Modelling of repairable items for production inventory with random deterioration

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1 IOSR Journal of Mahmaics IOSR-JM -ISSN: 78-78, p-issn: 9-7X. Volum, Issu Vr. IV Jan - Fb., PP -9 Modlling of rpairabl ims for producion invnory wih random drioraion Dr.Ravish Kumar Yadav, RajvKumar, ssocia Profssor,Dparmn of mahmaics,hindu Collg, Moradbad, Dva Mahavidhyal, Bijnoor bsrac: Kping in viw h concrn abou nvironmnal procion, h sudy incorpora h concp of rpairing in a producion invnory modl consising of producion sysm and rpairing sysm ovr infini planning horizon. his sudy prsns a forward producion and rvrs rpairing sysm invnory modl wih a im dpndn random drioraion funcion and incrasing xponnially dmand wih h fini producion ra is proporional o h dmand ra a any insan. h shorags allow and xcss dmand is backloggd. Exprssions for opimal paramr ar obaind.w also obaind Producion and rpairing schduling priod, maximum invnory lvl and oal avrag cos. Using calculus, opimum producion policy is drivd, which minimizs h oal cos incurrd I. Inroducion n invnory sysm h ffc of drioraion plays an imporan rol. Drioraion is drivd as dcay or damag such ha h im canno b usd for is original propos. Foods, pharmacuicals, chmicals, blood, drugs ar a fw xampls of such ims in which sufficin drioraion can ak plac during h sorag priod of h unis and h imporanc of his loss mus b akn ino accoun whn analyzing h sysm. Whn dscribing opimum policis for drioraing ims Ghar and Schradr 9 proposd a consan ra of drioraion and consan ra dmand. In rcn yar, invnory problm for drioraion ims hav bn widly sudid afr Ghar and Schradr 9, Covr and Philip 97 formulad h modl for variabl drioraion ra wih wo paramrs Wibull disurbaion Goswami and Chaudhuri 99, Bos al 99 assumd ihr insananous or fini producion wih diffrn assumpion on h parn of drioraion. Balkhi and Bnkhroo 99 considrd a producion a producion lo siz invnory modl wih arbirary producion and dmand ra dpnds on h im funcion. Bhunia and Maii s 977 modl o formula a producion invnory modl. Chang and Dv 999 invsigad an EOQ modl allow shorag and backlogging. I is assumd ha h backlogging ra is variabl and dpndn on h lngh of waiing im for h nx rplnishmn. Rcnly, many rsarchrs hav modifid invnory policis by considring h im proporional parial backlogging ra such as Wang, Prumal, ng al, Skouri and Papachrisos c. Schrady 97 firs sudid h problm on opimal lo sizs for producion/procurmn and rcovry. For issus in h grning procss, Nahmias and Rivra 979 sudid an EPQ varian of Schrady s modl 97 wih a fini rcovry ra. Richr 99a, 99b, 997 and Richr and Dobos 999 invsigad a was disposal modl by considring h rurnd ra as a dcision variabl. Dobos and Richr, invsigad a producion/rmanufacuring sysm wih consan dmand ha is saisfid by noninsananous producion and rmanufacuring for singl and mulipl rmanufacuring and producion cycl. Dobos and Richr xndd hir prvious modl and assumd ha h qualiy of collcd rurnd ims is no always suiabl for furhr rpairing. Konsanaras and Skouri prsnd a modl by considring a gnral cycl parn in which a variabl numbr of rproducion los of qual siz wr followd by a variabl numbr of manufacuring los of qual siz. hy also sudid a spcial cas whr shorags wr allowd in ach manufacuring and rproducion cycl and similar sufficin condiions, as h non-shorags cas, ar givn.el Saadany and Jabr xndd h modls dvlopd by Dobos and Richr, by assuming ha h collcion ra of rurnd ims is dpndn on h purchasing pric and h accpanc qualiy lvl of hs rurns. ha is, h flow of usd/rurnd ims incrass as h purchasing pric incrass, and dcrass as h corrsponding accpanc qualiy lvl incrass. lamri dvlopd a gnral rvrs logisics invnory modl. Chung and W dvlopd an invnory modl on shor lif-cycl drioraing produc rmanufacuring in a grn supply chain modl In his papr w prsn a ralisic invnory modl in which h producion ra dpnds on h dmand and dmand is an xponnially incrasing funcion im and drioraion is random funcion says ha drioraion of an im dpnds upon h flucuaion of humidiy, mpraur, c. I would b mor rasonabl and ralisic if w assum h drioraion funcion o dpnd upon a paramr "" in addiion o im DOI:.979/ Pag

