Opimizaion of Piodi Rviw Invnoy Modl Innaional Jounal of Compuaional Sin 99-6669 (Pin) 99-6677 (Onlin) Global Infomaion Publish 7, Vol., No., 69-89 Opimizaion of Piodi Rviw Invnoy Modl wih Emgny od in wo-ehlon Supply Chain using Dfnial Evoluion Paswaana Kani * and Shimizu Yoshiaki Dpamn of Poduion Sysms Engining, oyohashi Univsiy of hnology, oyohashi 44-858, Japan {kani, shimizu}@s.ps.u.a.jp Absa. his pap sudis h invnoy modl wih a gula and an mgny supply mod in a wo-lvl supply hain sysm. Rgula mod is usd fo nomal plnishmns whil h mgny mod is usd fo fas dlivy bu subjd o high os. his sysm is opad und unainy in boh dmand and gula lad ims wh im lag bwn gula and mgny lad im is no sid o on piod. W popos h modls fo aquiing h mgny od: ) manufau modl : pla boh gula and mgny od o h manufau; ) ousouing modl : pla all mgny od o an ousouing ompany o an xnal mmb in h hain; 3) pioiy modl : giv pioiy o mak h mgny od o h manufau fis bu h manufau ould no supply, hn pla h od o h ousouing ompany. W dmin h maximum pofi of h supply hain by using dfnial voluion (DE). h suls show ha whn h mgny opions a availabl, h pofiabiliy of h supply hain is inasd ompad wih h gula modl. Howv ah alnaiv of mgny mod pfoms wll und dfn ondiions. Kywods: Supply hain, mgny od, dfnial voluion, invnoy onol. Inoduion Und unainy in usom dmand, invnoy sysms wih wo supply mods, whih a a gula and an mgny mod, bom inasingly ommon in pai. his is h as whn a * Cosponding Auho. l.& Fax: 8-53-44673, Email: kani@s.ps.u.a.jp. GLOBAL INFORMAION PUBLISHER 69
Innaional Jounal of Compuaional Sin gula supply mod is usd piodially fo sok plnishmn, whil a sonday mod o mgny mod is ypially uilizd whn a sok ou is likly o ou a h ail. Low volums, sho lad ims and high aquisiion os p uni, haaiz mgny ods. h high os is usually a onsqun of h inasd anspoaion os []. h main anion of plaing mgny ods is o du h dlay in ming h dmand und h bakodd modl o o du amoun of los sals in full-los modl. h invnoy modl posd by h availabiliy of wo supply mods has bn sudid sin 96s. Baanki [] sudid a singl-piod of piodi viw invnoy sysm wih fixd gula lad im of on and immdialy mgny plnishmn. Bulinskaya [3], Fukuda [4] and Vino [5] xndd h singl-piod analysis o h n-piod as und h assumpion ha supply lad ims a a mulipl of a viw piod and h dfn of gula and mgny lad-im is always qual o on piod. hn Chiang and Guiz [6] fis sudid a piodi invnoy sysm wih long viw piod (many viw piods in on yl) by assuming ha h lad-im is sho han h viw piod. hy invsigad h as in whih mgny ods of a piodi viw und od-up-o-r poliy and allowd h plamn ih of a gula od o an mgny od (no boh). Chiang [7] xndd his pvious wok by allowing possibl gula and mgny od a piodi viw poh and xndd h pvious wok o mo gnal ass wh lag vaiabl os and/o fixd od oss w inud. A un sah of mgny mod of Bylka [8] psnd a piodi viw apaiad lo sizing modl wih limid paial baklogging a sok ous modl. h minimum os is obaind by onsiding h sysm as a dis-im Makov dision poss. Howv, his poblm sill si o h assumpion of and lad-im of gula and mgny mod, spivly. Ou wok dfs fom hos liaus in wo folds. Fisly, in all ali sahs of an mgny mod was limid o a singl-hlon invnoy sysm (a singl ompany) and all mgny od was assumd o b saisfid wihou onsiding h poduiviy and pfoman of h mgny supply mod. hfo, his pap inodus h alnaivs of mgny supply mods o h supply hain sysm and valuas h bnfi basd on h pfoman of h whol sysm. Sondly, o mak h modl mo sophisiad, boh gula and mgny lad ims a allowd o b abiay lnghs (h dfn of gula and mgny lad im is no sid o on piod) and h invnoy sysms a opad und boh dmand and gula lad-im vaiaions. Hn, h modl is fd o h mixd-ing nonlina pogamming (MINLP) poblm blonging o NP-had lass in soluion, whih is vy omplx and dfiul o solv in gnal. Hn, w sl a dfnial voluion (DE), whih is a kind of voluional sah mhods and known as a paial and ffiv mhod o solv suh poblms (s [9] and []). h s of h pap is oganizd as follows. h dsipion and fomulaion of h modls a givn in sion. In sion 3, h onp and algoihm of DE is psnd. hn h numial xampl, ompuaional suls and onlusions a psnd in sion 4, 5 and 6 spivly. 7 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl Mhodology Fomulaion In his sion, w sudy a piodi viw modl of a mak-o-sok in a wo-lvl supply hain sysm ha onsiss of a singl manufau and a singl ail. As a basi oopaion of all mmbs along h supply hain, w assum ha h is full infomaion shaing in h sysm. W xamin h alnaivs fo mgny od: ) manufau modl : pla boh gula and mgny od o h manufau; ) ousouing modl : pla an mgny od o h ousouing ompany o xnal mmb in h hain; 3) pioiy modl : giv h pioiy of plaing an mgny o h manufau fis, bu h manufau ould no supply, hn pla an mgny od o h ousou. Fo manufau and pioiy modls, h manufau is allowd o supply mgny od only whn h manufau sill an supply h gula od a las 9% svi lvl. hn w valua h bnfi of ah mgny modl by ompaing h pofiabiliy wih gula modl (no mgny mod). hn h appopia ondiion fo using ah alnaiv modl is dmind. W shall xplain h fomulaions of gula and mgny modls in h following subsions.. Rgula Modl h following noaions will b usd in h gula modl. = Piod indx in (,,, ) p = Duaion of ah piod FD = Foas dmand a h manufau D = Cusom dmand a h ail lm = Man dlivy lad-im of aw maial lm = Ral dlivy lad-im of aw maial l = Man dlivy lad-im of podu l = Ral dlivy lad-im of podu p = Poduion a p day a h manufau Bs = Bginning aw maials on hand a h manufau s = Amoun of aw maials on hand bfo iving a nw plnishmn Es = Ending aw maial on hand a h manufau Bm = Bginning invnoy on hand of podu a h manufau (xluding safy sok) Em = Ending invnoy on hand of podu a h manufau (xluding safy sok) B = Bginning invnoy on hand of podus a h ail. E = Ending invnoy on hand of podu a h ail Bss = Bginning safy sok on hand a h manufau Ess = Ending safy sok on hand a h manufau Qm = Oding quaniy of aw maial Qp = Poduion quaniy of podu bfo iving h nw plnishmn o duing [, + lm ) Qp = oal poduion quaniy of podu in on piod o duing [, + p ) Q = Oding quaniy of podu GLOBAL INFORMAION PUBLISHER 7
Innaional Jounal of Compuaional Sin Qsm = Sals volum of gula od a h manufau Qs = Sals volum of h ail S = oal quaniy of shoag a h ail. f = Quaniy o fill h us of safy sok a piod = Uni puhasing os of podu fom gula od a h ail m = Uni puhasing os of aw maial a h manufau a = Uni adminisaion os of podu a h ail h = Uni holding os of aw maial a h manufau ha will b alulad fom h % of uni puhasing of aw maial m o ( m h )/ hm = Uni holding os of podu a h manufau ha will b alulad fom h m % of uni puhasing of aw maial m and uni poduion os p o (( m + p ) h m )/ h = Uni holding os of podu a h ail ha will b alulad fom h % of uni puhasing of podu and adminisaion os a o ( + a ) h ))/ = Uni anspoaion os of podu fom h gula od a h manufau p = Uni poduion os a h manufau s = Uni oppouniy los os o shoag os a h ail i = Sals pi p uni of podu a h ail Dision Vaiabl of gula modl ss = Safy sok lvl a h manufau S = Dis lo sizing a h manufau R = ag sok lvl a h ail.. Manufau h invnoy lvls of boh manufau and ail a viwd a vy im inval ov oally piods planning hoizons ( =,,, ) in ah yl. Foas dmand, nd usom dmand, dlivy lad im of aw maial fom h suppli o h manufau and dlivy lad im of podu fom h manufau o h ail a ah piod a appoximad by nomal disibuion. A h bginning of ah piod, h manufau ods aw maial o h suppli, who has unlimid apaiy, hn podus a singl kind of podu wih a fixd poduion a p day and supply hs podus o h ail. Du o unainis in boh nd usom dmand and lad-im of aw maial, h manufau has o hold h appopia amoun of safy sok ( ss ) and also did h lo sizing poliy o od aw maial ( S ). h safy sok will b usd only whn a nomal invnoy lvl anno saisfy h ail dmand and i mus b filld as soon as possibl af having usd. Rgading o S, h a 6 piod planning hoizons, h a 3 possibl oding poliis fo h manufau o sl. Fo xampl, h fis poliy may follow lo-fo-lo poliy o mak an od in vy piod, h sond possibl poliy may ombin od of piod and hn us lo-fo-lo fo h s fou piods and so on. Af h manufau has sld h bs pan of S, h amoun of invnoy on hand a 7 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl h bginning of h piod will b hkd. If h amoun on hand is lss han h sum of h foasd dmand and h amoun o fill bak h safy sok ha usd in h pvious piod, h manufau will mak an od. Ohwis, no od will b issud as shown in Equaion (). { FD + f B Bs, } Qm = max () L assum ha w always sa viwing h invnoy a im. hn h manufau an sa poduing h podus a h lad-im ona + lm. Whn la dlivy is oud ( lm > lm ), h manufau sill an sa h poduion a im + lm h bginning aw maial on hand Bs is availabl. Ohwis, h manufau has o wai unil i ivs aw maial fom h suppli a im + lm. h poduion quaniy duing la-im dlivy ( Qp ) and h amoun of aw maial on hand bfo iving a nw plnishmn ( s ) an b alulad by using Equaions () and (3). Consqunly, w an dmin