An EOQ Model for Weibull Deteriorating Items with Linear Demand and Partial Backlogging in Fuzzy Environment
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1 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg ville Online Inernionl Journl of ompuer Siene nd Moile ompuing Monly Journl of ompuer Siene nd Informion enology IJSM Vol. 4 Issue. eemer 5 pg.4 54 ISSN 88X n EOQ Model for Weiull eerioring Iems wi Liner emnd nd Pril Bklogging in Fuzzy Environmen J.Suj P.Prvi ss. Professor ep. of Memis Quid-E-Mill Gov. ollege for Women enni Indi Hed & sso.professor ep. of Memis Quid-E-Mill Gov. ollege for Womenenni Indi sr --- In is pper we developed n EOQ model for Weiull deerioring iems wi liner demnd re in fuzzy environmen. Sorges re llowed nd prilly klogged. Holding os deeriorion os ordering os sorge os nd opporuniy os re ssumed s ringulr fuzzy numers. e purpose of is pper is o minimize e ol os funion in fuzzy environmen. Grded men represenion signed disne nd enroid meods re used o defuzzify e ol os funion nd e resuls oined y ese meods re ompred wi e elp of numeril exmple. Sensiiviy nlysis is lso rried ou o dee e mos sensiive prmeers of e sysem. Keywords--- EOQ model wo prmeer Weiull deerioring iems Sorges ringulr fuzzy numer Grded men represenion meod Signed disne meod enroid meod. I. INROUION In e rdiionl invenory models one of e ssumpions ws e iems preserved eir pysil rerisis wile ey were kep sored in e invenory. is ssumpion is evidenly rue for mos iems u no for ll. However e deerioring iems re suje o oninuous loss in eir msses or uiliy rougou eir lifeime due o dey dmge spoilge nd penly of oer resons. Owing o is f onrolling nd minining e invenory of deerioring iems eomes llenging prolem for deision mkers. Hrris 95 [] developed e firs invenory model Eonomi Order Quniy wi ws generlized y Wilson 94 [] wo gve formul o oin eonomi order quniy. Wiin 957 [] onsidered e deeriorion of e fsion goods e end of e presried sorge period. Gre nd Srder 96 [4] developed model for n exponenilly deying invenory. ve nd Pel 98 [5] were e firs o sudy deerioring invenory wi liner inresing demnd wen sorges re no llowed. Some of e reen work in is field s een done y ung nd ing 99 [6]; Wee 995 [7] sudied n invenory model wi deerioring iems. ng nd ye 999 [8] developed n invenory model wi imevrying demnd nd pril klogging. Skouri e l. 9 [9] developed n invenory model wi rmp-ype demnd re pril klogging nd Weiull's deeriorion re. Misr V.K. nd Sing L.S. [] developed n deerioring invenory model wi ime dependen demnd nd pril klogging. over nd Pilip [] exended Gre nd Srder s onsn deeriorion re o wo-prmeer Weiull disriuion. In. K. Jggi S. Preek. Srm nd Nidi [] presened fuzzy invenory model for deerioring iems wi ime-vrying demnd nd sorges..k. ripy nd U. Misr [] [] ey developed n invenory model for Weiull deerioring iems wi prie dependen demnd nd ime vrying olding os. Buni nd Mii 994 [4] developed n wo wreouse invenory model for liner rend in demnd 5 IJSM ll Rigs Reserved 4
2 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg Buni nd Mii 998 [5] developed n wo wreouse invenory model for deerioring iems wi liner demnd nd sorges. Goswmi nd nduri 99 [6] developed n EOQ model for deerioring iems wi liner rend in demnd. In onvenionl invenory models unerinies re reed s rndomness nd re eing ndled y pplying e proiliy eory. However in erin siuions unerinies re due o fuzziness nd su ses re diled in e fuzzy se eory wi ws demonsred y Zde in [7]. Kuffmnn nd Gup [8] provided n inroduion o fuzzy rimei operion nd Zimmermnn [9] disussed e onep of e fuzzy se eory nd is ppliions. onsidering e fuzzy se eory in invenory modelling renders n ueniiy o e model formuled sine fuzziness is e loses possile ppro o reliy. s reliy is impreise nd n only e pproximed o erin exen sme wy fuzzy eory elps one o inorpore unerinies in e formulion of e model us ringing i loser o reliy. Prk [] pplied e fuzzy se oneps o EOQ formul y represening e invenory rrying os wi fuzzy numer nd solved e eonomi order quniy model using fuzzy numer operions sed on e exension priniple. Vujosevi e l. [] used rpezoidl fuzzy numer o fuzzify e order os in e ol os of e