Fuzzy Inventory Model for Deteriorating Items with Time-varying Demand and Shortages

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1 Americn Journl of Operionl Reserch 0 6: 8-9 DO: 0.59/j.jor Fuzzy nvenory Model for Deerioring ems wih ime-vrying Demnd nd horges hndr. Jggi * rl Preek Anuj hrm Nidhi Deprmen of Operionl Reserch Fculy of Mhemicl ciences Universiy of Delhi Delhi 0007 ndi enre for Mhemicl ciences Bnshli Universiy Bnshli 040 Rjshn ndi Absrc Fuzzy se heory is primrily concerned wih how o quniively del wih imprecision nd unceriny nd offers he decision mker noher ool in ddiion o he clssicl deerminisic nd probbilisic mhemicl ools h re used in modeling rel-world problems. he presen sudy invesiges fuzzy economic order quniy model for deerioring iems in which demnd increses wih ime. horges re llowed nd fully bcklogged. he demnd holding cos uni cos shorge cos nd deeriorion re re ken s ringulr fuzzy numbers. Grded Men Represenion igned Disnce nd enroid mehods re used o defuzzify he ol cos funcion nd he resuls obined by hese mehods re compred wih he help of numericl exmple. ensiiviy nlysis is lso crried ou o explore he effec of chnges in he vlues of some of he sysem prmeers. he proposed mehodology is pplicble o oher invenory models under unceriny. eywords nvenory Deeriorion horges Fuzzy Vrible ringulr Fuzzy Number Grded men represenion mehod igned disnce mehod enroid mehod. nroducion n convenionl invenory models uncerinies re reed s rndomness nd re being hndled by pplying he probbiliy heory. However in cerin siuions uncerinies re due o fuzziness nd such cses re diled in he fuzzy se heory which ws demonsred by Zdeh in[]. uffmnn nd Gup[] provided n inroducion o fuzzy rihmeic operion nd Zimmermnn[4] discussed he concep of he fuzzy se heory nd is pplicions. onsidering he fuzzy se heory in invenory modeling renders n uheniciy o he model formuled since fuzziness is he closes possible pproch o reliy. As reliy is imprecise nd cn only be pproximed o cerin exen sme wy fuzzy heory helps one o incorpore uncerinies in he formulion of he model hus bringing i closer o reliy. Prk[0] pplied he fuzzy se conceps o EOQ formul by represening he invenory crrying cos wih fuzzy number nd solved he economic order quniy model using fuzzy number operions bsed on he exension principle. Vujosevic e l.[5] used rpezoidl fuzzy number o fuzzify he order cos in he ol cos of he invenory model wihou bckorder nd go fuzzy ol cos. Yo nd Lee [7] inroduced bckorder invenory model wih fuzzy order * orresponding uhor: ckjggi@yhoo.com hndr. Jggi Published online hp://journl.spub.org/jor opyrigh 0 cienific & Acdemic Publishing. All Righs Reserved quniy s ringulr nd rpezoidl fuzzy numbers nd shorge cos s crisp prmeer. Gen e l.[4] expressed heir inpu d s fuzzy numbers nd hen he inervl men vlue concep ws inroduced o solve he invenory problem. hng e l.[0] considered he bckorder invenory problem wih fuzzy bckorder such h he bckorder quniy is ringulr fuzzy number. hng[] discussed he fuzzy producion invenory model for fuzzify he produc quniy s ringulr fuzzy number. Lee nd Yo[5] proposed he invenory wihou bckorder models in he fuzzy sense where he order quniy is fuzzified s he ringulr fuzzy number. Yo e l.[9] ssumed o be he order quniy nd he ol demnd re s ringulr fuzzy numbers nd obined he fuzzy invenory model wihou shorges. Wu nd Yo[] fuzzified he order quniy nd shorge quniy ino ringulr fuzzy numbers in n invenory model wih bckorder nd hey obined he membership funcion of he fuzzy cos nd is cenroid. Yo nd hing[8] considered he ol cos of invenory wihou bckorder. hey fuzzified he ol demnd nd cos of soring one uni per dy ino ringulr fuzzy numbers nd defuzzify by he cenroid nd he signed disnce mehods. Du e l.[7] developed model in presence of fuzzy rndom vrible demnd where he opimum is chieved using grded men inegrion represenion. hng e l.[] developed he mixure invenory model involving vrible led-ime wih bckorders nd los sles. Firs hey fuzzify he rndom led-ime demnd o be fuzzy rndom vrible nd hen fuzzify he ol demnd o be he ringulr fuzzy number

