A Production Inventory Model with Shortages, Fuzzy Preparation Time and Variable Production and Demand *

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1 Amricn Journl of Orions srch,,, 8-9 h://d.doi.org/.46/or.. Pulishd Onlin Jun (h:// A Producion Invnory Modl wih Shorgs, Fuzzy Prrion Tim nd Vril Producion nd Dmnd * Nirml Kumr Mhr, Um Kumr Br, Mnornn Mii Drmn of Mhmics, Pnskur Bnmli ollg, Pnskur, Indi Drmn of Mhmics, Bngl Insiu of Tchnology & Mngmn, Sninikn, Indi Drmn of Alid Mhmics, Vidysgr Univrsiy, Midnor, Indi Emil: {nirml_hridoy, r_um, mmii5}@yhoo.co.in civd Frury 8, ; rvisd Mrch, ; ccd Mrch 6, ABSTAT A roducion invnory modl is formuld for singl im. Hr, dmnd vris wih h on-hnd invnory lvl nd roducion ric. Shorgs r llowd nd fully ckloggd. Th im g wn h dcision nd cul commncmn of roducion is rmd s rrion im nd is ssumd o cris/imrcis in nur. Th s-u cos dnds on rrion im. Th fuzzy rrion im is rducd o cris inrvl rrion im using nrs inrvl roimion nd following h inrvl rihmic, h rducd rolm is convrd o muli-ociv oimizion rolm. Mhmicl nlysis hs n md for singl ociv cris modl (Modl-I). Numricl illusrion hv n md for oh cris (Modl-I) nd fuzzy (Modl-II) modls. Modl-I is solvd y gnrlizd rducd grdin chniqu nd muli-ociv modl (Modl-II) y Glol riri Mhod. Snsiiviy nlyss hv n md for som rmrs of Modl-I. Kywords: Fuzzy Prrion Tim; Inrvl Numr; Muli-Ociv; Glol riri Mhod. Inroducion Afr h dvlomn of EOQ modl y Hrris [] in 95, lo of rsrchrs hv ndd h ov modl wih diffrn ys of dmnds nd rlnishmn. A dild lirur is vill in h ook such s Hdly nd Whiin [], Trsin [], Silvr nd Prson [4], c. In clssicl invnory modls, dmnd is normlly ssumd o consn. Bu, now--dys, wih h invsion of h muli-nionls undr h WTO grmn in h dvloing counris lik Indi, Bngldsh, c., hr is srong comiion mongs h sllrs o cur h mrk i.. o llur h cusomrs hrough diffrn cics. ommon rcics in his rgrd is o hv rciv dcoriv gorgous disly of h ims in h show room o cr sychologicl rssur or o moiv h cusomrs o uy mor. Furhrmor, dmnd of n im dnds on uni roducion cos, i.. i vris invrsly wih h uni roducion cos. This mrking olicy is vry usful for fshionl goods/fruis c. Thr is som lirur on h invnory modls wih sock-dndn dmnd. Svrl uhors lik Mndl nd Phudr [5], Urn [6], Bhuni nd Mii [7], nd ohrs sudid h ov-mniond y of invnory modls. * This work is suord y Univrsiy Grns ommission, Gov. of Indi undr h rsrch grnd No. F.PSW-96/5-6 (EO). Eisnc of d-im (h im g wn lcmn of ordr nd h cul rci of i) is nurl hnomnon in h fild of usinss. So fr, mos of h rsrchrs hv dl wih ihr consn or sochsic ld-im. Elmnry discussion on ld-im nlysis r now--dys vill in h ooks lik Nddor [8], Mgson [9], Foo, Krici nd Ku [] nd ohrs. In rcic, i is difficul o rdic h ld-im dfinily/riclssly nd somims, h s rcords r lso no vill o form roiliy disriuion for h ld-im. Hnc, h only lrniv vill o DM is o dfin h ld-im rmr imrcisly y fuzzy numr. Gnrlly, ld im is ssocid wih EOQ modl i.. insnnous rocurmn or urchs of h lo. Bu, in roducion sysm,h scnrio is diffrn. Hr h im g wn h dcision roducion nd h cul commncmn of roducion mrs known s rrion im for h nlysis of invnory conrol modls. This rrion im mns h im o collc h rw-mrils, o rrng skilld/unskilld lors, o g mchin rdy for roducion, c., nd hnc influncs h s-u cos of h sysm. For h firs im, Mhr nd Mii [,] formuld nd solvd roducion invnory modls for drioring/rkl im wih imrcis rrion im. Though muliociv dcision mking (MODM) rolms hv n formuld nd solvd in mny ohr oyrigh Scis. AJO

