Fuzzy Optimal Replenishment Policy for Weibull Deteriorating Items with Ramp Type Demand and Partial Backlogging Under Permissible Delay in Payments

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1 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr 5 Fuzzy Opimal Rplnishmn Poliy for Wibull rioraing ms wih Ramp yp mand and Parial Baklogging Undr Prmissibl lay in Paymns J.Sujaha P.Parvahi Ass. Profssor parmn of ahmais Quaid-E-illah Gov. ollg for WomnAuonomous hnnai amil Nadu ndia Had & Asso. Profssor parmn of ahmais Quaid-E-illah Gov. ollg for WomnAuonomous hnnai amil Nadu ndia Absra -- n his papr w dvlopd a fuzzy opimal rplnishmn poliy for nvnory modls of Wibull rioraion ims wih Ramp yp mand undr prmissibl dlay in paymns. rioraion of ims bgins on hir arrival in sok. h shorags ar allowd and parially bakloggd. All h invnory oss involvd in his modl ar akn as riangular fuzzy numbrs. Rplnishmn yl lngh h im a whih shorag bgins ar akn as dision variabls. Gradd man ingraion rprsnaion mhod is usd o dfuzzify h modl. h drivd modl is illusrad by a numrial xampl and is snsiiviy analysis is arrid ou. Kywords-- Fuzzy numbrs and fuzzy onps parial baklogging Ramp yp mand Wibull disribuion of wo paramrs.. NROUON h invnory sysm is aking an imporan par of os onrolling in businss. For h las fw yars rsarhrs in his ara hav xndd invsigaion ino various modls wih onsidraions of im shorag im drioraion dmand parns im ordr yls and hir ombinaions. rioraion of physial goods is on of h imporan faors in any invnory and produion sysm. h drioraing ims wih shorags hav rivd muh anion of svral rsarhrs in h rn yars baus mos of h physial goods undrgo day or drioraion ovr im. Shorags ar lassifid ihr omplly bakloggd shorags or parially bakloggd shorags. omplly bakloggd shorags ar h shorags whih ar duly fulfilld by h vndor during h shorag priod. Somims h shorags may no b omplly bakloggd and only a par of h dmand an b m by h vndor during h shorag priod whih is rmd as parially bakloggd. n som invnory sysms suh as fashionabl ommodiis h lngh of h waiing im for h nx rplnishmn would drmin whhr h baklogging will b apd or no. hrfor h baklogging ra should b variabl and dpnds on h waiing im for h nx rplnishmn. hang and y [] dvlopd a modl for drioraing ims wih im varying dmand and shorags in whih h baklogging ra is assumd o b invrsly proporional o h waiing im for h nx rplnishmn. Skouri and Papahrisos [] sudid a muli priod invnory modl using h xponnially drasing baklogging ra proposd by Abad. Rnly many rsarhrs hav modifid invnory poliis by onsidring h im proporional parial baklogging ra suh as Wang [] ng and Yang [4] San Jos al. [5] and Wu al. [] h dmands of fashionabl goods inras up o h rain lvl and afr ha h dmand boms sady. Suh yp of dmand funions ar known as ramp yp of dmand. W know ha dmand is no always a monooni funion i.. ims lik fashionabl goods ovr a planning priod and afr a pariular im i boms sady. Hill [7] rsolvd h indisiplin of im dpndn dmand parn by onsidring h dmand as h ombinaion of wo diffrn yps of disiplind dmand in wo sussiv im priods ovr h nir im horizon and rmd i as ramp yp im dpndn dmand parn. Wu al. [8] dvlopd an EOQ modl wih ramp yp dmand ra for ims wih Wibull drioraion. andal and Pal [9] dvlopd an ordr-lvl invnory modl for drioraing ims wih ramp yp dmand. Wu and Ouyang [] xndd hir modl by inorporaing h onp of shorags followd by SSN: 78 All Righs Rsrvd 5 JARE 548