2 8 hndr. Jggi e l.: Fuzzy nvenory Model for Deerioring ems wih ime-vrying Demnd nd horges nd derive he fuzzy ol cos. By he cenroid mehod of defuzzificion hey esime he ol cos in he fuzzy sense. Wee e l.[6] developed n opiml invenory model for iems wih imperfec quliy nd shorge bckordering. Lin[] developed he invenory problem for periodic review model wih vrible led-ime nd fuzzified he expeced demnd shorge nd bckorder re using signed disnce mehod o defuzzify. Roy nd mn[] discussed fuzzy coninuous review invenory model wihou bckorder for deerioring iems in which he cycle ime is ken s symmeric fuzzy number. hey used he signed disnce mehod o fuzzify he ol cos. Gni nd Mheswri[6] developed n EOQ model wih imperfec quliy iems wih shorges where defecive re demnd holding cos ordering cos nd shorge cos re ken s ringulr fuzzy numbers. Grded men inegrion mehod is used for defuzzificion of he ol profi. Ameli e l.[] developed new invenory model o deermine ordering policy for imperfec iems wih fuzzy defecive percenge under fuzzy discouning nd inflionry condiions. hey used he signed disnce mehod of defuzzificion o esime he vlue of ol profi. Nezhd e l.[] developed periodic review model nd coninuous review invenory model wih fuzzy seup cos holding cos nd shorge cos. Also hey considered he led-ime demnd nd he led-ime plus one period s demnd s rndom vribles. hey use wo mehods in he nme of signed disnce nd possibiliy men vlue o defuzzify. Uhykumr nd Vllihl[9] developed n economic producion model for Weibull deerioring iems over n infinie horizon under fuzzy environmen nd considered some cos componen s ringulr fuzzy numbers nd using he signed disnce mehod o defuzzify he cos funcion. n his pper n invenory model for deerioring iems wih shorges is considered where demnd holding cos uni cos shorge cos nd deeriorion re re ssumed s ringulr fuzzy numbers. For defuzzificion of he ol cos funcion Grded Men Represenion igned Disnce nd enroid mehods re used. By compring he resuls obined by hese mehods we ge he beer one s n esime of he ol cos in he fuzzy sense.. Preliminries n order o re fuzzy invenory model by using grded men represenion signed disnce nd cenroid o defuzzify we need he following definiions. Definiion. By Pu nd Liu[8 Definiion.]. A fuzzy se on R membership funcion is is clled fuzzy poin if is x x 0 x where he poin is clled he suppor of fuzzy se. Definiion. A fuzzy se b where 0 nd < b defined on R is clled level of fuzzy inervl if is membership funcion is b x b x 0 oherwise Definiion. A fuzzy number b c A where < b < c nd defined on R is clled ringulr fuzzy number if is membership funcion is x x b b c x A b x c c b 0 Oherwise When b c we hve fuzzy poin c c c c. he fmily of ll ringulr fuzzy numbers on R is denoed s F b c b c b c R. N he -cu of A b c F 0 is A A L AR. Where A L b A R c c b A. N nd re he lef nd righ endpoins of Definii on.4 f A b c is ringulr fuzzy number hen he grded men inegrion represenion of A is defined s w A L h R h h dh 0 P A wa wih 0 < h w nd 0 < A PA / 4b c hdh w A. h h b c h c dh 0 hdh. 4 Definii on.5 f A b c is ringulr fuzzy

3 Americn Journl of Operionl Reserch 0 6: number hen he signed disnce of A is defined s Definiion.6 he cenroid mehod on he ringulr d A0 da L AR 0 = b c is defined s fuzzy number A b c Figure. b c A. 6 -cu of ringulr fuzzy number. Assumpions nd Noions he mhemicl model in his pper is developed on he bsis of he following ssumpions nd noions.. Noions i D is he demnd re ny ime per uni ime. ii A is he ordering cos per order. iii is he deeriorion re 0 iv is he lengh of he ycle. v Q is he ordering Quniy per uni. vi h is he holding cos per uni per uni ime. vii is he shorge os per uni ime. viii is he uni os per uni ime. ix x is he ol invenory cos per uni ime. D is he fuzzy demnd. xi is he fuzzy deeriorion re. xii h is he fuzzy holding cos per uni per uni ime. xiii is he fuzzy shorge os per uni ime. xiv xv ime. is he fuzzy uni os per uni ime. is he ol fuzzy invenory cos per uni xvi is he defuzzify vlue of by pplying Grded men inegrion mehod xvii d is he defuzzify vlue of by pplying igned disnce mehod xviii is he defuzzify vlue of by pplying enroid mehod.. Assumpions i Demnd D b is ssumed o be n incresing funcion of ime i.e. where nd b re posiive consns nd 0 0 b. ii Replenishmen is insnneous nd led-ime is zero. iii horges re llowed nd fully bcklogged. 4. Mhemicl Model Le be he on-hnd invenory ime wih iniil invenory Q. During he period[0 ] he on-hnd invenory deplees due o demnd nd deeriorion nd exhused ime. he period[ ] is he period of shorges which re fully bcklogged. A ny insn of ime he invenory level is governed by he differenil equions. 4.. risp Model d D 0 4. d Wih 0 Q nd 0. d D 4. d Wih 0. he soluion of equion 4. nd 4. is given by nd b b Qe e b 4. b 4.4

4 84 hndr. Jggi e l.: Fuzzy nvenory Model for Deerioring ems wih ime-vrying Demnd nd horges By using 0 we hve b e b b Q 4.5 Now 4. becomes 6 b 4.6 Neglecing higher powers of. ol verge no. of holding unis H during period[0 ] is given by b d H 4.7 ol no. of deeriored unis D during period[0 ] is given by Q D ol Demnd 0 b d b Q D 4.8 ol verge no. of shorge unis during period[0 ] is given by b d 4.9 ol cos of he sysem per uni ime is given by D H h A b b hb h A Fuzzy Model Due o uncerinly in he environmen i is no esy o define ll he prmeers precisely ccordingly we ssume some of hese prmeers nmely b h my chnge wihin some limis. Le h h h h b b b b re s ringulr fuzzy numbers. ol cos of he sysem per uni ime in fuzzy sense is given by b b h b h A 4. We defuzzify he fuzzy ol cos by grded men represenion signed disnce nd cenroid mehods. i By Grded Men Represenion Mehod ol os is given by

5 Americn Journl of Operionl Reserch 0 6: Where 6 4 A h h b b 6 8 i b b 4 A h hb b 6 8 b 4 A h hb b 6 8 b b 4 6 o minimize ol cos funcion per uni ime following equions: Equion 4. is equivlen o nd he opiml vlue of 0 nd 4. nd cn be obined by solving he 4. 0 h h b b b b h hb b b h hb b b b 4.4

6 86 hndr. Jggi e l.: Fuzzy nvenory Model for Deerioring ems wih ime-vrying Demnd nd horges b b 4 b 6 b b 4 6A h h b b 6 8 b b h hb b b 4 h hb b 6 8 b b o be convex he following condiions mus be sisfied Furher for he ol cos funcion nd he second derivives of he ol cos funcion re compliced nd i is very difficul o prove he convexiy mhemiclly. hus he convexiy of ol cos funcion hs been esblished grphiclly Figure A. ii By igned Disnce Mehod ol cos is given by Where d d i d d d d d 4 4 A h h b b 6 8 b b 4 A h hb b 6 8 b 4 A h hb b 6 8 b b

7 Americn Journl of Operionl Reserch 0 6: he ol cos funcion d o minimize ol cos funcion per uni ime following equions: d d d d 4 hs been minimized following he sme process s hs been sed in cse i. d 4.8 d he opiml vlue of nd cn be obined by solving he 0 nd d 0 Equion 4.9 is equivlen o h h b b b b h hb b 0 4 b h hb b b b nd b b b 4 b b 4 4A h h b b 6 8 b b 4 0 h hb b b 4 h hb b 6 8 b b Furher for he ol cos funcion d o be convex he following condiions mus be sisfied d 0 d

8 88 hndr. Jggi e l.: Fuzzy nvenory Model for Deerioring ems wih ime-vrying Demnd nd horges nd d he second derivives of he ol cos funcion d d d re compliced nd i is very difficul o prove he convexiy mhemiclly. hus he convexiy of ol cos funcion hs been esblished grphiclly Figure B. iiiby enroid Mehod ol cos is given by Where he ol cos funcion 4 A h h b b 6 8 b b 4 A h hb b 6 8 b 4 A h hb b 6 8 b b hs been minimized following he sme process s hs been sed in cse i. o minimize ol cos funcion per uni ime following equions: 4.4 he opiml vlue of nd cn be obined by solving he 0 nd Equion 4.5 is equivlen o h h b b b b h hb b 0 b h hb b b b

9 Americn Journl of Operionl Reserch 0 6: nd b b b b b 4 A h h b b 6 8 b b 4 0 h hb b 6 8 b 4 h hb b 6 8 b b o be convex he following condiions mus be sisfied Furher for he ol cos funcion 0 nd he second derivives of he ol cos funcion re compliced nd i is very difficul o prove he convexiy mhemiclly. hus he convexiy of ol cos funcion hs been esblished grphiclly Figure Numericl Exmple onsider n invenory sysem wih following prmeric vlues. risp Model A Rs 00 /order Rs 0 /uni h Rs. 5/uni/yer 00 unis/yer b. unis/yer.0/yer Rs 5 /uni/yer. he soluion of crisp model is = Rs =. 749 yer =.966 yer. Fuzzy Model b h 57 he soluion of fuzzy model cn be deermined by following hree mehods. By Grded Men Represenion Mehod we hve. When b h ll re ringulr fuzzy numbers = Rs =.6908 yer =.98 yer.. When b re ringulr fuzzy numbers = Rs =. 75 yer =.9560 yer.. When b re ringulr fuzzy numbers = Rs =. 75 yer =.9596 yer. 4. When b nd re ringulr fuzzy numbers yer. = Rs =. 70 yer =.960

10 90 hndr. Jggi e l.: Fuzzy nvenory Model for Deerioring ems wih ime-vrying Demnd nd horges 5. When nd b re ringulr fuzzy numbers = Rs =. 7 yer =.96 yer. By igned Disnce Mehod we hve. When b h ll re ringulr fuzzy numbers d = Rs = yer =.966 yer.. When b re ringulr fuzzy numbers d = Rs =. 78 yer =.95 yer.. When b re ringulr fuzzy numbers d = Rs =. 709 yer =.9576 yer. 4. When b nd re ringulr fu zzy numbers d = Rs =. 706 yer =.9587 yer. 5. When nd b re ringulr fuzzy numbers d = Rs =. 7 yer =.9599 yer. By enroid Mehod we hve. When b h ll re ringulr fuzzy numbers unis/yer yer. = Rs =. 669 yer = 95. When b re ringulr fuzzy numbers = Rs =. 7 yer =.9487 yer.. When b re ringulr fuzzy numbers = Rs = yer =.9557 yer. 4. When b nd re ringulr fuzzy numbers yer. = Rs =. 709 yer = When nd b re ringulr fuzzy numbers = Rs =.7 yer =.9587 yer. 6. ensiiviy Anlysis ble. ensiiviy Anlysis on prmeer A sensiiviy nlysis is performed o sudy he effecs of chnges in fuzzy prmeers b nd on he opiml soluion by king he defuzzify vlues of hese prmeers. he resuls re shown in below bles. yer yer Rs ble indices h s he vlue of increses fuzzy cos drsiclly. b unis/yer ble. ensiiviy Anlysis on prmeer b increses significnly bu nd decreses yer yer Rs ble indices h s he vlue of b increses fuzzy cos grdully. increses regulrly bu nd decreses

11 Americn Journl of Operionl Reserch 0 6: ble. ensiiviy Anlysis on prmeer yer yer Rs ble indices h s he vlue of increses fuzzy increses slighly bu nd decreses cos grdully. f we plo he ol cos funcion wih some nd s.. =.65 o wih equl inervl vlues of =.84 o hen we ge sricly convex grph of ol cos funcion given below. nd s.. =.65 o wih equl inervl vlues of =.84 o hen we ge sricly convex grph of ol cos funcion given below. Figure. ol Fuzzy os Vs. nd Figure A. ol Fuzzy os Figure B. ol Fuzzy os Vs. nd d Vs. nd f we plo he ol cos funcion d wih some nd s.. =.65 o wih equl inervl vlues of =.84 o hen we ge sricly convex grph of ol cos funcion d given below. f we plo he ol cos funcion wih some 7. onclusions his pper presens fuzzy invenory model for deerioring iems wih llowble shorges in which demnd is n incresing funcion of ime. he de mnd deeriorion re invenory holding cos uni cos nd shorge cos re represened by ringulr fuzzy numbers. For defuzzificion grded men signed disnce nd cenroid mehod re employed o evlue he opiml ime period of posiive sock nd ol cycle lengh which minimizes he ol cos. By given numericl exmple i hs been esed h grded men represenion mehod gives minimum cos s compred o signed disnce mehod nd cenroid mehod. A sensiiviy nlysis is lso conduced on he prmeers fuzziness. b nd o explore he effecs of b nd Finding ugges h he chnge in prmeers will resul he chnge in fuzzy cos wih some chnges in nd.wih he increses vlues of hese prmeers will nd. resul in increse of fuzzy cos bu decreses imilrly wih he decreses vlues of hese prmeers will nd. A fuure sudy would be o exend he proposed model for resul in decrese of fuzzy cos bu increses

12 9 hndr. Jggi e l.: Fuzzy nvenory Model for Deerioring ems wih ime-vrying Demnd nd horges finie replenishmen re sock ous which re prilly bcklogged price dependen demnd sock dependen demnd nd mny more. ANOWLEDGEMEN he uhors would like o hnk nonymous referees for heir vluble nd consrucive commens nd suggesions h hve led o improvemen on he erlier version of he pper. he firs uhor would like o cknowledge he suppor of he Reserch Grn No.: DenR/R&D/0/97 provided by he Universiy of Delhi Delhi ndi for conducing his reserch. he hird uhor would like o hnk Universiy Grns ommission for providing Junior Reserch Fellowship vide leer No. JRF/AA/68/ REFERENE [] Arnold ufmnn Mdn M Gup nroducion o Fuzzy Arihmeic: heory nd Applicions Vn Nosrnd Reinhold New York 99. [] Ajn Roy Guru P mn Fuzzy coninuous review invenory model wihou bckorder for deerioring iems Elecronic Journl of Applied isicl Anlysis vol. no. pp [] Hung. hng Jing Yo Ling Y Ouyng Fuzzy mixure invenory model involving fuzzy rndom vrible led-ime demnd nd fuzzy ol demnd Europen Journl of Operionl Reserch vol. 69 no. pp [4] Hns J Zimmermnn Fuzzy e heory nd s Applicions rd Ed. Dordrech: luwer Acdemic Publishers 996. [5] Huey M Lee Jing Yo Economic order quniy in fuzzy sense for invenory wihou bckorder model Fuzzy es nd ysems vol. 05 pp [6] Hui M Wee Jons Yu Mei hen Opiml invenory model for iems wih imperfec quliy nd shorge bckordering nernionl Journl of Mngemen cience vol. 5 pp [7] Jing Yo Huey M Lee Fuzzy invenory wih bckorder for fuzzy order quniy nformion ciences vol. 9 pp [8] Jing Yo Jershn hing nvenory wihou bckorder wih fuzzy ol cos nd fuzzy soring cos defuzzified by cenroid nd signed disnce Europen journl of Operions reserch vol. 48 pp [9] Jing Yo n hng Jin u Fuzzy nvenory wihou bckorder for fuzzy order quniy nd fuzzy ol demnd quniy ompuer nd Operions Reserch vol. 7 pp [0] Prk Fuzzy-se heoreic inerpreion of economic order quniy EEE rnscions on ysems Mn nd yberneics M-7 pp [] weimei Wu Jing Yo Fuzzy invenory wih bckorder for fuzzy order quniy nd fuzzy shorge quniy Europen Journl of Operionl Reserch vol. 50 no. pp [] Lofi A. Zdeh Fuzzy ses nformion nd onrol vol. 8 no. pp [] M Ameli A Mirzzdeh nd M A hirzi Economic order quniy model wih imperfec iems under fuzzy inflionry condiions rends in Applied ciences Reserch vol. 6 no. pp [4] Misuo Gen Ysuhiro sujimur Dzhong Zheng An pplicion of fuzzy se heory o invenory conrol models ompuers nd ndusril Engineering vol. pp [5] Mirko Vujosevic Dobril Perovic Rdivoj Perovic EOQ Formul when invenory cos is fuzzy nernionl Journl Producion Economics vol. 45 pp [6] Ngoor A Gni. Mheswri Economic order quniy for iems wih imperfec quliy where shorges re bckordered in fuzzy environmen Advnces in Fuzzy Mhemics vol. 5 no. pp [7] Pnkj Du Dbjni hkrbory Akhil R Roy A single-period invenory model wih fuzzy rndom vrible demnd Mhemicl nd ompuer Modeling vol. 4 no.8-9 pp [8] P M Pu nd Y M Liu Fuzzy opology neighborhood srucure of fuzzy poin nd Moore- mih onvergence Journl of Mhemicl Anlysis nd Applicion vol. 76 pp [9] R Uhykumr M Vllihl Fuzzy economic producion quniy model for weibull deerioring iems wih rmp ype of demnd nernionl Journl of regic Decision sciences vol. no. pp [0] n hng Jing Yo Huey M Lee Economic reorder poin for fuzzy bckorder quniy Europen Journl of Operionl Reserch vol. 09 pp [] nchyi hng Fuzzy producion invenory for fuzzy produc quliy wih ringulr fuzzy number Fuzzy e nd ysems vol. 07 pp [] heli Nezhd him M Nhvndi Jmshid Nzemi Periodic nd coninuous invenory models in he presence of fuzzy coss nernionl Journl of ndusril Engineering ompuions vol. pp [] Yu J Lin A periodic review invenory model involving fuzzy expeced demnd shor nd fuzzy bckorder re ompuers & ndusril Engineering vol. 54 no. pp