2 84 N. K. MAHAPATA ET A. rs lik ir olluion, srucurl nlysis (cf. o []), rnsorion (cf. i nd i [4]), c., ill now fw rs on MODM hv n ulishd in h fild of invnory conrol. Pdmnhn nd Vr [5] formuld n invnory rolm of drioring ims wih wo ocivs imizion of ol vrg cos nd wsg cos in cris nvironmn (I is n nvironmn ll inu d r ssumd o drisic, rcisly dfind nd givn.) nd solvd y non-linr gol rogrmg mhod. oy nd Mii [6] formuld n invnory rolm of drioring ims wih wo ocivs, nmly, mimizing ol vrg rofi nd imizing ol ws cos in fuzzy nvironmn. Mhr, oy nd Mii [7], Mhr, Ds, Bhuni nd Mii [8] formuld muli-ociv muli-im invnory rolms undr som consrins. In his r, roducion invnory conrol sysm for singl im is considrd. Hr dmnd is dndn on uni roducion cos nd currn sock lvl. Shorgs r llowd nd ckloggd fully. Prrion im in roducion of h nw consignmn is llowd nd cris/fuzzy in nur. Th su cos is dndn on rrion im. Th cris rolm for imizing vrg cos is solvd y gnrlizd rducd grdin mhod. Th rolm imizing vrg ol cos wih fuzzy rrion im is convrd o muli-ociv imizion rolm wih h hl of inrvl rihmic nd hn i is solvd y glol criri mhod o g ro-oiml soluion. Mhmicl drivion nd nlysis lso hv n md for oh singl nd muliociv modls. Furhr, h snsiiviy nlysis is includd nd wo numricl illusrion r rovidd.. Inrvl Arihmic Throughou his scion lowr nd ur cs lrs dno rl numrs nd closd inrvls rscivly. Th s of ll osiiv rl numrs is dnod y. An ordr ir of rcks dfins n inrvl A, :, nd r rscivly lf nd righ limis of A. Dfiniion.: *,,.,/ inry orion on h s of osiiv rl numrs. If A nd B r closd inrvls hn A* B * : A, B dfins inry orion on h s of closd inrvls. In h cs of division, i is ssumd h B. Th orions on inrvls usd in his r my licily clculd from h ov dfiniion s A,, B, B, nd < AB,,, k, k, for k ka k, k, for k <, k is rl numr.. Ordr lions wn Inrvls Hr, h ordr rlions which rrsn h dcisionmkr s rfrnc wn inrvl coss r dfind for imizion rolms. h uncrin coss for wo lrnivs rrsnd y inrvls A nd B rscivly. I is ssumd h h cos of ch lrniv is known only o li o h corrsonding inrvl. Th ordr rlion y h lf nd righ limis of inrvl is dfind in Dfiniion-.. Dfiniion.: Th ordr rlion wn A, nd B, is dfind s A B if nd A < B if A B nd Th ordr rlion rrsns h DM s rformnc for h lrniv wih h lowr imum cos, h is, if A B, hn A is rfrrd o B... Formulion of h Muli-Ociv Prolm A gnrl non-linr ociv funcion wih som inrvl vlud rmrs is s follows: Minimiz Z( ) suc o k k n i i i A B i i,,,, n.,,,, n,,, i ci ci Ai i, i nd B. Now, w hii h formulion of h originl rolm () s muli-ociv non-linr rolm. Sinc h ociv funcion Z nd h consrins conin som rmrs rrsnd y inrvls, i is nurl h h soluion s of () should dfind y rfrnc rlions wn inrvls. Now from Equion () h righ nd lf limis z, z nd cnr z of h inrvl ociv funcion Z rscivly my licid s k n k n i i ( ) i, ( ) i i i z c z c z ( ) z( ) z( ) Thus h rolm () is rnsformd ino Minimiz z, z suc o k i i k i i,,, n,,,,, n () oyrigh Scis. AJO