2 Modlling of rpairabl ims for producion invnory wih random drioraion.his modl is dvlopd for drioraing im by assuming ha h drioraion ra is uniform and h fini producion ra is proporional o h dmand ra & h dmand ra incrasing xponnially. Rpairabl Ims ar collcd a im of producion run and rpairs a im of no producion no shorag complly. hs rpaird ims as good as nw and consumd a im of shorag. Whn shorags is maximum producion sar and ims consumd from boh h channls forward producion and rpaird ims as wll. W driv an xprssions for diffrn cos associad in h modl and oal avrag cos.w driv quaions, soluion of hs quaions givs h opimal cycl and opimal cos of rpairabl ims. Fig.. Flow of invnory in h ingrad supply sysm II. ssumpion and Noaion h mahmaical modl of h producion invnory problm wih rpairabl sysm considrd hrin is dvlopd on h basic of h following assumpions-: a. singl im is considrd ovr a prscribd priod of unis of im, which is subjc o a im dpndn Random drioraion ra. b. Driora D is known and incrasing xponnially D,, is iniial dmand, is a consan govrning h incrasing ra of dmand. c. Producion ra P a any insan dpnds on h dmand ha is, a im, >, P a bd, a >, b and P > D. d. Drioraion of h unis is considrd only afr hy hav bn rcivd ino h invnory.. Ims ar rurnabl and ar rpaird. Rpaird ims ar as good as nw ons and hy ar usd during h shorag priod of forward producion. f. h im horizon of h invnory sysm is infini. Only a ypical planning schdul of lngh is considrd, all rmaining cycls ar idnical. g. Shorags ar allowd and backloggd. h. h producion im inrval for forward producion coincids wih h collcion im inrval for rvrs rpairing sysm. Noaions for producion sysm and rpairing sysm: I = Invnory lvl a any im, h ims drioraion ra is random. I m = Maximum invnory lvl. I b = Maximum shorags lvl. C = Sup cos for nw cycl. C S = Shorag cos pr uni. 7 K = h oal avrag cos of sysm. 8 C HP =Holding cos pr uni pr uni of im during h producion. 9 C DP =Drioraing cos pr uni pr uni of im during h producion. P cp = Producion cos pr im. C HC = Holding cos pr uni pr uni of im during h collcing and consuming procss for h rpairing sysm. C DC = Drioraing cos pr uni pr uni of im during h collcing and consuming procss for h rpairing sysm. C HR = Drioraing cos pr uni pr uni of im during h rpairing procss for h rpairing sysm. C DR =Holding cos pr uni pr uni of im during h rpairing procss for h rpairing sysm. I c = Invnory lvl during h collcing procss for h rurnabl ims. I = Invnory lvl during h rpairing procss for h rurnabl ims. 7 z =Fracion of h producion lo siz <z<. 8 R c =Ra of collcion of rurnabl ims. 9 M =Ra of rpair of rurnabl ims o b rpaird. DOI:.979/ Pag