h oal poduion quaniy ( Qp ) and h nding aw maial on hand ( Es ) as shown in Equaions (4) and (5), spivly. Qp ( lm lm ) p = Bs lm > lm and Bs lm > lm and Bs lm lm > ( lm lm ) p ( lm lm ) p () s Bs ( lm lm ) p = Bs Qp + (( p lm ) p) Qp + s + Qm Qp = ( p lm ) p Qm lm > lm and Bs lm > lm and Bs lm lm Bs and Qm Bs and Qm > ( lm lm ) p ( lm lm ) p Bs > and ( s + Qm ) > ( p lm ) p Bs > and ( s + Qm ) ( p lm ) p > ( p lm ) p ( p lm ) p (3) (4) ( s + Qm ) (( p lm ) p) Bs > and s + Qm > ( p lm ) p Es = ( p lm ) p Bs and Qm > ( p lm ) p ohwis By oding aw maial in a big bah, h manufau nomally has h bginning aw maial on hand. Hn h manufau may b abl o podu h podus mo han lo-fo-lo as. Evnually, his suls in high apabiliy o supply ail s dmand bu i may inu high holding os. Ending invnoy lvl ( Em ) is h amoun of finishd podus ha xds h ail s dmand and lfs fom fulfill h us of safy sok fom h pvious piod. Ohwis, no invnoy on hand of h podu a h nd of piod. Em an b alulad as follows: (5) GLOBAL INFORMAION PUBLISHER 73
Innaional Jounal of Compuaional Sin Qp + Bm Q Qp + Bm Q ss+ Bss Em = Q < ( Qp and ( Qp ohwis Q < ( Qp + Bm + Bm ) and Bss + Bm ) and Bss ss < ss - Q ) > (ss Bss ) (6).. Rail h ail maks a gula od o h manufau piodially. h gula od quaniy ( Q ) is dmind by ompaing h nding sok lvl ( E ) a h viw im wih h dsid ag sok lvl R, whih is qual o R E. Du o som fluuaion in h sysm, h manufau may fail o supply all quimns o h ail ( Qsm < Q ). Sin w aim a maximizing h pofi of h supply hain ha onsidd manufau and ail as on niy, h shoag os paid by h manufau o h ail is ngld in his modl and only h oppouniy los os is hagd a h ail fo ah uni unsaisfid nd usom dmand. As mniond abov, h ail and h manufau should saisfy a las 9% svi lvl ( β, βm 9%). In h oh wods, h sok ou ould no xd % of h od quaniy a boh mmbs. hs onsains should b aid ou in all modls. his sah onsids h sok ou a h ail as h shoags (full los) aoding o wo asons. Fis, in oday high ompiiv mak, h usoms hav a plny of hois o aqui h podus, spially whn viwing h ail as dpamn sos, supmaks o onvnional sos. Consqunly, h siuaion ha h usoms buy h sam podu fom h sond shop h is h sok ou a h fis shop is qui ommon in h al wold businss. Sond, non of h liaus assoiad wih h mgny od analyzd h unsaisfid dmand as a full-los modl. Fom h abov analysis, w an dsib h pofi of h supply hain Πs ha onsiss of pofis of h manufau Π and h ail Π. m Π m = = Qsm = m Qm = = h Es p Qp = = hm ( Em Qsm + Ess ) (7) Π = = Qs i = Qsm = h E = a Qs = s S (8) 74 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl. Emgny Modl In addiion o h opaion and os paams dfind in gula modl, addiional paams fo mgny modls f o h following noaions. D = Cusom dmand duing h mgny lad im [, + l ) D = Cusom dmand duing h upoming im inval [, + l ) o h dmand duing h gula lad im D = Cusom dmand duing h upoming im inval [ + l, + l ) o h dmand duing h im inval bwn wo supply mods, so D = D D. D3 = Cusom dmand duing h upoming im inval [ + l, + p ) o h dmand ha ous af iv h gula plnishmn unil h nd of viw piod l = Emgny lad im I = Sok on hand lvl of podus a h ail bfo iving mgny plnishmn o a h nd of upoming im [, + l ) I = Sok on hand lvl of podus a h ail bfo iving gula plnishmn o a h nd of upoming im [, + l ) I = Invnoy on hand lvl of podus a h ail af iving h gula od S = Quaniy of shoag a h ail bfo iving mgny plnishmn o duing h upoming im [, + l ) S = Quaniy of shoag a h ail bfo iving gula plnishmn o duing h upoming im [, + l ) Qm = Sals volum of mgny od a h manufau. Qo = Emgny oding quaniy fom h ousouing ompany. = Uni puhasing os of podu fom mgny od a h ail = Uni anspoaion os of podu fom h mgny modl Addiional Dision Vaiabl fo mgny modl = Rod poin a h ail Nomally, h ail maks a gula od o h manufau piodially. Howv, und mgny modls, h invnoy lvl is low han h od poin, h mgny od is plad. Boh gula and mgny ods a viwd and plad a h viw im. h gula od quaniy is h amoun of podus ha ais h invnoy up o R lvl, whih is qual o R. h mgny od quaniy is dmind by ompaing h nding sok lvl a h viw im wih h dsid od poin, whih is qual o E as shown in Figu. h mgny od is haaizd by a sho ( l < l ) and fixd lad-im, and no shoag bu hag high uni puhasing os ( > ). H, w allow h gula and mgny o hav abiay lngh, and no si o h assumpion of on piod dfn bwn gula and mgny lad-im. Fo boh gula and mgny od, no fixd oding oss a onsidd. h a invnoy holding oss and shoag oss ha a hagd basd on h n invnoy a h nd of ah yl. Uni holding os a h ail dpnds on whh h mgny is oud o no, so i an b alulad as follows: GLOBAL INFORMAION PUBLISHER 75