3 N. K. MAHAPATA ET A Bsic Fuzzy Ss nd Fuzzy Numrs Fuzzy s: A Fuzzy s A in univrs of discours X is dfi s h following s of irs A nd, ( ) : X A, : X, A is g clld h mmrshi funcion of h fuzzy s A nd ( ) A is clld h mmrshi vlu or dgr of mmrshi of X in h fuzzy s A. Th lrgr ( ) A is h srongr grd of mmrshi form in A. Fuzzy numr: A fuzzy numr is fuzzy s in h univrs of discours X h is oh conv nd norml. Figur shows fuzzy numr A of h univrs of discours X h is oh conv nd norml. -cu of fuzzy numr: Th -cu of fuzzy numr A is dfind s cris s : ( ), X, A A A is non-my oundd closd inrvl conind in X nd i cn dnod y A A( ), A( ). A ( ) nd A ( ) r h lowr nd ur ounds of h closd inrvl rscivly. Figur shows fuzzy numr A wih -cus A A ( ), A( ), A A ( ), A ( ). I is sn h if hn A A nd A A. Fuzzy numrs r rr snd y wo y of mmrshi funcions: ) inr mmrshi funcions.g. Tringulr fuzzy numr (TFN), Trzoidl fuzzy num- r, Picwis inr fuzzy numr c. ) Non-linr mmrshi funcions.g. Prolic fuzzy numr (PFN), Prolic fl fuzzy numr, Eonnil fuzzy numr nd ohr non-linr fuzzy numr..4. Tringulr Fuzzy Numr (TFN) TFN is h fuzzy numr wih h mmrshi funcion ( ) A, coninuous g : ( ):, A for < for ( ) A for c c for c < Figur shows Tringulrm f uzzy numr A of h univrs of discours X h is oh conv nd norml..5. Th Nrs Inrvl Aroimio n Hr w wn o roim fuzzy numr y cris on. Su os A nd B r wo fuzzy numrs wih -cus r A ( ), A ( ) B ( ), B ( ) r- nd scivly. Thn h disnc wn A nd B is d AB, Figur. Fuzzy numr A wih α-cus. Figur. Tringulr fuzzy numr. A ( ) ( ) d B A( ) B( ) d Givn A is fuzzy numr. W hv o find closd inrvl d A which is nrs o A wih rsc o mric d. W cn do i sinc ch inrvl is lso fuzzy numr wih consn -cu for ll,. Hnc d A,. Now w hv o imiz d A, d A A ( ) ( ) d A( ) ( ) d wih rsc o nd. In ordr o imiz d A, d A, i is sufficin o imiz h funcion D, d A, d A. Th firs ril driv- ivs r nd Solving, D A ( )d D, A ( )d, D, D nd * w g A ( )d nd Agin sinc * A( )d. oyrigh Scis. AJO

4 86 N. K. MAHAPATA ET A. D * * * *, D, H,, D, * * >, > nd * * * * D * * D,.. 4> So D, i.. d A, d A is glol imum. Thrfor h inrvl d A A, ( )d ( )d A is nrs inrvl roimion of fuzzy numr A wih rsc o mric d. A,, fuzzy num r. Th α-lvl inrvl of A is dfind s A A ( ), A ( ). Whn A is TFN hn A ) nd ( ) ( A. By nrs inrvl roimion mhod lowr nd ur limis of h inrvl r rscivly A ( )d nd ( )d A ( )d ( )d Thrfor, inrvl numr considring A s TFN is,.. Assumions nd Noions for h Proosd Modl Th following noions nd ssumions r usd in dvloing h modl. ) Producion sysm involvs only on non-drioring im. ) Shorgs r llowd nd fully ckloggd. ) Tim horizon is infini. 4) q invnory lvl ny im. 5) QmQ s h mimum invnory (shorgs) lvl in n cycl. 6) rrion im for h n roducion,which is fuzzy numr. 7) S-u cos which is of h form,, nd con- r wo sns so chosn o s fi h s-u cos. 8) roducion cos r uni im. 9) D of dmnd which dnds on roducion ric nd sock i.., D(, q) q if q > if q, r osiiv rl consns. ) cycl lngh of cycl. ) holding cos r uni r uni im. ) shorg cos r uni r uni im. ) r-roducion im, i.., im whn roducion for n cycl is dcidd. 4) K roducion r which is of h form K D nd. 4. Mhmicl Formulion Producion invnory sysm involvs only on im. A cycl srs wih shorgs im nd im mim um shorgs lvl is Q s nd h im roducion rocss srs o cklog h shorg quniis fully nd fr im h shorgs rchd o zro, invno ry ccumuls u o im of moun Q m. A h im roducion rocss sod, ccumuld invnory dclins du o dmnd nd rchs o zro im. Th ov roducion invnory sysm is shown in Figur. Th diffrnil quions govrning h sock sus for his modl r givn y D if dq K D if d K D if D if wih h oundry condiions q() q( ) q( ). Ths quions cn rwrin in h following form. if dq ( ) if d ( ) ( q ) if ( q ) if For simliciy w k. Solving firs nd scond quions of diffrnil Equion () in h inr- vls nd rscivly wih h hl q() q, w g of oundry condiions, from firs quion of () q () ( ) ( ), from scond quion of (), from firs quion wih Qs ( ) ( ), from scond quion So im, ( ) () () oyrigh Scis. AJO

5 N. K. MAHAPATA ET A. 87 Tol roducion cos d ( q)d P ( ) Now h vrg ol cos ( AT ) Figur. Invnory gins wih ckloggd shorgs nd nd wih no invnory. Similrly, solving hird nd fourh diffrnil Equions of (8) in h inrvl nd rscivly wih h hl of h oundry condiions ( ) q ( ), w g q { }, from hird quion of () q () ( ), from fourh quion of () ( ) wih Qm ( ) A im, ( ) ( ), using h rlion (4) Tol shorgs cos S S is givn y q ()d q ()d,, (5) (6) Now h ol holding cos ( H ) during h inrvl, nd, is givn y H q ()dq ()d ( ) ( ) (Holding os S u os T, Shorgs os Producion os) T ( ) ( ) ( ) ( ) 4.. ris Modl (Modl-I) Hnc, h roosd modl cn sd s Minimiz AT(, ) ( ) ( ) ( ) ( ) 4.. Fuzzy Modl (Modl-II) As in his cs, rrion im is fuzzy numr which is rlcd y n rori inrvl numr, nd so, in his cs is rlcd y, (for dils s Andi I) nd h cris rolm (7) coms fuzzy oimizion rolm which, using inrvl rihmic, coms muli-ociv nonlinr rogrmg rolm s follows: (7) oyrigh Scis. AJO

6 88 N. K. MAHAPATA ET A. Minimiz AT, AT (8) AT H S P AT H S P nd AT AT AT. 5. Mhmicl Anlysis 5.. Modl-I (ris Modl) Th ociv funcion AT of Modl-I is funcion of nd s, nd r dndn vril (dnd on nd ). To chiv oiml nd, h ril drivivs of h vrg ol cos AT wih rsc o nd r s o zro. Th rsuling quion cn solvd simulnously o oin h oiml nd. nd AT T ( ) ( ) ( ) ( ) AT T T ( ) ( ) T i.. T T (9) () AT cn rovd s sricly conv sinc ll h rincil ors of is Hssin mri r sricly osiiv. AT T, T ( ) ( ) ( ) AT T T T T using (), nd T ( ) > ( ) > AT AT AT T T T () () AT using ()nd () > Hnc, h soluion of Modl-I is glol imum of AT. 5.. Modl-II (Fuzzy Modl) In Modl-II, h ociv funcions AT nd AT (nd hnc AT ) r funcions of nd s, nd r dndn vril (dnd on nd ). As of Modl-I, i cn sn h AT nd AT (nd hnc AT ) r sricly conv. Dfiniion 5.: A muli-ociv oimizion rolm is conv if ll h ociv funcions nd h fsil rgion r conv. * Dfiniion 5.:, is sid o Pro oiml soluion o h MONP iff hr dos no is nohr, in h fsil rgion such h AT, AT, for ll ii (, ) nd i i oyrigh Scis. AJO

7 N. K. MAHAPATA ET A. 89 AT, < AT, for ls on ind (, ). An ociv vcor AT, AT is Pro-oiml if hr dos no is nohr ociv vcor AT, AT such h ATi AT i for ll i, nd AT AT for ls on ind,. Thrfor, AT, AT is Pro-oiml if h dcision vcor corrsonding o i is Pro oiml. Thorm 5. : h muli-ociv oimizion rolm conv. Thn vry loclly Pro-oiml soluion is lso glolly Pro-oimum (cf. Miinn [9])). Thrfor h rolm givn y (7) is sri cly conv muli-ociv imizion rolm nd hnc osssss glol Pro-oiml soluions. Hnc, h rolm (7) is solvd y Glol riri Mhod. 5.. Glol riri Mhod Th Muli-Ociv Non-linr Progrmg (MONP) rolms r solvd y Glol riri Mhod convring i o singl ociv oimizion rolm. Th soluion rocdur is s follows. S-. Solv h muli-ociv rogrmg rolm (6) s singl ociv rolm usin g only on ociv im ignoring. S-. From h ov rsul, formul y-off mri s follows: AT AT m AT AT m S-. From h y-off mri of S-, dr h idl ociv vcor, sy AT, AT m m nd h corrson ding vlu of AT, AT. Hr, h idl ociv vcor is usd s rfrnc oin. Th rolm is hn o solv h following uiliry rolm: AT(, ) AT G Minimiz m AT AT AT(, ) AT m AT AT 6. Numricl Illusrions Th Equion (7) is non-linr in nd nd difficul o oin h oiml vlus of nd y () < nd G is known s glol criri. An usul vlu of is. 6.. ris Modl (Modl-I) lculus mhod. Hnc, w imiz (7) y grdin sd non-linr oimizion mhod. Using h oiml vlus of nd, h corrsonding vlus of,,, Qs, Q m nd imum vrg ol cos (AT) r oind. To illusr h roosd invnory modl (Modl-I), following inu d r considrd..6,.8,,,.7,.5,.5, 5,,, 5 in rori unis. Solving h Modl-I, following oiml vlus r oind: AT 56.5,.669, 6.999, Q 7.877, Q s 6.. Fuzzy Modl (Modl-II) m To illusr h roosd invnory Modl-II, following inu d r considrd..5,.8,.8,,,.7,.5,.5, 5,,, 5 in rori unis. For h ov inu d, formul y-off mri s follows: AT AT Solving h Modl-II using Glol riri Mhod, following oiml vlus r oind: AT 69.4, AT 7., AT 8.6, , 7.947, G Discussion Th Modl-I is singl ociv oimizion rolm nd is solvd using gnrlizd rducd grdin mhod. Th Modl-II is muliociv rolm nd is solvd y Glol riri Mhod. In Modl-II, h vrg cos imizion rolm wih inrvl ociv funcion ws convrd ino h muli-ociv rolm whos ocivs r o imiz h cnr ( AT ) nd righ limi ( AT ) of h inrvl ociv funcion. Ths wo ociv funcions cn considrd s h imizion of h vrg cs nd h wors cs. Th soluion ss of roducion invnory rolms wih inrvl ociv funcions r dfind s h Pro oiml soluions of h corrsonding muli ociv rolm. Thrfor, h soluion s dfind in his r includs h oiml soluions gins oh h wors nd h vrg css. oyrigh Scis. AJO