3 Modlling of rpairabl ims for producion invnory wih random drioraion = im whn producion sops and also h im whn collcing procss for rurnabl ims sops. his vry im rpairing of collcd ims sar. = im priod whn rpairing of rurnabl ims sops and also h im whn accumulad invnory of producion sysm vanishs. = im whn shorags is maximum.= + + = Priod of im whn producion sars again during h priod of shorag. = is h cycl im. I S = Maximum invnory lvl of rpaird ims. P cc = Cos of purchasing h rurnabl ims pr uni. 7 P cr =Rpair cos of rpaird ims pr uni. III. Mahmaical Modl : Iniially, h invnory lvl is sar wih zro. h forward producion invnory lvl sars a im = and i rachs a maximum invnory lvl I m uni afr im. ha im producion is soppd and h invnory lvl is dcrasing coninuously and rachs zro a im, a his im shorags sar dvloping a im i rachs o maximum shorag lvl I b. his im frsh producion sar o rmov backlog by h im. h bginning of ach cycl, h invnory is zro. h producion sars a h vry bginning of h cycl. s producion progrsss h invnory of finishd goods pils up vn afr ming h mark dmand, drioraion. h bginning of ach cycl, h procss of collcing rurnabl ims in a spara sor also bgins. a poin whr h producion from h forward producion sysm sops; h collcion procss of rurnabl ims also sops a h sam poin.i is assumd hr is no collcion of usd ims onc h rpairing of collcd ims sars. his vry poin h rpairing of rusabl ims bgin a a consan ra. h accumulad invnory producd from h advancd producion sysm in h manwhil sars ging consumd and ulimaly bcoms nil. h accumulad invnory of rpairing producs, which ar assumd o b as good as h nwly producd producs, is consumd whn h shorags from h forward producion sysm bgin o surfac. hrafr, producion sars whn shorags is maximum in forward invnory sysm and shorags ar gradually clard afr ming dmand by rpairabl ims and producd im from forward sysm simulanously and h cycl nds wih zro invnoris. Hr our aim is o find ou h opimal valus of,,,, I m & I b ha minimiz h oal avrag cos K ovr h im planning horizon cycl,. Forward producion sysm Rpairing Sysm Fig.. Invnory of producion and rpairing sysm h diffrnial quaion govrning h sock saus during h priod can b wrin as di a b I, I =, I =I m, d DOI:.979/ Pag

4 Modlling of rpairabl ims for producion invnory wih random drioraion di I, I =I m, I =, d di, I =o, I =I b, d,. di a b, I = I b,i =, d. Diffrnial quaions rprsning rpairing sysm in collcing im & consuming im dic Rc Ic, I c = d dic M Ic, I c = Bz, d Whr B=Producion lo siz during producion sysm=producion- Drioraion Pd Pd a b d {a b a b } d b a b { } { } Diffrnial quaions rprsning invnory of rpaird ims. di M I I =, I =I s, d,.7, di I,I =I s,i I d, =I b, 8 Soluion of quaion,,. and by adjusing h consan of ingraion using boundary condiion ar givn by b I a 9, I, I, b I a, Solving and DOI:.979/ Pag

5 Modlling of rpairabl ims for producion invnory wih random drioraion Ic Rc, Ic Rc Bz, Bz RC Ic Bz M M, Soluion of quaion 7 and 8 by adjusing h consan of ingraion using boundary condiion ar givn by I M, I, Ib, h invnory lvl of producion sar iniially a im uni = o = a maximum lvl I m is obaind by quaion 9 I m a b 7 and afr im h producion is soppd and sock lvl is dcrasing coninuously and bcom zro a im is obaind. = a ha im shorags ar dvlop and raching o I b a im b I I a.8 b b hus by quaion 7 w obsrvd ha and ar dpndn so hy ar rlad by h quaion R 9 and by quaion 8 and ar dpndn o ach ohr so rlad by h quaion R oal amoun of driorad unis I DP of producion invnory, is givn by I I d I d DP DOI:.979/ Pag

6 Modlling of rpairabl ims for producion invnory wih random drioraion DOI:.979/ Pag a b 7 b oal amoun of driorad unis DC I of collcd ims of rpairabl invnory channl in, is givn by DC C C c c I I d I d R d Bz M M d Bz R M M oal amoun of driorad unis DR I of Rpaird ims of rpairabl invnory channl is givn by

7 Modlling of rpairabl ims for producion invnory wih random drioraion DOI:.979/ Pag During priod, oal invnory of producd ims HP I in forward producion channl can b obaind as HP I I d I d 7 b a a 7 During priod, oal invnory of collcd ims HC I can b obaind as HC C C c c I I d I d R d Bz M M d R Bz M M DR I I d I d M d d M 8

8 Modlling of rpairabl ims for producion invnory wih random drioraion During priod, oal invnory of rpaird ims I can b obaind as I I d I d HR M d d M oal amoun of shorag unis I s during h priod, is givn by HR I S I d I d P=Producion cos +Collcion cos +Rpaird cos CP CC c CR P P a b d P R d P Md a b 7 b PCP a PCCRc PCRM Hnc h oal avrag cos of h invnory sysm is K = sup cos +producion cos+ drioraion cos + invnory carrying cos + shorag cos C P CDPI DP CDCI DC CDR IDR CHPI HP CHCI HC CHRI HR CSIS 8 and puing h valu of I DP, I DC, I DR, I HP, I HC, I HR and I S w ging h oal avrag cos of h invnory sysm. In many cass and IV. h pproximaion Soluion Procdur ar xrmly small hnc o us Maclaurin sris for approximaion By using quaion.9 h oal avrag cos of sysm b C [ ] DP K C PCP a PCC Rc PCR M a b Bz Rc M C DC M 9 DOI:.979/ Pag

9 Modlling of rpairabl ims for producion invnory wih random drioraion C DR M 8 C HP a b a Rc Bz M C HC + M H HR M C S a b nd b C [ ] DP K C PCP a PCC Rc PCR MR a b R R R BzR R R Rc M C DC R M R R C DR R R M R 8 C HP a b a R R DOI:.979/ Pag

10 Modlling of rpairabl ims for producion invnory wih random drioraion R R R R R R R Rc Bz M C HC + R M R R H HR R R M R C S R a b ccording o quaion conain four variabls,, and and hs ar dpndn variabl and rlad by quaion 9 and. lso w hav K >, hnc h opimum valu of and which minimiz oal avrag cos ar h soluions of h quaions K K and. Providd ha hs valus of saisfy h condiions K K, and K K K. Now diffrniaing wih rspc o and, w g K C b [ ] DP C PCP a PCC Rc PCR MR R a b R R R R BzR R R Rc M C DC R M R R R M C DR R R R R 8 DOI:.979/ Pag

11 Modlling of rpairabl ims for producion invnory wih random drioraion C HP a a b R R R R R R Rc BzR C R R HC R R M R M R R H HR R R M R R C S R a b R [ C ] DP PCP a b PCC Rc PCR MR b a R R R R R BzR R Rc R R R R M C DC R R M R M R M C DR R R R R M DOI:.979/ Pag

12 Modlling of rpairabl ims for producion invnory wih random drioraion C HP a b a R R R R R R R R R R R R R R R Rc BzR M C HC R M R M + R M H HR R R R R M K C b [ ] DP C PCP a PCCRc PCRMR R a b R R R R BzR Rc C DC R R M R R M R R M C DR R R R R 8 DOI:.979/ Pag

13 Modlling of rpairabl ims for producion invnory wih random drioraion C HP a a b R R R R R R R R R R Rc BzR M C HC M R R R M H HR R R R R C S R a b R R R R R C DR R R H R R + HR R R R C S R R a b Hr w obain wo simulanous non-linar quaion in of and can b find ou opimal valu by using som suiabl compuaional numrical mhod and h opimum valu of,, I m, I b and minimum oal avrag cos K can b obaind from quaions. V. Spcial Cass: Cas I : If b = hn h discussd modl convr o producion invnory modl in which producion ra is consan and indpndn on h dmand. Cas II : If hn h discussd modl rducs o producion invnory modl wih ou drioraion Cas III : If, b h modl rduc o uniform producion ra and consan dmand. DOI:.979/ Pag