Innaional Jounal of Compuaional Sin h ( Q + ( Qm + Qo ) ( Q + Qm + Qo ) = h ( + a ) a ) + a h Qm + Qo ohwis > (9) Invnoy lvl (unis) R R - E - E im (piod) +l +l +p +p+l +p+p Fig.. Invnoy sysm of h ail und an mgny od poliy Sin w onsid sok ou as los, oppouniy los os is hagd fo ah uni unsaisfid h nd usom dmand. In od o dmin h nding invnoy lvl ( E ) and oal sok ou quaniy a h ail ( Qs ), w hav o do h alulaion basd on wo disjoin im invals, whih a bfo iving gula plnishmn [, + l ) and af iving gula plnishmn [ + l, + p ). Duing [, + l ), mgny od is plad ( Q > ), i s nssay o dmin invnoy lvl ( I ) and sok ou quaniy ( S ) bfo iv an mgny plnishmn. hn w an us hs wo valus o alula invnoy lvl ( I ) and sok ou quaniy ( S ) bfo iving a gula plnishmn as shown blow. { B D } I = max, () { D B } S = max, () B D Q and B > D I = ( I + Q ) D Q > and ( I + Q ) > D () ohwis 76 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl Q and B > D D B Q and B D S = (3) S Q > and ( I + Q ) > D D ( I + Q ) + S Q > and ( I + Q ) D Duing [ + l, + p ) o af iving h plnishmn fom gula mod, h invnoy lvl a h ail aiss o I lvl, whih is qual o I + Q. hn h nding invnoy lvl ( E ) and oal sok ou quaniy a h ail S an b alulad as follow: { I D3 } E = max, (4) { D3 I } S = S + max, (5) his pap offs h alnaiv opions o aqui h mgny od, whih a manufau, ousouing and pioiy modls. h dail of ah modl is dsibd as follows:.. Manufau Modl In his modl, h ail puhass boh gula and mgny od fom h manufau und h onsains ha h manufau has o supply all mgny od wih fas and fixd lad im ( l ) and also quis o supply h gula od a las 9% of h oding quaniy. h manufau an g high vnu fom slling mgny od, bu in od o aiva fas and fixd lad im, spial anspoaion os ( ) is hagd fo ah uni in mgny od. Fom h abov analysis, w obain a nw fomula fo h manufau s pofi and ail s pofi as follows: Π m = Π = = Qsm = = = hm ( Em Qsm Qs i = + = + Ess ) Qsm Qm = = h = Qm = E pm = Qm h Qm = Es s = = S a p Qs Qp (6) (7).. Ousouing Modl Und unainis of h sysm, h mgny od is oud igulaly wih unpdiabl quaniy. hfo, i may b mo osly fo h manufau o hold high invnoy o supply GLOBAL INFORMAION PUBLISHER 77
Innaional Jounal of Compuaional Sin boh gula and mgny by islf. So in ousouing modl, w assum ha h ail will mak h gula od o h manufau whil making an od of mgny o h ousouing ompany. No ha w onsid ousouing ompany as h xnal mmb, so h pfoman and finanial saus of his ompany is no inludd in h modl. h pofi of h manufau an b alulad in h sam way as h gula modl whil pofi of h ail an b alulad in h sam way as h manufau modl...3 Pioiy Modl Evn hough mgny od an absob h poblm of shoag du o h fluuaion of dmand and lad ims, puhasing all mgny quaniy fom h xnal mmb (ousouing ompany) may sul in h duion of manufau s vnu ha lads o h duion of supply hain s pofi. On h opposi way, all mgny od is ddiad o h manufau, i also may no b an onomial poliy sin h mgny od is unpdiabl. hfo, und pioiy modl, h ail givs h pioiy of plaing h mgny od o h manufau fis, bu h manufau ould no supply hn pla an mgny od o h ousouing ompany. h manufau is allowd o supply mgny od only whn h manufau an sill mainains 9% svi lvl of gula od ( β m 9%). h xpnss o aqui h mgny od fom boh manufau and ousouing ompany a s a h sam valu. h pofi of h manufau an b alulad in h sam way as h manufau modl whil pofi of h ail an b alulad as follows: Π = i = Qs = Qsm h = EI = s = S ( Qm + Qo ) a = Qs (8) 3 Dfnial Evoluion Whn boh dmand and lad-im a unain, and boh gula and mgny lad-im an hav abiay lnghs, dmining global opimal soluion nds xnsiv ompuaional load. I is oo omplx and im onsuming o wok wih i by using onvnional opimizaion mhods. Insad, dfnial voluion (DE) is poposd o ompu h soluions and valua h pfoman of h sysms. DE is a sohasi di sah wih vo populaion ha has abiliy o handl non-dfniabl, non-lina, NP-had and mulimodal os funion. DE us slfoganizing shm o ak h dfn vo of wo o mo vos o a h muan vos. 78 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl So ha a fw inpu is quid fom h us and i ass o implmn DE o solv h poblms. h basi sps of implmning DE a xplaind as follows: 3. Suu and oding Dision vaiabls und gula modl onsis of oding poliy of aw maial (S), safy sok lvl of podu ( ss ) and ag sok lvl of podu (R). hn, od poin () is add o b h foh dision vaiabl und mgny modl. Fo S, h manufau has o did whh o mak h od a h bginning of vy piod o ombin h od in a big bah. hfo, h binay oding is sld o psn h valu of S. On h oh hand, ss, R and a onsidd as h amoun of podus (unis) a h manufau and h ail, so h ing oding is sld o psn hs h valus. 