8 9 N. K. MAHAPATA ET A. Tl. Snsiiviy nlysis of Modl-I. rmr μ % chngs * % of AT Q s Q m No soluion Snsiiviy Anlysis A snsiiviy nlysis of h dcision vrils nd for Modl-I hv n md whn ch of h rmrs,,, ing chngd from 5% o 5%. Th rliv chng of AT, Q s, Q m r lso kn ino ccoun nd rs nd in Tl. From Tl, i is sn h AT chngs ridly wih h rid chng of h rm r, u hs chng r oo slow in cs of rmr nd. Agin, AT incrss nd dcrss wih dcrs nd incrs in rrion im rscivly, u his chng is slow which grs wih h rl hnomnon. Figur 4 shows h AT vrsus grh. From his grh i is rvld h iniilly AT dcrss ridly wih h incrs in nd fr som im i gins o incrs slowly. Hr, i is inrsing o no h ffc of sh rmr of h dmnd fu ncion on AT. 9. Prcicl Imlicions In his rrion im hs n considrd for n invnory modl wih fini roducion r. In rl lif, onc roducion is disconinud, i.., onc h lour forc is dismnld, suly of rw mrils is disrud, mchinris is k in disordr, c., i is ovious h o sr h n roducion, som im is rquird o ring h ov mniond hings in ordr. Th cos rld o ov fcors dnds on h im g wn h dcision o sr h rrion nd cul commnc of roducion. If h g is smll, hn vryhing will hv o rrngd hurridly nd i coss mor. Hnc, su Figur 4. Grh of AT vrsus. cos dcrss wih h incrs of rrion im. Th r rion im for h n roduc ion run is vry im- orn. This dcision influncs svrl coss lik su cos, roducion cos, c. Such rl lif modls r considrd hr king diffrn nvironmns ino considrion. Th fuzzy numrs r dscrid y linr/non-linr y mmrshi funcions. Fuzzy numr dscri ing rrion im is hn roimd o n inrvl numr. Following his, h fuzzy rolm is convrd ino muli-ociv invnory rolm h ociv funcions r rrsnd y righ limi nd cnr of inrvl funcion which r o imizd. Th roosd modl hs rod r of liciliy. Two invnory modls r rsnd wih diffrn y of rrion im. If dcision mkr (DM) knows dfinily oyrigh Scis. AJO

9 N. K. MAHAPATA ET A. 9 h im rquird o rsr h n roducion run, h/ sh my do Modl-I. Similrly, for fuzzy rrion im, DM my cc h Modl-II. DM my slc ny of hs modls ccording o h rviling nvironmn in his/hr roducion cnr.. onclusions Th rsn r rooss soluion rocdur for roducion invnory modl wih roducion cos nd on hnd invnory dndn dmnd r nd rrion im. Hr, shorgs r llowd nd ckloggd fully. ik ld im, im whn h dcision is kn for h rrion of n roducion run i.. rrion im hs n considrd for roducion invnory modl. In rl lif, su cos dcrss wih h incrs of rrion im. This considrion is kn ino ccoun in Modl-I. In Modl-II, rrion im is kn s imrcis vi fuzzy numr. Th fuzzy numr is d s- crid y linr/non-linr y mmrshi funcion. Fuzzy numr dscriing rrion im is hn roimd o n inrvl numr. Following his, h rolm is convrd ino muli-ociv invnory rolm h ociv funcions r rrsnd y cnr nd righ limi of inrvl funcion which r o imizd. To oin h soluion of h muli-ociv invnory rolm, Glol riri Mhod hs n usd. Th roosd dmnd hs rod r of liciliy. Dmnd of commodiy dcrss wih h incrs in roducion cos u incrss wih h incrs of sock of h dislyd commodiy nd vic vrs. Hr, hough h formulion of h modl nd h soluion rocdur r qui gnrl, h modl is siml roducion modl wih dmnd dndn roducion r. Th uni roducion cos which is ssumd hr o consn, in rliy, vris wih h rrion im nd roducd quniy. Morovr, im dndn roducion r, rilly los sls, inflion, c., cn incorord o h modl o mk i mor rlisic. Hr, dmnd is sockdndn. Th rsn nlysis cn rd for h dynmic dmnd lso. Though h rolm hs n rsnd in cris nd fuzzy nvironmn, i cn lso formuld nd solvd in fuzzy-sochsic