invenory model wiou korder nd go fuzzy ol os. Yo nd Lee [] inrodued korder invenory model wi fuzzy order quniy s ringulr nd rpezoidl fuzzy numers nd sorge os s risp prmeer. ng [] disussed e fuzzy produion invenory model for fuzzify e produ quniy s ringulr fuzzy numer. Yo nd ing [4] onsidered e ol os of invenory wiou korder. ey fuzzified e ol demnd nd os of soring one uni per dy ino ringulr fuzzy numers nd defuzzify y e enroid nd e signed disne meods. Gni nd Meswri[5] developed n EOQ model wi imperfe quliy iems wi sorges were defeive re demnd olding os ordering os nd sorge os re ken s ringulr fuzzy numers. Grded men inegrion meod is used for defuzzifiion of e ol profi. Uykumr nd Vllil [6] developed n eonomi produion model for Weiull deerioring iems over n infinie orizon under fuzzy environmen nd onsidered some os omponen s ringulr fuzzy numers nd using e signed disne meod o defuzzify e os funion. In is pper n invenory model for Weiull deerioring iems nd liner demnd wi sorges is onsidered were ordering os olding os deeriorion re sorge os nd opporuniy os re ssumed s ringulr fuzzy numers. For defuzzifiion of e ol os funion Grded Men Represenion Signed disne nd enroid meods re used. By ompring e resuls oined y ese meods we ge e eer one s n esime of e ol os in e fuzzy sense. II. FUZZY PRELIMINRIES In order o re fuzzy invenory model y using grded men represenion signed disne nd enroid o defuzzify we need e following definiions. efiniion. fuzzy se on x x x R is lled fuzzy poin if is memersip funion is were e poin is lled e suppor of fuzzy se efiniion. fuzzy se inervl if is memersip funion is were x x oerwise nd < defined on R is lled level of fuzzy IJSM ll Rigs Reserved 4
3 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg IJSM ll Rigs Reserved 44 efiniion. fuzzy numer were < < nd defined on R is lled ringulr fuzzy numer if is memersip funion is oerwise x x x x Wen we ve fuzzy poin e fmily of ll ringulr fuzzy numers on R is denoed s R F N. e -u of F N is R L Were L nd R re e lef nd rig endpoins of. efiniion.4 If is ringulr fuzzy numer en e grded men inegrion represenion of is defined s w w d d R L P Wi w nd w o 6 4 d d P efiniion.5 If is ringulr fuzzy en e signed disne of is defined s R L d d efiniion.6 e enroid meod on e ringulr fuzzy numer is defined s Figure. -u of ringulr fuzzy numer
4 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg III. NOIONS N SSUMPIONS e proposed invenory model ving following noions nd ssumpions:. Noions I : e invenory level ime. W : e mximum invenory level for e ordering yle. IB : e mximum moun of demnd klogged for e ordering yle. Q : e eonomi order quniy for e ordering yle. : leng of e ordering yle. : deeriorion os $/per uni. : sorge os $/per uni /per uni ime. : opporuniy os$/ per uni /per uni ime. : olding os $/per uni/ per uni ime. : ordering os of invenory $/ per order : fuzzy deeriorion os $/per uni. : fuzzy sorge os $/per uni/ per uni ime. : fuzzy opporuniy os$/ per uni /per uni ime. : fuzzy olding os $/per uni/ per uni ime. : fuzzy ordering os of invenory $/ per order. : ol invenory os per uni ime. : ol fuzzy invenory os per uni ime. : defuzzify vlue of : defuzzify vlue of d : defuzzify vlue of. ssumpions: y pplying Grded men inegrion meod y pplying Signed disne meod y pplying enroid meod i e invenory sysem involves only one iem nd e plnning orizon is infinie. ii Replenismen ours insnneously n infinie re. iii e demnd re weni weni were > > nd is iniil demnd. iv e deeriorion of ime s follows y Weiull prmeers wo disriuion were is e sle prmeer nd prmeer. is e spe v uring e sorge period e klogging re is vrile nd is dependen on e leng of e wiing ime for e nex replenismen. e longer e wiing ime is e smller e klogging re would e. Hene e proporion of usomers wo would like o ep klogging ime is deresing wi e wiing ime wiing for e nex replenismen. o ke re of is siuion we ve defined e klogging re o e wen invenory is negive. e klogging prmeer is posiive onsn. 5 IJSM ll Rigs Reserved 45