3. Low and upp bound In od o gna iniial populaion, low and upp bound of ah dision vaiabl should b s as follows: Fo S: As mniond abov, h manufau has o did whh o ombin h od o no. So S is psnd by binay oding wh mans h od is mad and mans h od is ombind wih h pvious piod. Sin h manufau always maks h od a h fis piod, h numb of bi quimn is qual o planning hoizon minus. Fo xampl, wih 6 piods planning hoizons, i quis 5 bis o psn h soluion, mans lo-fo-lo poliy is usd fo piod, and 3, hn h od is ombind fom piod 4 o 6 and h od is plad a piod 4. Fo ss, and R: hs dision vaiabls india safy sok lvl and invnoy lvl, so hs vaiabls a psnd by ing oding. L D, FD, σ D, σ FD a h man and sandad dviaion of h nd usom dmand and foas dmand, lm, l, σ lm and σ l a h man and sandad dviaion of h dlivy lad im of aw maial and podu, z is h numb of sandad dviaions fom h man osponding o pobabiliy spid by h svi lvl. - Low bound of ss is s o o no invnoy is kp a h manufau. - Upp bound of ss is alulad by h using Equaion (9). In od o ovom h onsain of 9% svi lvl and o mak su ha all sahing spa is boundd, upp bound of ss, and R s o ov 99.99% svi lvl (lags spa) whih is osponding o z = 4. (s [] fo fuh infomaion). z ( lm σ FD) + ( FD σ lm) (9) GLOBAL INFORMAION PUBLISHER 79
Innaional Jounal of Compuaional Sin - Low bound of is s o o no mgny od is quid. Upp bound of is h amoun o avoid sok ou duing lad im dmand z ( l σ D ) + ( D σ l ) () - Low bound of R should b a las o qual o h xpd dmand duing h viw im plus dlivy lad im ona, whih is qual o D ( p + l). - Upp bound of R an b dmin in h sam way as [], whih is qual o: D ( p+ l) + z ( l σ D) + ( D σl) () 3.3 Algoihm of DE his sah uss DE/and//bin sagy o solv h poblm. Noaion and/ mans on dfn s of vo ( vos) is andomly hosn among h populaions o b muad. hn, bin mans h indpndn binomial xpimn is usd fo ossov shm. Basi algoihm of DE/and//bin onsiss of following sps:. Randomly gna h iniial populaion o yild h ag vo x G ( i, M ) i, =,... () Wh M and G psn populaion siz and gnaion, spivly. Exampl of h iniial populaion of ag vo is shown in Figu. i = i =... i = M j =............... Binay Coding fo oding poliy : Od is plad : Od is no plad (ombin od wih fom piods) j = n 95 838 3 34 873 88 945 69 Ing Coding fo safy sok lvl Ing Coding fo ag sok lvl Ing Coding fo od poin Fig.. Illusaion of iniial populaion. Gna muan vo by adding h wighd dfn bwn wo ag vos o h hid ag vo. (hs h vos a hosn andomly among h populaion) ( x x ) vi,g = x3,g + F,G,G + (3) 8 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl Wh F is a al and onsan fao [, ] whih onols h ampliaion of h dfnial vaiaion ( x, G x, G ). Oiginal DE algoihm is only apabl o solv h poblm wih oninuous vaiabl, so v i,g+ a gnad in m of h al numb. In od o handl h binay and ing vaiabls, w us ounding off hniqu o onv v i,g+ bak o binay and ing valus. Fo xampl, h fis dimal poin is ga han o qual o fiv, w ound up,.g..9 will bom. Ohwis, w ound down,.g.. will bom. Moov, by adding wighd dfn of wo vos o h hid vo, i dos no guaan ha valu of muan vos would b lying insid h sahing bounday. Bound violaion in his pap an b paid by using fo bound sagy. So, h valu of muan vo is byond h upp bound, w s i qual o h upp bound and h valu is low han h low bound, w s i qual o h low bound. 3. Apply h ossov opaion o gna h ial vo by mixing som lmns of h ag vo wih h muan vo hough ompaison bwn andom valu and ossov onsan as shown in Figu 3. u v ( and( j) CR) ( and( j) > CR) o j = nb( i) and j nb( i) (,,... ). ji, G+ ji, G+ = fo j = n x ji, G Wh and ( j) is h j h valuaion of a unom andom numb gnao, CR is h ossov onsan [, ], and nb (i) a andomly hosn indx in {,,, n} whih nsus ha ial vo gs a las on paam fom muan vo. (4) j = x i,g v i,g+ u i,g+ and()<=cr and()<=cr............ j = n 945 69 38 nb(n) = j 945 5 5 Fig. 3. Exampl of ossov poss fo n = 7 4. If h ial vo is b han h ag vo, h ial vo plas h ag vo. Ohwis, h ag vo is maind. ha will sl h mmbs of h nw populaion a h nx gnaion. 5. Chk h p-spid sopping ondiion. If i is saisfid ( G = G max ), sop and un h ovall bs vo as h final soluion. Ohwis, go bak o Sp by inmning h gnaion numb by. GLOBAL INFORMAION PUBLISHER 8
Innaional Jounal of Compuaional Sin 4 Numial Exampl and Rsuls h numial xampls in his pap w undakn aoding o h following poposs. - Evalua h bnfi of using mgny od modls in ompaison wih h gula modl and xamin h ff of aquiing h mgny od fom dfn sous ( manufau, ousouing and pioiy modls) und dfn ondiions (vaiaion in sandad dviaion of dmand, holding os podu and anspoaion os of mgny od a h manufau, holding os of podu and shoag os a h ail). - Implmn DE o solv a al wold poblm by dmining h appopia oding poliy and invnoy lvl of ah mmb ha maximizs h pofi of h supply hain. In onsuing h modls, som inpu paams a s as follows: End usom dmands p day of ah ail and dlivy lad-im a andomly gnad und h nomal disibuion, so D, FD = 5 p day, σ D, σ FD = 5, 5 and days, ( lm, σ lm ) = Nomal (3,), ( l, σ l ) = Nomal (3, ), l = day, p =7 days and = 6 piods. Cos paams of manufau and ail a s as follows: =$5 p uni, p m =5 p uni, h =5% of aw maial valu, hm =%, % and 4% of podu valu, =$, $, $5 and $ p uni, =$5, $35, $45 and $55 p uni, s =$, $, $3 and $4 p uni, h =5%, 3%, 45% of podu valu, i =$55 p uni, β and β m =9%. Du o h nau of poblm dpndn, ky paams ( M, CR and F ) in DE should b hosn wisly and appopialy o g fas onvgn and avoid h poblm of pmau onvgn. Aoding o ou pliminay xpimns, an appopia valu of M, CR, F and G max a s as follows: M = 5, F =.75 and CR =.5, G max is qual o 5, gnaions und gula modl and qual o, gnaions und mgny modls. Du o h sohasi nau of h poblm in his sudy, w aid ou ials wih dfn sd valus fo ah poblm. So h suls in abl,, 3 and Figu 4, 5, 6 a h avag valu of ouoms. 4. Snsiiviy Analysis Rgading Sandad Dviaion of Dmand and Uni anspoaion Cos of Emgny Od abl shows h ompuaional sul of dision vaiabls ( S, ss, and R ) of ah mgny modl and Figu 4 shows h pn impovmn of supply hain pofi gaind fom using mgny modls ( manufau modl, ousouing modl and pioiy modl ) as ompad wih h gula modl und vaious ss of sandad dviaion of dmand (σ dnos boh sandad dviaion of foas and usom dmands) and uni anspoaion os of mgny od ( ). I s la fom Figu 4 ha vy mgny modl und all opions an gna high pofi han h gula modl and mgny modls bom mo ffiv as σ inass and dass. Und σ = 5 and 5 unis p day, manufau modl oupfoms h oh modls in mos 8 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl ass, xp whn is los o $/uni, as shown in Figu 4 (a) and (b). his is du o h fa ha h manufau an g high bnfi fom slling mgny od wih low. Moov, h suls of dision vaiabls und manufau modl in abl suggs ha h manufau an op wih boh gula and mgny od by making h od following lo-fo-lo poliy and hold a bi high amoun of safy sok (in ompaison wih oh mgny modls). A h sam im, ail also shown o giv h oopaion of using manufau modl by sing qui high valu of (spially und = $/uni). Howv whn inass, h manufau an g lss bnfi fom slling mgny od. As a onsqun, pn pofi impovmns of manufau and pioiy modls das unil qual o zo (no b usd) whn inass los o. hfo, i s mo bnfiial o ddia all mgny od o h ousouing ompany and s low safy sok lvl a h manufau (s abl und ousouing modl ). abl. Compaison of dision vaiabls und dfn s of sandad dviaion of dmand and uni anspoaion os of mgny od σ Manufau Modl Ousouing Modl Pioiy Modl S ss R S ss R S ss R 3 368 945 98 399 98 344 37 3 67 3 98 399 954 37 333 5 37 335 98 399 864 36 358 7 4 374 98 399 87 3 357 5 94 487 8 6 73 3 75 33 338 838 88 36 6 73 3 86 5 38 5 84 5 393 6 73 3 78 393 835 33 343 6 73 3 665 98 36 5 758 94 83 64 69 38 767 75 965 698 878 396 64 69 38 74 79 37 5 67 73 38 64 69 38 654 447 38 664 39 335 64 69 38 663 46 355 No: h highligh aa psns h bs poliy und ah ondiion. GLOBAL INFORMAION PUBLISHER 83
Innaional Jounal of Compuaional Sin 8 6 4 % Pofi impovmn 5 8 6 4 5 5 Emgny anspoaion os, [$/uni] σ (b) σ = 5 () σ = (a) = 5 8 6 4 Manufau modl Ousou Modl Pioiy Modl Fig. 4. Compaison of pofi impovmn wih sandad dviaion of dmand and uni anspoaion os of mgny od Und σ =, pn pofi impovmn is qui snsiiv o, so h bs alnaiv of implmning mgny modl an b dividd ino h ass as illusad in Figu 4 (). h fis as is whn is low (lss han ), as xpd, h manufau an g high bnfi fom slling mgny od und his ondiion. hfo, by making h od of aw maial in a big bah and hold high amoun of safy sok (as shown in h fis olumn of abl ), h manufau modl boms h mos pfabl alnaiv. h sond as is whn inass (bu lss han ), h bnfi fom slling mgny od of h manufau dass. his is du o h fa ha puhasing som mgny od fom ousouing mod an avoid aying oo muh amoun of safy sok a h manufau. hfo, h pioiy modl boms h mos ffiv alnaiv und his ondiion. Lasly, whn is vy high (los o ), h manufau an gain jus a fw bnfis fom slling mgny. Hn, h ousouing modl is always oupfomd h oh alnaivs in his ondiion. 