nvironmn rrsning roducion cos nd invnory coss hrough roiliy disriuion. EFEENES [] F. Hrris, Orions nd os (Fcory Mngmn Sris), A.W. Shw o., hicgo, 95, [] G. Hdly nd T. M. Whiin, Anlysis of Invnory Sysms, Prnic Hll, Englwood liffs, 96. []. J. Trsin, Princils of Invnory nd Mrils Mngmn, Elsvir Norh Hollnd Inc., Nw York, 98. [4] E. A. Silvr nd. Prson, Dcision Sysm for In- vnory Mngmn nd srch, John Wily, Nw York, 985. [5] B. N. Mndl nd S. Phudr, A No on Invnory Modl wih Sock-Dndn onsumion, OP- SEAH, Vol. 6, 989, [6] T.. Urn, Drisic Invnory Modls Incororing Mrking Dcisions, omur nd Indusril Enginring, Vol., No., 99, doi:.6/6-85(9)95-i [7] A. K. Bhuni nd M. Mii, An Invnory Modl for Dcying Ims wih Slling Pric, Frquncy of Advrismn nd inrly Tim Dndn Dmnd wih Shorgs, IAPQ Trnscions, Vol., 997, [8] E. Nddor, Invnory Sysm, John Wily, Nw York, 966. [9] D. Mgson, Sock onrol Whn d-tim n No B onsidrd onsn, Journl of h Orionl srch Sociy, Vol., 979,. 7-. [] B. Foo, N. Krici nd H. Ku, Hurisic Policis for Invnory Ordring Prolms wih ong nd ndom Vrying d Tims, Journl of Orions Mngmn, Vol. 7, No. -4, 988, doi:.6/7-696(8)98-5 [] N. K. Mhr nd M. Mii, Invnory Modl for Brkl Im wih Uncrin Prrion Tim, Tmsui Oford Journl of Mngmn Scincs, Vol., No., 4,. 8-. [] N. K. Mhr nd M. Mii, Producion-Invnory Modl for Drioring Im wih Imrcis Prrion Tim for Producion in Fini Tim Horizon, Asi Pcific Journl of Orions srch, Vol., No., 6, [] S. S. o, Muli Ociv Oimizion in Srucurl Dsign wih Uncrin Prmrs nd Sochsic Proc- ss, AIAA Journl, Vol., 984. [4]. i nd K. K. i, A Fuzzy Aroch o h Mulio- civ Trnsorion Prolm, omurs & Orions srch, Vol. 7, No.,, doi:.6/s5-548(99)7-6 [5] G. Pdmnhn nd P. Vr, EOQ Modls for Prishl Ims undr Sock-Dndn Slling, Euron Journl of Orions srch, Vol. 86, No., 995, doi:.6/77-7(94)-j [6] T. K. oy nd M. Mii, Muli-Ociv Invnory Modls of Drioring Ims wih Som onsrins in Fuzzy Environmn, omurs nd Orions srch, Vol. 5, No., 998, doi:.6/s5-548(98)9-x [7] N. K. Mhr, T. K. oy nd M. Mii, Muli-Ociv Muli-Im Invnory Prolm, Procdings of h Sr on cn Trnds nd Dvlomns in Alid Mhmics, Howrh, Mrch, [8] N. K. Mhr, K. Ds, A. K. Bhuni nd M. Mii, Muliociv Invnory Modl of Drioring Ims Wih m Ty Dmnd Dndn Producion, Su nd Uni oss, Procdings of h Nionl Symosium on cn Advncs of Mhmics nd Is Alicions oyrigh Scis. AJO

10 9 N. K. MAHAPATA ET A. in Scinc nd Sociy, Univrsiy of Klyni, Klyni, - Novmr, [9] K. M. Miinn, Non-inr Muli-Ociv Oimizion, Kluwr s Inrnionl Sris, Boson, 999. Andi I Sinc rrion im is fuzzy numr which is rlcd y n rori inrvl numr,, hrfor from Equions (4) nd (6), rlcing y,, w hv,,,,,,,, i i, i i, i i i Tol shorgs cos S is givn y S S S, S y y y Now h S-u cos is givn y.,, Now h ol holding cos ( H ) during h inrvl, nd, is givn y H H H, H, sy ( ) ( ) Tol Producion cos is givn y P P, P P ( ) Now h rssions of S, H nd P my oind from h rssions of S, H nd P on rlcing h suffics y nd y rscivly. Thrfor, h ol cos coms AT H S P ( H, H ) (, ) ( S, S ) ( P, P ) AT, AT, nd AT H S P AT H S P AT AT AT. oyrigh Scis. AJO