5 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg risp Model: IV. MHEMIL MOEL Figure : Grpil represenion of e invenory sysem We onsider e deerioring invenory model wi liner demnd. Replenismen ours ime = wen e invenory level ins is mximum W. From = o e invenory level redues due o demnd nd deeriorion. ime e invenory level ieves zero en sorge is llowed o our during e ime inervl is prilly klogged. nd ll of e demnd during sorge period s e invenory level redues due o demnd re s well s deeriorion during e invenory inervl e differenil equion represening e invenory sus is governed y di I d ; Were nd di I d ; uring e sorge inervl [ ] e demnd ime is prilly klogged e frion. erefore e differenil equion governing e moun of demnd klogged is di d ; wi e oundry ondiion I nd I W were 8 is given y. e soluion of equion 7 nd ; I IJSM ll Rigs Reserved 46
6 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg IJSM ll Rigs Reserved 47 log log I ; Mximum moun of demnd klogged per yle is oined y puing in = in Equion. erefore log I IB Mximum invenory level for e yle is oined y puing e oundry ondiion W I in equion 9. erefore W Hene e eonomi order quniy per yle is log IB W Q e ol os per yle onsiss of following os omponens i Invenory olding os over e yle is given y H= d I ii eeriorion os over e yle is given y d W iii Sorge os over e yle is given y d I S d log log log
7 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg IJSM ll Rigs Reserved 48 iv e opporuniy os due o los sles per yle is d O log v Ordering os per yle O = ol os of e sysem per uni ime is given y O S H log log Fuzzy model: ue o unerinly in e environmen i is no esy o define ll e prmeers preisely ordingly we ssume some of ese prmeers nmely my nge wiin some limis. Le ; ; ; ; re s ringulr fuzzy numers. ol os of e sysem per uni ime in fuzzy sense is given y log log We defuzzify e fuzzy ol os y Grded men represenion Signed disne nd enroid meods.
8 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg IJSM ll Rigs Reserved 49 i By Grded Men Represenion Meod ol os is given y e neessry ondiion for o e minimize is nd. Solving ese equions we find e opimum vlues of nd sy * nd * for wi os is minimum nd e suffiien ondiion is log log log log log log
9 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg IJSM ll Rigs Reserved 5 e opiml soluion of e equions 4 n e oined y using pproprie sofwre. is s een illusred y e following numeril exmple. ii By signed disne Meod ol os is given y 4 log log e neessry ondiion for o e minimize is nd. Solving ese equions we find e opimum vlues of nd sy * nd * for wi os is minimum nd e suffiien ondiion is log log log log
10 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg IJSM ll Rigs Reserved 5 e opiml soluion of e equions 8 n e oined y using pproprie sofwre. is s een illusred y e following numeril exmple. iii By enroid Meod ol os is given y d d d d e neessry ondiion for d o e minimize is d nd d. Solving ese equions we find e opimum vlues of nd sy * nd * for wi os is minimum nd e suffiien ondiion is d d d d log log d log log d log log
11 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg d d e opiml soluion of e equions n e oined y using pproprie sofwre. is s een illusred y e following numeril exmple. V. NUMERIL EXMPLE Exmple: risp Model: Le us onsider n invenory sysem wi e following d: = = =5 = = e soluion of risp model : Q 9. 9 Fuzzy Model: Le us ke re ll ringulr fuzzy numers. e soluion of fuzzy model n e deermined y following ree meods. By Grded men Inegrion represenion meod we ve Q 9.74 By enroid meod we ve d Q 9.85 By Signed disne meod we ve Q SENSIIVIY NLYSIS o sudy e effes of nges in e sysem prmeers e sensiiviy is nlyzed. e resuls re sown in elow les le. Sensiiviy nlysis on Prmeer efuzzify Fuzzify vlue of Vlue of prmeer IJSM ll Rigs Reserved 5
12 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg le. Sensiiviy nlysis on Prmeer efuzzify Fuzzify vlue of Vlue of prmeer le. Sensiiviy nlysis on Prmeer efuzzify Fuzzify vlue of Vlue of prmeer le.4 Sensiiviy nlysis on Prmeer efuzzify Fuzzify vlue of Vlue of prmeer le.5 Sensiiviy nlysis on Prmeer efuzzify Fuzzify vlue of Vlue of prmeer Oservions: From le s we inrese e prmeer e opimum vlues of nd derese. By is effe e ol os inreses. From le s we derese e prmeer e opimum vlues of derese nd inrese. By is effe e ol os dereses. From le s we derese e prmeer e opimum vlues of derese nd inrese. By is effe e ol os dereses. 4 From le 4 s we inrese e prmeer e opimum vlues of effe e ol os inreses. 5 From le 5 s we inrese e prmeer e opimum vlues of effe e ol os inreses. nd derese. By is nd derese. By is 5 IJSM ll Rigs Reserved 5