2 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr 5 invnory. Singh al. [] dvlopd an EOQ invnory modl wih Wibull disribuion drioraion ramp yp dmand and parial baklogging. Giri al. [] dvlopd an onomi ordr quaniy modl wih Wibull drioraion disribuion shorags and ramp yp dmand. h supplirs offr dlay in paymn o h railrs o buy mor ims and h railrs an sll h im bfor h losing of h dlay im. As a rsul h railrs sll h ims and arn inrss. Usually hr is no inrs harg if h ousanding amoun is paid wihin h prmissibl dlay priod. his provids opporuniis o h railrs o aumula rvnu and arn inrs by slling hir ims during h dlay priod. his prmissibl dlay in paymn provids bnfi o h supplir by araing nw usomrs who onsidr i o b a yp of pri rduion and rduion in slls ousanding as som usomrs mak paymns on im in ordr o ak advanag of prmissibl dlay mor frqunly. n his dirion Goyal [] xndd h EOQ modl undr h ondiions of prmissibl dlay in paymns. hung Yuan y [4] dvlopd an invnory modl for drioraing ims wih sok dpndn dmand and parial baklogging undr ondiions of prmissibl dlay in paymns. R.Uhayakumar al [5] hav dvlopd a drminisi invnory modl wih sok and im dpndn dmand undr prmissibl dlay in paymns. n h risp invnory modls all h paramrs in h oal os ar known and hav dfini valus. Bu in h praial siuaion i is no possibl. Hn fuzzy invnory modls fulfil ha gap. iffrn fuzzy invnory modls our du o fuzzy various os paramrs in h oal os. Rsarhrs rlad o his ara ar: Zimmrmann [] and Kaufmann and Gupa [7] Kaprzyk and Saniwski [8] Bllman and Zadh [9] L and Yao [] Yao and Su [] hn and Ouyang [] ahaa and Goswamy [] Vijayan and Kumaran [4]. n his papr an opimal rplnishmn poliy for invnory modl of drioraing ims wih ramp yp dmand undr prmissibl dlay in paymns has bn dvlopd. wo paramrs Wibull disribuion drioraion ra has bn akn in his sudy. Shorags ar allowd wih parial baklogging. Baklogging ra is an xponnial drasing funion of im. h invnory oss ar akn as riangular fuzzy numbr. Gradd man ngraion rprsnaion mhod is applid for dfuzzifiaion. h proposd modl is illusrad wih numrial xampls.. FUZZY PRELNARES finiion : L X dnos a univrsal s. hn h fuzzy subs A of X is dfind by is mmbrship funion x : X [ ] whih A assigns a ral numbr x in h inrval [] A o ah lmn x X whr h valu of A x a x shows h grad of mmbrship of x finiion : A fuzzy s A on R is onvx if + - for all x x R and [ ]. finiion : A fuzzy s in h univrs of disours X is alld as a fuzzy numbr in h univrs of disours X riangular fuzzy numbr W onsidr h siuaion whr fuzzy numbrs ar rprsnd by riangular mmbrship funions. h fuzzy numbr A is said o b riangular fuzzy numbr if i is fully drmind by of risp numbrs suh ha whos mmbrship funion rprsning riangl an b dnod by X = h Funion Prinipl h funion prinipl was inrodud by hn o ra fuzzy arihmial opraions. his prinipl is usd for h opraion for addiion subraion mulipliaion and division of fuzzy numbrs. Suppos A a a a and B b b b ar wo riangular fuzzy numbrs. hn i Addiion: A B a b a b a b whr a a a b b b ar any ral numbrs. iisubraion: A B a b a b a b whr a a a b b b ar any ral numbrs. ii ulipliaion: A B a b ab ab Whr a a a b b b ar all non zro posiiv ral numbrs iv ivision: A a a a whr b b b B b b b ar all non zro posiiv ral numbrs v Salar ulipliaion: For any ral numbr K KA Ka Ka Ka K KA Ka Ka Ka K Gradd an ngraion Rprsnaion hod f A a a a is riangular fuzzy numbr hn h gradd man ingraion rprsnaion of A is givn by a 4a a P A SSN: 78 All Righs Rsrvd 5 JARE 549

3 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr 5. NOAONS AN ASSUPONS h proposd invnory modl having following n oaions and assumpions:. Noaions = Fuzzy uni purhas os in $. =Fuzzy invnory holding os pr uni pr im uni xluding inrs hargs in $. = Fuzzy rplnishmn os pr yl in $. = Fuzzy shorag os $ pr uni pr yar. l = Fuzzy opporuniy os du o las sals $ pr uni. = Fuzzy inrs whih an b arnd $ pr uni im. = Fuzzy inrs hargs payabl pr $ pr im uni = invnory holding hargs pr $ uni pr yar. = h lngh of h ordr yl. = im a whih shorag sars = prmissibl dlay priod for sling aouns in im unis. 4. Ramp-yp dmand ra f is givn by f H ; > Whr Havisid s uni funion for H for is wll known 5. uring h fixd rdi priod h uni os of gnrad sals rvnu is dposid in an inrs baring aoun. h diffrn bwn sals pri and uni os is raind by h sysm o m h day-o-day xpnss of h sysm. A h nd of h rdi priod h aoun is sld and inrs hargs ar payabl on h aoun in sok.. h drioraion of im as follows by Wibull paramrs wo disribuion whr is h sal paramr and is h shap paramr. 7. Shorags ar allowd and unsaisfid dmand is bakloggd a a ra whr h baklogging paramrs is a posiiv onsan. V. OEL FORULAON h invnory sysm is dvlopd as follows: Q unis of ims arriv a h invnory sysm a h bginning of ah yl. uring h im inrval [ ] h invnory lvl is drasing only du o dmand ra. h invnory lvl is dropping o zro owing o dmand and drioraion during h im inrval [ ]. Finally a shorag ours du o dmand and parial baklogging during h im inrval [ ]. h bhaviour of h invnory modl is dmonsrad in Fig.. = h lif im of h ims pr yl. Q = h invnory lvl a im S Q = niial invnory lvl of ah ordring yl. = h ordr siz pr yl. = Fuzzy oal invnory os whn =Fuzzy oal invnory os whn = Fuzzy oal nvnory os whn. Assumpions. h lad-im is zro.. h im-horizon of h sysm is infini. Fig. Graphial rprsnaion of h invnory sysm L Q dno h on hand invnory of h sysm a any im plion du o dmand and drioraion will our simulanously. h diffrnial quaion ha dsribs h insananous sa of Q is givn by. hr is no rpair or rplamn of h driorad invnory during a givn yl. dq Q ; d SSN: 78 All Righs Rsrvd 5 JARE 55

4 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr 5 dq Q d dq d ; ; wih h iniial and boundary ondiions Q = S and Q = whr S is h i n i i a l invnory.h soluions of quaions and subj o h ondiions 4 ar rspivly Q S Q ; Q ; ; Subsiuing in quaions 5 and and hn quaing w g ii Numbr of unis ha driora during h priod is givn by Q d d h fuzzy os du o drioraion during h priod = S h maximum bakloggd invnory B is obaind a = hn from 7 B Q hus h ordr siz during oal im inrval is Q S B Q i oal invnory fuzzy holding os during h priod is givn by H Q Q d d Q d iii Fuzzy rplnishmn os pr yl = iv h fuzzy shorag os during h priod is givn by S Q d v h fuzzy opporuniy os du o los sals during h priod is givn by d L l l as i : SSN: 78 All Righs Rsrvd 5 JARE 55

5 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr 5 55 SSN: 78 All Righs Rsrvd 5 JARE Fig.. nvnory lvl as a funion of im for as: i uring h prmissibl dlay priod whn h aoun is no sld h railr slls h goods and oninus o aumula sals rvnu and arns h inrs wih fuzzy ra. h inrs arnd dnod E. hrfor h oal inrs arnd in h yl priod is d E Afr h rdi priod h buyr has o pay h inrs for h goods sill in sok wih fuzzy ra. h inrs payabl dnod. hrfor h inrs payabl in any yl is d Q d Q d Q h oal avrag fuzzy os of h invnory sysm pr uni im is givn by - E H + + S + LS + R l

6 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr 5 55 SSN: 78 All Righs Rsrvd 5 JARE o minimiz oal avrag os pr uni im h opimal valus of and an b obaind by solving h following quaions simulanously Providd hy saisfy h following ondiions : And h opimal soluion of h quaions 8 an b obaind by using appropria sofwar. his has bn illusrad by h following numrial xampl. as ii: Fig.. nvnory lvl as a funion of im for as: ii n his as h priod of dlay in paymn is mor han h priod wih no drioraion bu lss han h priod wih posiiv invnory. W an inrs payabl dnod by as follows and fuzzy annual ra is.h inrs payabl in any yl [ ] is d Q nrs arnd in h yl priod [ ] is d E h oal avrag fuzzy os of h invnory sysm pr uni im is givn by - E H + + S + LS + R

7 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr SSN: 78 All Righs Rsrvd 5 JARE l o minimiz oal avrag os pr uni im h opimal valus of and an b obaind by solving h following quaions simulanously Providd hy saisfy h following ondiions : And h opimal soluion of h quaions 5 an b obaind by using appropria sofwar. his has bn illusrad by h following numrial xampl. as iii : Fig. 4. nvnory lvl as a funion of im for as iii: n his as h priod of dlay in paymn is mor han priod wih posiiv invnory. n his as h railr arns inrs on h sals rvnu wih an fuzzy annual ra o h prmissibl dlay priod and no inrs is payabl during h priod for h im kp in sok. nrs arnd for h im priod [ ]. h inrs arnd dnod by E is givn by oal inrs arnd during h yl = nrs arnd up o + nrs arnd during d d E h oal avrag fuzzy os of h invnory sysm pr uni im is givn by R + H + + S + LS - E l