4. Snsiiviy Analysis Rgading Holding Cos of Podus a h Manufau and Uni Puhasing Cos of Emgny Od abl and Figu 5 show h ompuaional sul und dfn holding os of podus a h manufau ( hm ) and uni puhasing os of mgny od ( ). I an b noid fom Figu 5 ha h pn pofi impovmn of supply hain inass as dass bu qui sady und vaiaion of hm. Whn is vy hap, i is qui aaiv and mo onomial fo ail o hold lss sok and puhas mo mgny od. In h opposi way, manufau an an low bnfi und low, so i is b o aqui h mgny od fom boh manufau and ousouing ompanis. h suls fom h hid olumn in abl wih =$5/uni suggs ha by puhasing maial in a big bah, sing high od poin wih moda amoun of safy sok and ag sok lvl, pioiy modl oupfoms h ohs alnaivs. 84 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl abl. Compaison of dision vaiabls und dfn ss of holding os of podu a h manufau and uni puhasing os of mgny od Manufau Modl Ousouing Modl Pioiy Modl hm S ss R S ss R S ss R 5 7 67 39 97 33 954 38 337 35 4 63 37 93 39 34 7 344 45 8 4 333 964 94 358 55 67 363 55 5 4 334 77 43 3764 6 44 364 5 3 67 3 98 399 954 37 333 35 9 4 335 98 989 3396 89 344 45 8 4 33 96 935 354 986 36 55 4 37 334 57 34 374 99 366 4 5 45 5 33 88 33 866 37 358 35 45 36 36 866 988 346 85 79 34 45 3 37 33 86 9 3454 985 36 55 77 5 334 45 64 37 987 366 No: h highligh aa psns h bs poliy und ah ondiion. % Pofi impovmn 9 9 8 8 7 7 6 6 5 5 4 4 3 3 5 35 45 55 5 35 45 55 9 8 7 6 5 4 3 Emgny Puhasing os [$/uni] 5 35 45 55 Manufau modl Ousou Modl Pioiy Modl (a) hm = % (b) hm = % () hm = 4% Fig. 5. Compaison of pofi impovmn wih holding os of podu a h manufau and uni puhasing os of mgny od Moov, i is obviously sn fom Figu 5 ha h pn pofi impovmn of ousouing modl and pioiy modl das damaially as inass, bu slighly dass und manufau modl. his is du o h fa ha high suls in high inom a h manufau. So h manufau maks h od of aw maial in a big bah o inas poduiviy and hold high amoun of safy sok o pvn shoag (s h fis blok olumn of abl ). As a sul, h manufau modl is bom h mos pfabl und high. In h opposi way, puhasing mgny od wih high fom ousouing ompany is onsid- GLOBAL INFORMAION PUBLISHER 85
Innaional Jounal of Compuaional Sin d as loss of inom a h manufau ha finally lads o h loss of inom of h supply hain sysm. Consqunly, ousouing modl will nv bn usd (% pofi impovmn = ) whn is vy high (.g., a $55/uni in his sudy). 4.3 Snsiiviy Analysis of Holding Cos of Podus and Oppouniy Los Cos of h Rail abl 3 and Figu 6 show h ompuaional sul und dfn holding os of podus a h ail ( h ) and uni oppouniy los os of h ail ( s ). I an b noid fom Figu 6 ha h pn pofi impovmn of supply hain inass as holding os of podus a h ail ( h ) and uni shoag os of h ail ( s ) inas. h suls fom Figu 6 (a) and (b) illusa ha und low h (5% and 3% of podu s valu), pioiy modl is h mos advanag alnaiv. Whn h holding os is low, h ail an hold a lag amoun of invnoy of podus (s h hid blok olumn of abl 3 wih h =5% and 3% of podu s valu) and h mgny mod is no usd so ofn. Consqunly, lss pofi impovmn is oud und low h, spially whn h = 5% podu's valu, as shown in Figu 6 (a). Howv, h mgny modl boms mo pfabl as h and s inas. Whn h =45%, h manufau modl is h mos pfabl alnaiv and an gna almos % pofi impovmn und s =3 p uni as shown in Figu 6 (). As xpd, whn h is high, h ail pfs o od an mgny od han holding a lo of invnoy. Hn, h mgny oding is oud qui ofn wih high quaniy. Aoding o h sul fom h fis blok olumn of abl 3 wih h =45% of podu valu, h ail ss low ag sok lvl and high od poin. I is vidn ha h manufau an g h advanag of holding high sok o supply boh gula and mgny od. hfo, manufau modl oupfoms h oh alnaivs whn h is high. 86 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl abl 3. Compaison of dision vaiabls und dfn ss of holding os and oppouniy los of shoag a h ail Manufau Modl Ousouing Modl Pioiy Modl h s S ss R S ss R S ss R 5 43 8 333 979 98 3596 3 359 43 87 3379 976 4 3598 3 359 3 4 35 347 976 4 3598 3 359 3 68 38 954 7 36 978 46 38 3 67 3 98 399 954 37 333 3 48 35 345 96 358 3455 947 356 334 45 47 38 84 568 87 39 63 8 98 39 4 888 577 35 697 398 95 3 5 49 96 69 9 37 74 434 955 8 6 4 8 6 4 % Pofi impovmn 8 6 4 8 6 4 3 3 3 Oppouniy los os a h ail s [$/uni] 8 6 4 8 6 4 Manufau modl Ousou Modl Pioiy Modl (a) h = 5% (b) h = 3% () h = 45% Fig. 6. Compaison of h pofi impovmn wih holding os and oppouniy los of shoag a h ail 5 Conlusions W hav invsigad a piodi viw invnoy of wo-hlon supply hain sysm in whih boh gula and mgny ods an b plad piodially. h alnaivs fo making h mgny ods a onsidd as manufau, ousouing and pioiy modls. W fomula h poblm und boh dmand and lad-im vaiaions, and also allows boh gula and mgny lad-im o b abiay lnghs. In addiion, his sudy is no sid o h assumpion mad in h pvious sudis ( [-5] and [8]) ha h dfn of gula and mgny lad-im is always qual o on piod. So h poblm is fd o h ing nonlina pogamming poblm, whih is vy omplx and dfiul o solv. o onn wih suh poblm, DE was applid o dmin h opimal maial oding poliy and safy sok lvl of GLOBAL INFORMAION PUBLISHER 87
Innaional Jounal of Compuaional Sin h manufau, od poin and ag sok lvl a h ail ha maximizs h pofi of h supply hain sysm. DE is known as an ffiv algoihm o handl wih non-dfniabl, nonlina and oh omplx mahmaial funions. Also DE is a simpl and saighfowad sagy baus i quis only a fw ky paams, whih a no dfiul o sl, o obain good soluions. Moov, insad of using pu binay oding basd on h onp of GA and al valu oding of oiginal DE, mixd binay-ing oding is implmnd in h psn DE algoihm. h suls show ha DE is a good mhod ha has fas onvgn popy and div h opimal soluion ffiinly (h ompaison bwn GA and DE in [3] shows h supioiy of DE o GA). Howv, a limid numb of sahs hav onsidd abou DE s bhavio in alwold appliaions, spially in h filds of supply hain managmn and indusial ngining. hough as sudy, h main bnfis of using mgny mod and h pfoman of ah alnaiv mod an b summaizd as follows: W an ahiv pofi impovmn of h supply hain sysm by implmning any alnaivs of mgny mods in whih ah alnaiv is suiabl o implmn in dfn ondiions. Howv, h mgny mod will bom lss aaiv whn sandad dviaion of dmand, shoag os and holding os of podus a h ail das, and whn anspoaion os and puhasing os of mgny od inas. Anoh insing obsvaion is ha h manufau modl and pioiy modl a qui snsiiv o σ and. In fa, hs wo mods will nv bn usd und a vy low σ wih high (i.., σ = 5 unis/day wih = $/uni in his sudy). On h oh hand, h ousou modl will nv bn usd whn h is vy high (i.., = $55/uni in his sudy). Sin his is h fis sp of applying h mgny onp o h supply hain sysm and inodu DE algoihm o solv poblm in indusial ngining fild, h a sill lf a lo of poblms fo fuh sudy. Fo xampls, i is insing o xnd a singl wo-hlon supply hain sysm o h supply hain nwok and sudy abou h hybid of DE wih h oh opimizaion appoahs. Aknowldgmns his sah is suppod by oyoaki Sholaship Foundaion. his suppo is aknowldgd gafully. Rfns. Vlahos, D. and agaas, G.: An invnoy sysm wih wo supply mods and apaiy onsains. In. J. Poduion Eonomis 7 () 4-58. Baankin, E.W.: A dlivy-lag invnoy modl wih an mgny povision. Naval Rsah Logisis Qualy 8 (96) 85-3 88 GLOBAL INFORMAION PUBLISHER
Opimizaion of Piodi Rviw Invnoy Modl 3. Bulinkaya, EV.: Som suls onning opimum invnoy poliis. hoy of Pobabiliy Appliaions 9 (964) 389-43 4. Fukuda, Y.: Opimal poliis fo h invnoy poblm wi ngoiabl lad-im. Managmn Sin (964) 69-78 5. Vino, A.F. J.: h saus of mahmaial invnoy hoy. Managmn Sin (966) 745-777 6. Chiang, C. and Guiz, G.J.: A piodi viw invnoy sysm wih wo supply mods. Euopan jounal of Opaional Rsah 94 (966) 57-547 7. Chiang, C.: Opimal plnishmn fo a piodi viw invnoy sysm wih wo supply mods. Euopan Jounal of Opaion Rsah 49 (3) 9-44 8. Bylka, S.: unpik poliis fo piodi viw invnoy modl wih mgny ods. In. J. Poduion Eonomis 93-94 (5) 357-373 9. Son, R. and Pi, K.: Dfnial Evoluion-A simpl and ffiin huisi fo global opimizaion ov oninuous spa. Jounal of Global Opimizaion (997) 34-359.Angia, R. and Babu, B.V.: Opimizaion of poss synhsis and dsign poblms: A modid dfnial voluion appoah. Chmial Engining Sin 6 (6) 477-47.sin, R.J.: Pinipls of invnoy and maial managmn, Pni Hall, Englwood Clfs, Nw Jsy (996).Chiadamong N. and Paswaana, K.: A ompaaiv sudy of supply hain modls und h adiional nalizd and oodinaing poliis wih inniv shms. Compu & Indusial Engining 5 (6) 367-384 3.Paswaana K., Shimizu Y. and Chiadamong, N.: Evoluional opimizaion on maial oding and invnoy onol of supply hain hough inniv shm. Poding of h s Innaional Symposium on Shduling, 8-4 July 6, okyo, Japan, 8-86 GLOBAL INFORMAION PUBLISHER 89