Optimization of Periodic Review Inventory Model with Emergency order in Two-Echelon Supply Chain using Differential Evolution

Transcription

1 Opimizaion of Piodi Rviw Invnoy Modl Innaional Jounal of Compuaional Sin (Pin) (Onlin) Global Infomaion Publish 7, Vol., No., 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 , 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: , kani@s.ps.u.a.jp. GLOBAL INFORMAION PUBLISHER 69

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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 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

13 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 nb(n) = j 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

14 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

15 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 No: h highligh aa psns h bs poliy und ah ondiion. GLOBAL INFORMAION PUBLISHER 83

16 Innaional Jounal of Compuaional Sin % Pofi impovmn Emgny anspoaion os, [$/uni] σ (b) σ = 5 () σ = (a) = 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

17 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 No: h highligh aa psns h bs poliy und ah ondiion. % Pofi impovmn Emgny Puhasing os [$/uni] 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

18 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

19 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 % Pofi impovmn Oppouniy los os a h ail s [$/uni] 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

20 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 () Baankin, E.W.: A dlivy-lag invnoy modl wih an mgny povision. Naval Rsah Logisis Qualy 8 (96) GLOBAL INFORMAION PUBLISHER

21 Opimizaion of Piodi Rviw Invnoy Modl 3. Bulinkaya, EV.: Som suls onning opimum invnoy poliis. hoy of Pobabiliy Appliaions 9 (964) Fukuda, Y.: Opimal poliis fo h invnoy poblm wi ngoiabl lad-im. Managmn Sin (964) Vino, A.F. J.: h saus of mahmaial invnoy hoy. Managmn Sin (966) Chiang, C. and Guiz, G.J.: A piodi viw invnoy sysm wih wo supply mods. Euopan jounal of Opaional Rsah 94 (966) Chiang, C.: Opimal plnishmn fo a piodi viw invnoy sysm wih wo supply mods. Euopan Jounal of Opaion Rsah 49 (3) Bylka, S.: unpik poliis fo piodi viw invnoy modl wih mgny ods. In. J. Poduion Eonomis (5) Son, R. and Pi, K.: Dfnial Evoluion-A simpl and ffiin huisi fo global opimizaion ov oninuous spa. Jounal of Global Opimizaion (997) Angia, R. and Babu, B.V.: Opimizaion of poss synhsis and dsign poblms: A modid dfnial voluion appoah. Chmial Engining Sin 6 (6) 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) 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