13 J.Suj e l Inernionl Journl of ompuer Siene nd Moile ompuing Vol.4 Issue. eemer- 5 pg VI. ONLUSION is pper proposed n EOQ model for Weiull deerioring iems wi liner demnd re in fuzzy environmen. Sorges re llowed nd prilly klogged. e deeriorion os ordering os olding os sorge os opporuniy os re represened y ringulr fuzzy numers. For defuzzifiion grded men signed disne nd enroid meod re employed o evlue e opiml ime period of posiive sok nd ol yle leng wi minimizes e ol os. By given numeril exmple i s een esed grded men represenion meod gives minimum os s ompred o signed disne meod nd enroid meod. sensiiviy nlysis is lso ondued on e prmeers o explore e effes of fuzziness. e proposed model n e exended for sok dependen demnd nd prie dependen demnd in fuzzy environmen. REFERENES []. F. Hrris. Operions nd os W Sw o. igo 95. []. R.H.Wilson 94 sienifi rouine for sok onrol. Hrv Bus Rev :6 8 doi:.86/5-7x-9-4. []..M. Wiin 957 e eory of invenory mngemen nd ediion. Prineon Universiy Press Prineon [4]. P.M. Gre nd G.F. Srder 96 model for n exponenilly deying invenory. J Ind Engineering 4:8 4 [5]. U.ve nd L.K. Pel 98 Si poliy invenory model for deerioring iems wi ime proporionl demnd. J Oper Res So :7 4 [6]. K.J.ung nd P.S. ing 99 eurisi for replenismen for deerioring iems wi liner rend in demnd. J Oper Res So 44:5 4 [7]. H.M.Wee 995 deerminisi lo-size invenory model for deerioring iems wi sorges nd delining mrke. ompu Oper :45 56 [8]. H.J. ng nd.y. ye 999 n EOQ model for deerioring iems wi ime vrying demnd nd pril klogging. J Oper Res So 5:76 8 [9]. K. Skouri I. Konsnrs S. Pprisos nd I. Gns 9 Invenory models wi rmp ype demnd re pril klogging nd Weiull deeriorion re. Eur J Oper Res 9:79 9 []. V.K. Misr nd L.S. Sing eerioring invenory model wi ime dependen demnd nd pril klogging. ppl M Si 47:6 69 []. R. B. over nd G. S. Pilip n EOQ Model wi Weiull isriuion eeriorion IIE rnsions Vol. 5 No pp. -6. doi:.8/ [].. K. Jggi S. Preek. Srm nd Nidi Fuzzy invenory model for deerioring iems wi ime-vrying demnd nd sorges merin Journl of Operionl Reser vol. 6 pp.8-9. [].. K. ripy nd U. Misr n Invenory Model for Weiull eerioring Iems wi Prie ependen emnd nd imevrying Holding os pplied Memil Sienes 4 pp: 7 79 [4]..K. Buni nd M. Mii 994 wo wreouses invenory model for liner rend in demnd opser 8-9. [5]..K. Buni nd M. Mii 998 wo wreouses invenory model for deerioring iems wi liner demnd nd sorges Journl of e operionl Reser Soiey [6].. Goswmi nd K.S. uduri 99 n EOQ model for deerioring iems wi liner rend in demnd Journl of e operionl Reser Soiey 4: 5-. [7]. L.. Zde Fuzzy ses Informion onrol vol pp.8-5. [8]. rnold Kufmnn Mdn M Gup Inroduion o Fuzzy rimei: eory nd [9]. H. J. Zimmermn Using fuzzy ses in operionl reser Europen Journl of Operionl Reser vol. 98 pp.-6. []. K Prk Fuzzy-se eorei inerpreion of eonomi order quniy IEEE rnsions on Sysems Mn nd yerneis SM-7 pp []. Mirko Vujosevi oril Perovi Rdivoj Perovi EOQ Formul wen invenory os is fuzzy Inernionl Journl Produion Eonomis vol. 45 pp []. Jing S Yo Huey M Lee Fuzzy invenory wi korder for fuzzy order quniy Informion Sienes vol. 9 pp []. Snyi ng Fuzzy produion invenory for fuzzy produ quliy wi ringulr fuzzy numer Fuzzy Se nd Sysems vol. 7 pp [4]. Jing S Yo Jersn ing Invenory wiou korder wi fuzzy ol os nd fuzzy soring os defuzzified y enroid nd signed disne Europen journl of Operions reser vol. 48 pp [5]. Ngoor Gni S. Meswri Eonomi order quniy for iems wi imperfe quliy were sorges re kordered in fuzzy environmen dvnes in Fuzzy Memis vol. 5 no. pp. 9-. [6]. R Uykumr M Vllil Fuzzy eonomi produion quniy model for weiull deerioring iems wi rmp ype of demnd Inernionl Journl of Sregi eision sienes vol. no. pp [7]. L.. Zde nd R. E. Bellmn eision mking in fuzzy environmen Mngemen Siene vol pp IJSM ll Rigs Reserved 54
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