14 Modlling of rpairabl ims for producion invnory wih random drioraion VI. Conclusion In h proposd modl a producion invnory modl is formulad for random drioraing im wih a incrasing mark dmand ra wih im and producion ra is dpndn on h dmand. Rsul in his sudy can provid a valuabl rfrnc for dcision markrs in planning h producion and conrolling h invnory. h modl proposd hr in is rsolvd by using maclaurin sris and cos minimizaion chniqu is usd o g h approxima xprssion for oal avrag cos and ohr paramrs & som spcial cass of modl ar also discussd. W driv an xprssions for diffrn cos associad in h modl. W driv quaions, soluion of hs quaions givs h opimal cycl and opimal cos of rpairabl ims. fuur sudy will incorpora mor ralisic assumpion in h proposd modl. Rfrncs []. lamri,... hory and mhodology on h global opimal soluion o a Rvrs logisics invnory modl for drioraing ims and im varying ras. Compur & Indusrial Enginring,, -7. []. Balkhi, Z.. and Bnkhrouf, L. 99 bon h opimal rplnishmn schdul for an invnoy sysm wih drioraing ims and im varying dmand and producion ras.compurs & Indusrial Enginring, ; []. Bhunia,. K. and Maii, M. 998 wo warhous invnory modl for drioraing ims wih a linar rnd in dmand and shorags.jour of Opl. Rs. Soc., 9 : []. Chang, H. J. and Dy, C. Y. 999 n EOQ modl for drioraing ims wih im varying dmand and parial backlogging.j o ur, o f O pl. R s. So c., : 7-8. []. Chung, C. J., & W, H. M.. Shor lif-cycl drioraing produc rmanufacuring in a grn supply chain invnory conrol sysm. Inrnaional Journal of Producion Economics, 9, 9-. []. Covr, R.P., and Philip, G.C., 97, n EOQ Modl for Im wih Wibull Disribuion Drioraion, IIE ransacions,,, -, [7]. Dobos, I., & Richr, K., producion/rcycling modl wih saionary dmand and rurn ras.cnral Europan journal of Opraions Rsarch,, -. [8]. Dobos, I., & Richr, K., n xndd producion/rcycling modl wih saionary dmand and rurn ras. Inrnaional Journal of producion Economics, 9, -. [9]. Dobos, I., & Richr, K., producion/rcycling modl wih qualiy considraions. Inrnaional Journal of Producion Economics,, []. El Saadany. M.., & Jabr M. Y.., producion/ rmanufacuring invnory modl wih pric and qualiy dpndan rurn ra. Compurs and Indusrial Enginring, 8,. []. Ghar, P.M., and Schradr, G.F.,9, Modl for n Exponnially Dcaying Invnory, h Journal of Indusrial Enginring,,, 8-. []. Goswami, and Chaudhuri, K. S. 99 EOQ modl for an invnory wih a linar rnd in dmand and fini ra of rplnishmn considring shorags.in. J our. Sys ScL,, []. Konsanaras, I., & Skouri, K., Lo sizing for a singl produc rcovry sysm wih variabl s up numbrs. Europan Journal of Opraions Rsarch,, -. []. Prumal, V. and rivarignan, G. producion invnory modl wih wo ras of producion and backordrs.in. Jo ur of Mg. & Sys., 8 : 9-9. []. Pyk, D. 99 Prioriy Rpair and Dispach Policis for Rparabl-Im Logisics Sysms. Naval Rsarch Logisics v7 n - Fb. []. Raafa, F., Wolf, P.M. and Eldin, H.K. n Invnory Modl for Drioraing Im, Compurs & Indusrial Enginring,,, 89-9, 99. [7]. Richr, K, 99b. h xndd EOQ rpair and was disposal modl. Inrnaional Journal of Producion Economics, -, -7. [8]. Richr, K. 99, h EOQ rpair and was disposal modl wih variabl sup numbrs. Europan Journal of Opraional Rsarch, 9, -. [9]. Richr, K., & Dobos, I nalysis of h EOQ rpair and was disposal modl wih ingr s up numbrs. Inrnaional journal of producion conomics, 9-, -7. []. Schrady, D drminisic invnory modl for rpairabl ims. Naval Rsarch Logisics Quarrly,, []. ng, J.. & Chang, C... Economic producion quaniy modls for drioraing ims wih pric and sock-dpndn dmand. Compuaional Opraions Rsarch,, []. ng, J.. 99 drminisic rplnishmn modl wih linar rnd in dmand. O pns. Rs. L. 9 : -. DOI:.979/ Pag