Multi- item production inventory systems with budget constraints

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1 rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I Mul- m produon nvnory sysms wh bud onsrans ZAI.. AKI parmn of Sass & Opraons Rsarh Coll of Sn Kn Saud Unvrsy.O. ox 55Ryadh 5 SAUI ARAIA E-Mal: zbalkh@ksu.du.sa Wb: hp://fauly.ksu.du.sa/588/dfaul.aspx Absra. In hs papr an nfn m horzon mul-m produon nvnory modl undr rsour onsrans s onsdrd. h xsn of rsour onsrans mpls h us of nar rorammn n ordr o drmn h opmal produon ras for ah m.for any produ ah of produon dmand and droraon ras n any yl as wll as all os paramrs ar rad as known and arbrary funons of m. Shora for ah produ s allowd bu s parally baklod. All os omponns ar affd by boh nflaon and m valu of mony. h objv s o fnd h opmal produon and rsarn ms for ah produ n any yl so ha h ovrall oal nvnory os for all produs s mnmzd. A formulaon of h problm s dvlopd and rorous opmzaon hnqus ar usd o show h unqunss and lobal opmaly of h soluon. Kywords. Mul-m produon nar prorammn Invnory onrol Varyn aramrs Opmaly. Inroduon Kpn nvnory n sors has s own varous oss whh may somms b mor han h valu of h ommody bn arrd ou n sors. As xampls nular and boloal wapons blood n blood banks and som knds of snsv mdaons. owvr any nvnory sysm mus answr h follown wo man qusons. ow muh o ordr or o produ for ah nvnory yl?. Whn o ordr or o produ a nw quany?. Answrn hs wo qusons for ran nvnory sysm lads o h so alld Opmal Invnory ols whh Mnmz h oal Invnory Coss of hs sysm. I s xpd ha all sysms n whh onrolln and manan nvnory s an mporan faor an raly bnf from rsarh rsuls so as o mnmz hr rlvan nvnory os opraons. For xampl aordn o Nahmas ook roduon and Opraons Analyss 997 h nvsmn n nvnors n h Und Sas hld n h manufaurn wholsal and ral sors durn h frs quarr of 995 was smad o b $.5 rllon. hrfor hr s a ra nd o prform spal rsarh on nvnory onrol ISSN: manamn for hos an sysms. h manud of nvnory nvsmn mpls ha hr s ra ponal for mprovn h ffny and prforman of nvnors for varous sors n nral and n an manufaurn sysms as s h as n mos frs lass ounrs. In fa many lassal nvnory modls onrn wh snl m. Amon hs ar Rsh al ] who onsdrd a lassal lo sz nvnory modl wh lnarly nrasn dmand. on al 9 onsdrd an nvnory modl n whh h produon ra s unform and fn whr h nrodud hr produon pols for lnarly nrasn dmand.a nw nvnory modl n whh produs drora a a onsan ra and n whh dmand produon ras ar allowd o vary wh m has bn nrodud by alkh and nkhrouf 7]. In hs modl an opmal produon poly ha mnmzs h oal rlvan os s sablshd. Subsqunly alkh 6] 8] 9] ] ] and ] and alkh al ] hav nrodud svral nvnory modls n ah of whh h dmand produon and droraon ras ar arbrary funons of ms and n som of whh shoras ar allowd bu ar omplly baklod. In ah of h las mnond svn paprs losd forms of h oal nvnory os was sablshd a soluon produr was nrodud and h ondons ha uaran h opmaly of h soluon for h ISN:

2 rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I onsdrd nvnory sysm wr nrodud Rnly alkh ] 5] and Alamr alkh ] hav nrodud mor advand nvnory modls wh smlar bu wh mor rlaxd and nral assumpons. houh so many paprs hav dal wh snl m opmal nvnory poly and houh h lraur onrnd wh mul- m ar spars h analyss of mul-m opmal nvnory pols s almos paralll o ha of snl m. h mul-m nvnory lassal nvnory modls undr rsour onsrans ar avalabl n h wll known books of adly and Whn ] and n Nador ]. n-aya and Raouf 5] hav dvlopd an approah for mor rals and nral snl prod for mul-m wh budary and floor or shlf spa onsrans whr h dmand of ms follows a unform probably dsrbuon subj o h rsrons on avalabl spa and bud. haaharya 6] has sudd wo-m nvnory modl for droran ms. nard and Roy ] hav usd dffrn approahs for h drmnaon of opmal nvnory pols basd on h noon of ffn poly and xndd hs noon o mul-m nvnory onrol by dfnn h onp of famly and ara ms. Kar al ] oband som nrsn rsuls abou mul droran ms wh onsran spa and nvsmn. Rosnbla ] has dsussd mul-m nvnory sysm wh budary onsran omparson bwn h aranan and h fxd yl approah whras Rosnbla and Rohblum 5] hav sudd a snl rsour apay whr hs apay was rad as a dson varabl. alkh and Foul ] and ] hav rad h problm of mul-m nvnory onrol bu whou rsour onsrans. For mor dals abou mul-m nvnory sysm h radrs ar advsd o onsul h survy of Yasmn and Ernu 7] and h rfrns hrn. Our man onrn hr s o sablsh opmal nvnory pols for dffrn produs ha omps for svral lmd rsours and whh ar manufaurd by any mul-m produon sysm. hs mans ha w hav drmn h opmal soppn and rsarn produon ms for ah produd m so ha h oal rlvan nvnory os of all ms undr som vn rsour onsrans s mnmum. h papr s oranzd as follows. Frs w xplan h produr of fndn h opmal produon ras for any sysm ha produd svral ms whh omp for rsour onsrans.hn w nrodu our assumpons and noaons bfor buldn h mahmaal modl of h undrlyn problm. h soluon produr of h dvlopd modl s sablshd n son 5 and h opmaly of h oband soluon s provd n son 6. An llusrav xampl by whh w vrfy our horal rsuls s provdd n son7. Fnally w nrodu a onluson n whh w summarz h man rsuls of h papr as wll as our proposals for furhr rsarh. Opmal produon ras undr rsour onsrans A onssn and praal as whh w ar on o nrodu n hs papr s abou mul-m produs sysms. Suh produs ar usually rlad o svral ompon faors suh as lmd rsours lk buds raw marals manpowr manufaurn apay hnoloal and dmand rsrons E.. Also ah of hs produs has s own produon ondons prof and os. Whn hss known faors and ondons and an objv of maxmzn oal n profs w an asly fnd h opmal numbr of uns o b produd from ah of hos produs by a smpl nar roram. For mor dals suppos ha w hav m ms and K dffrn rsour and hnoloal onsrans as follows. s h numbr of avalabl uns from rsour j j=. K a s h numbr of uns onsumd j from rsour j by m = m. b h numbr of uns o b produd from m hr ar dson varabls and r s h un n prof of m. hn our problm n hs sa s o fnd h opmal valus of =.m so ha h oal n prof of all produd ms say Z s maxmum.ha s w nd o solv h follown problm j ISSN: ISN:

3 rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I Maxmz Z= m a j j m r subj o ; j..... K ; m u n raly ah of h paramrs r a vars prhaps prodally wh m whh n urn allows o vary prodally wh m. hrfor h mor nral as s o allow h opmal produon ras whh rsul from solvn h abov o vary arbrarly wh m. Assumpons and noaons Our assumpons and noaons for h modl o b rad hr ar as follows:. m dffrn ms ar produd n an arbrary yl Infn m horzon as and hld n sok. All ms ar subj o droraon whl hy ar ffvly n sok and hr s no rpar or rplamn of drorad ms.. h dmand ra produon ra droraon ra produon os shora os for bakordrd ms shora os for los ms s up os and holdn os ar all known and arbrary funons of m and all os omponns ar affd by h nflaon ra and m valu of mony.. All oss ar affd by nflaon ra and m valu of mony. W shall dno by r h nflaon ra and by r h dsoun ra rprsnn h m valu of mony so ha r = r r s h dsoun ra n of nflaon For = m h paramrs of h modl ar known funons of m and ar dnod as follows: : mand ra for m a m. : roduon ra for m a m. : roraon ra for m a m. I : Invnory lvl for m a m. : roduon os for m a m. j j h : oldn os for m pr un pr un of m a m. b : Shora os for m pr un pr un of m a m for bakordrd ms. l : Shora os for m pr un pr un of m a m for los ms. k : Sup os for m a m. = s h ra of bakordrd ms for m whr s h wan m for m up o h nw produon whn shoras sar o b baklod for hs m. No ha s a drasn funon of whh rfls h fa ha h lss wan m mpls mor bakordrd uns for m For m h proposd nvnory sysm opras as follows. h yl sars a m = and h nvnory aumulas a a ra - - I up o m = whr h produon sops. Afr ha h nvnory lvl sars o dras du o dmand and droraon a a ra - - I up o m = whr shoras sar o aumula a a ra up o m =.roduon rsars aan a m = and nds a m = and h nvnory aumulas aan wh a ra - o rovr boh h prvous shoras n h prod ] and o sasfy h dmand n h prod ]. h pross s rpad. In hs rsp and n ordr o rovr h bakordrd ms whn h prod ] and o sasfs h dmand n h prod ] w rqur ha ] ] h varaon of h undrlyn nvnory sysm for m for on yl s shown n F. blow. Modl buldn h hans of h nvnory lvl I for m s ovrnd by h follown dffrnal quaon di I ; d ISSN: ISN:

4 rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I di I ; d di ; d di ; d Wh h boundary ondons: I = I = I = I = rspvly. h soluons of h abov dffrnal quaons undr hr rlav boundary ondons ar u I { u u} du 5 u I u du ; 6 I u du ; 7 I { u u} du 8 Rspvly whr u du wh =. Nx w drv h prsn worh of ah yp of os for m. rsn worh of shora os of m for bakordrd ms WSC. Shoras our ovr wo prods ] and ]. whh w dno by rspvly. Now WSC WSC and WSC b r I d r b u u du d. Inran by pars w : WSC ] d 9 Smlarly WSC r b { u b Inran by pars w : WSC Whr r I d u} du d ]{ d ru b u du wh rsn worh of sora os of m for los ms WSC. In a small m prod d w los a fraon ] d hn: r WSC = l ] d Usn smlar hnqus w h follown: rsn worh of holdn os for m WC. Ims ar hld n sok n h wo prods ] and ].Usn nraon by pars as abov w oban WC } Whr d ]{ ] d ru u = h u du wh = o rsn worh of m produon os for m WC. Sn produon ours durn h wo prods ] and ] so w hav: WC = { r d r d} 5 No ha h las os nluds boh onsumd and drorad ms. ISSN: ISN:

5 rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I rsn worh of h s-up os for m WSUC.h s-up of nw produon ours w durn any yl. h frs s a = and h sond s a =.. hrfor h prsn worh of h s-up os r. r WSUC = k k r = k k 6 n h oal rlvan os pr un m for m as a funon of whh w shall dno by w s vn by w { WC WC WSC WSC w l WSC WC WSUC } { k k ] ] r r ]{ } d r d ] d ]{ } d d r d] 7 or... m m m m m hn h oal rlvan os for all ms say W s vn by W m w 8 d Our problm s o fnd h opmal valus of for =.m whh mnmz W vn by 8 subj o h follown onsran: =..m 9 d { } d d ] d and h las wo onsrans ar rspvly quvaln o d ] d C : : d d C = m. No ha onsrans 9 ar naural onsrans sn ohrws our problm would hav no mann. Consran oms from h fa ha for ah h nvnory lvls vn by 7 & 8 mus b qual a whras onsran oms from h fa ha h nvnory lvls vn by 5 & 6 mus b qual a hus our problm all s : fnd h opmal valus of =.m whh Mnmz W subj o 9 &. 5 Soluon produrs. o solv problm w frs nor 9. hs an b jusfd by h rasons ha f 9 do no hold hn h whol problm would hav no mann. owvr w shall no onsdr any soluons ha do no sasfy 9. hus our nw problm s: Mnmz W subj o & No ha s an opmzaon problm wh wo qualy onsrans so an b solvd by h aran hnqus. Now l m m C... m C... m b our aranan hn = + C C W h nssary ondons for havn opma ar: ISSN: ISN:

6 ... ; m d d d d d d d d d d d d Now from 7 & and for = m w hav: ' ] ' r d d d d. Ralln w hav: r r 5 d d ] ] 6 ] r l Sn h abov quaon an b smplfd o: ] ] 7 ] r l d dc d dc d dw d d Ralln and non ha ; ; h las quaon s quvaln o d l d r ] ' r r r d rk k ] 8 o fala fndn d d l V W hn from7&w hav d dc d dc V d dv d d hs vs: ] } ] ] { r V Whh lads o 9 ] ] ] r V W No ha 9 vs h mnmum oal os n rms of & Equaons 5789 ar 6m quaons wh 6m varabls Namly =-.m so ha h soluon of hs quaons f xss vs h ral pons of from whh s h orrspondn ral pon of W. 6 Opmaly and unqunss of h soluon In hs son w drv ondons ha uaran unqunss and lobal opmaly of any xsn soluon o problm. For ha purpos w frs sablsh suffn ondons undr whh h ssan marx of h rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I ISN: ISSN:

7 rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I aranan funon alulad a any ral pon of s posv smdfn. o ompu h ssan marx of w onsdr h follown noaons l l jl l ljl j= m and l= hn h rlad ompuaons showd ha for = m has h follown form y alkh and bkhrouf 7] Swar 6] and Em 7] hs symmr marx s posv sm-dfn f hus h abov arumns lad o h follown horm. horm. Any xsn soluon of s a mnmzn soluon o f hs soluon sasfs hrouh. Nx w shall show ha any mnmzn soluon of s unqu. o s hs w no from ha for a vn ah of an mplly b drmnd as a funon of.. f f f f Our arumn n shown h unqunss of any xsn soluon of s basd on h da ha h nral valu of W vn by 7 mus ond wh h mnmum valu of W vn by 9.ha s w mus hav V f f f / f - f f f f f f W = r V / f s akn from 9 and W f f f s akn from 7. No ha any mnmzn soluon of f xss s unqu hn lobal mnmum f quaon as an quaon of has a unqu soluon. hs fa has bn shown by alkh5] 6] 8] and 9].n h abov arumns lad o h follown horm horm. Any xsn soluon of for whh hrouh hold s h unqu and lobal opmal soluon o. Nx w shall vrfy our modl by h follown llusrav xampl 7 Illusrav Exampl and s Vrfaon W hav vrfd our modl by h follown llusrav xampl p a a p k k k l l l k h h h b b b In ordr o vrfy h horal rsuls of h nrodud modl fv dffrn ms wh dffrn paramr valus hav bn hosn o vrfy hs xampl. h numral rsuls ar shown by h abl- blow. From h abov numral rsuls on an asly ddu ha produon ra and h os paramrs hav h major nflun on h oal os W whras h nflun of nflaon and droraon ra on W s mnor. 7 Conluson In hs papr w hav onsdrd a nral mulm produon lo sz nvnory modl undr ISSN: ISN:

8 rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I lmd rsours. h xsn of rsour onsrans and som ohr hnoloal ondons mply h us of lnar prorammn hnqus n ordr o drmn h opmal produon ras for all ms bn produd so ha h oal n prof s maxmum. u n raly h paramrs vary from m o m. hs mpls ha h rsuln produon ras do vary wh m. hrfor h mor nral as whr h produon ras ar known and nral funons of m s rad Also w onsdrd h as whr ah of h dmand and droraon as wll as all os paramrs ar known and nral funons of m. Shoras ar allowd bu ar parally bakordrd. oh nflaon and m valu of mony ar norporad n all os omponns. h objv s o mnmz h ovrall oal rlvan nvnory os. W hav bul an xa mahmaal modl and nrodud soluon produrs by whh w ould drmn h opmal soppn and rsarn produon ms for any m n any yl. hn qu smpl and fasbl suffn ondons ha uaran h unqunss and lobal opmaly of h oband soluon ar sablshd. An llusrav xampl whh xplans h applably of h horal rsuls ar also nrodud and numrally vrfd. hs sms o b h frs m whr suh a nral mul m EQ s mahmaally rad and numrally vrfd. Aknowldmn. h auhor would lk o xprss hr hanks o h Rsarh Cnr n h Coll of Sn n Kn Saud Unvrsy for s fnanal suppor of hs rsarh undr proj Numbr Rfrns ]A.A. Alamr Z.. alkh. h ffs of larnn and forn on h opmal produon lo sz for droran ms wh m varyn dmand and droraon ras. Inrnaonal Journal of roduon Eonoms Vol. 7 pp ] Z. alkh On h opmaly of a nral produon lo sz nvnory modl wh varabl paramrs. rodns of h rd Inrnaonal Confrn on Compuaonal InllnCI ] Z. alkh and A. Foul. Improvn nvnory onrol for SAIC produs.rn Advans n Appld Mahmas and n Informaon Sns Vol.pp WSEAS rodns of 5 h Amran Confrn on Appld Mahmas ] Z. alkh and A. Foul. A mul-m produon lo sz nvnory modl wh yl dpndn paramrs. Inrnaonal Journal of Mahmaal Modls and Mhods n Appld Sns. Vol. pp ]Z. alkh. On h Opmaly of a Varabl aramrs Invnory Modl for roran Ims Undr rad Crd oly rodns of h h WSEAS Inrnaonal Confrn on Appld Mahmas pp ] Z. alkh. On h lobal opmal soluon o an nran nvnory sysm wh nral m varyn dmand produon and droraon ras. Europan Journal of Opraonal Rsarh Vol. pp ] Z. alkh Z. and. nkhrouf. A produon lo sz nvnory modl for droran ms and arbrary produon and dmand ras Europan Journal of Opraonal Rsarh Vol.9 pp ] Z. alkh. On h lobal opmaly of a nral drmns produon lo sz nvnory modl for droran ms lan Journal of Opraonal Rsarh Sass and Compur Sn Vol. 8 pp ] Z. alkh. h ffs of larnn on h opmal produon lo sz for droran and parally bakordrd ms wh m varyn dmand and droraon ras. Journal of Appld Mahmaal Modln Vol.7 pp ] Z. alkh Vwpon on h opmal produon soppn and rsarn ms for an EOQ wh droran ms. Journal of opraonal Rsarh Soy Vol.5 pp ] Z. alkh On a fn horzon produon lo sz nvnory modl for droran ms: An opmal soluon Europan Journal of Opraonal Rsarh Vol. pp. -. ] Z. alkh C. Goyal and. Gr. Vwpon Som nos on h opmal produon soppn and ISSN: ISN:

9 rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I rsarn ms for an EOQ modl wh droran ms Journal of Opraonal Rsarh Soy Vol.5 pp. -. ] Z. alkh. An opmal soluon of a nral lo sz nvnory modl wh drorad and mprf produs akn no aoun nflaon and m valu of mony Inrnaonal Journal of Sysm Sn Vol.5 pp ]Z. alkh. adj. A nralzd onom ordr quany modl wh droran ms and m varyn dmand droraon and oss. Inrnaonal ransaons n Opraonal RsarhVol.5pp ] M. n-aya and A. Raouf On h onsrand mul-m snl-prod nvnory problm Inrnaonal journal of roduon Manamn Vol. pp ].K. haaharya 5 roduon manufaurn and loss on mul-m nvnory Europan Journal of Opraonal Rsarh Vol.6 pp ] S. Em A mxd nr approah for opmzn produon plannn rodns of h h WSEAS Inrnaonal Confrn on Appld Mahmas Mah 8 pp ] G. adly J.M. Whn Analyss of nvnory sysms. rn all Company 96. 9] J.. on R.R. Sandropory and J.C. ayya On produon pols for lnarly nrasn dmand and fn produon ra Compuaonal and Indusral Ennrn Vol. pp ] M. Rsh M. Frdman and.c. arbosa On a nral soluon of a drmns lo sz problm wh m proporonal dmand Opraons Rsarh Vol. pp ] M.J. Rosnbla Mul-m nvnory sysm wh budary onsran: A omparson bwn h laranan and h fxd yl approah Inrnaonal Journal of roduon Rsarh Vol ] M.J. Rosnbla and U. G. Rohblum On h snl rsour apay problm for mul-m nvnory sysms Opraons RsarhVol. 8 pp ] G. W. Swar Inroduon o marx ompuaons. Aadm rss 97 7] A. Yasmn and Ernu S. S. Mul-m nvnory modls wh oordnad rplnshmns : A survy Inrnaonal Journal of Opraons and roduon Manamn Vol. 8 pp ISSN: ISN:

10 rodns of h s Inrnaonal Confrn on Manufaurn Ennrn Qualy and roduon Sysms Volum I Invnory vl I Of m =..m m F.h havor of mul-m produon lo sz nvnory modl aramrs of m a a p p abl- h h k k l l b b r Opmal rsuls for Im λ λ W oal Cos W =7.5 ISSN: ISN: