: c, sup d, f c Proof. A rge prt of ths e foows fro e 4 d the yss precedg t. Wht res to e proved s tht the f s s deed. For ths prt suppose w..o.g. tht >, pyg tht > for s. ote tht s, tes posse tegers,, Let,, 3, e cresg sequece of pre uers d et e the -vues for whch,...,, ow ote tht for s eough, c d hece,, c.

26 Proof of heore By e 5 we c evute the t s o the LHS of equty d sce the t exsts, the frst prt of the theore foows. For the secod prt te the t o the RHS of, d sce ths t exsts d t s depedet of /,, the c of the theore foows. Before gvg the proof of heore 3, we eed frst the foowg e. Le 6. If / R \Q,, the. c, Proof. Let c, for soe tegers,. r r Suppose tht there s ouded susequece,, r,,, such tht r r r s r d K, K for soe K >. Sce ths pes r r tht there re oy ftey y dfferet vues of,, there exsts secod susequece s s s, of tegers such tht, s d s s s s s s s s s s s However, we ssued tht / ws rrto, so there c e o ouded susequece, hece, s d c, s requred. Proof of heore 3 By Le 6 we c te ts o oth sdes of equty d sce oth ts exst, the t of / exsts d s equ to the stted vue. 6