8 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr 5 o minimiz oal avrag os pr uni im h opimal valus of and an b obaind by solving h following quaions simulanously Providd hy saisfy h following ondiions : Exampl-:onsidr an invnory sysm wih h following daa : α =.; β = ; = ; l = 789 ; = 9 ; = 55 ; = 57 ; =... ; =.4.5. ; = 5 ; µ =. ; = ; = 4/5 in appropria unis. hn w g h opimal valus as =.89 ; =.84 ; Q =.74 ; = 7.48 in appropria unis. Effs of oal os funion ovr rad rdi Priod And h opimal soluion of h quaions an b obaind by using appropria sofwar. his has bn illusrad by h following numrial xampl. V. NUERAL EXAPLES Exampl-: onsidr an invnory sysm wih h following daa: α =.; β = ; = 5; =55 ; l =789 ; =9 ; =55 ; = 57 ; =... ; =.4.5. ; µ =.; = ; = /5 in appropria unis. hn w g h opimal valus as =.755 ; =.9495 ; Q = 7.75 ; =.5 in appropria unis. Effs of oal os funion ovr rad rdi Priod Exampl-:onsidr an invnory sysm wih h following daa : α =.; β = ; =55; l =789; =9; =55 ; = 57 ; =... ;.4.5. ; = 5 ; µ =. ; = ; = 45/5 in appropria unis. hn w g h opimal valus as =.77 ; =.59 ; Q = 4.57 ; = 54.8 in appropria unis. = Effs of oal os funion ovr rad rdi Priod SSN: 78 All Righs Rsrvd 5 JARE 555

9 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr 5 V. SENSVY ANALYSS W hav prformd snsiiviy analysis by hanging paramrs and as -% - 4% -% +% +4% +% and kping h rmaining paramrs a hir original valus. h orrsponding hangs in h opimal lngh of im in whih hr is no invnory shorag lngh of h ordr yl h opimal ordr quaniy pr yl Q and h opimal oal avrag os. abl -: Snsiiviy analysis for as i Paramrs % hang Q α β µ abl -: Snsiiviy analysis for as ii Paramrs % hang Q α β µ SSN: 78 All Righs Rsrvd 5 JARE 55

10 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr abl -: Snsiiviy analysis for as iii Paramrs % hang Q α β µ From h abov abls w an onlud h following :. is inrsing o obsrv ha inrass in drioraion paramr α dras ordring quaniy Q and inras oal os of an invnory sysm.. And also inrass in drioraion paramr β dras ordring quaniy Q and oal os of an invnory sysm.. nras h paramr µ inras ordring quaniy Q and oal os of an invnory sysm. 4. nras in dlay priod rsuls in dras ordring quaniy Q and oal os of an invnory sysm. V. ONLUSON W hav dvlopd a fuzzy invnory modl. Whih is mor suiabl o day-o-day lif.wibull disribuion drioraion has also bn disussd. orovr h onps of parial baklogging of modlling 5-7. shorags and rad rdi priod ar also akn in our modl. rad rdi priod is an imporan asp for h smooh amosphr in any businss. By omparing h im a whih shorag bgins wih rad rdi priod hr ass hav bn sudid for minimizing h oal avrag invnory os. is obsrvd ha inniv of rdi priod is advanagous o h railr for lowring h oal os of an invnory sysm. All h oss inurrd in his modl ar adopd in fuzzy naur whih is mor gnral. h invnory oss ar akn as riangular fuzzy numbr. Gradd man ngraion rprsnaion mhod is applid for dfuzzifiaion. Snsiiviy analysis is arrid ou o luidad h proposd modl. REFERENES []. hang H.J. and y.y.999. An EOQ modl for drioraing ims wih im-varying dmand and parial baklogging. h Journal of h Opraional Rsarh Soiy []. Skouri K. And Papahrisos S. A oninuous rviw invnory modl wih drioraing ims imvarying dmand and linar rplnishmn os parially im varying baklogging Applid mahmaial SSN: 78 All Righs Rsrvd 5 JARE 557

11 nrnaional Journal of Advand Rsarh in ompur Enginring & hnology JARE Volum 4 ssu 9 Spmbr 5 []. Wang S.P. An invnory rplnishmn poliy for drioraing ims wih shorags and parial baklogging ompurs and Opraional Rsarh [4]. ng J.. and Yang H.L. 4 rminisi onomi ordr quaniy modls wih parial baklogging whn dmand and os ar fluuaing wih im Journal of h Opraional Rsarh Soiy [5]. San Jos L.A. Siilia J. and Laguna J.G. Analysis of an invnory sysm wih xponnial parial baklogging nrnaional Journal of Produion Eonomis 7-8. []. Wu K.S. Ouyang L.Y. and Yang.. An opimal rplnishmn poliy for non-insananous drioraing ims wih sok dpndn dmand and parial baklogging nrnaional Journal of Produion Eonomis [7]. Hill R Opimal EOQ modls for drioraing ims wih im-varying dmand Journal of h Opraional Rsarh Soiy [8]. WuJ.W. Lin.anB. and LW An EOQ modl wih ramp yp dmand ra for ims wih Wibull drioraion nrnaional Journal of nformaion and anagmn Sins 4-5. [9]. andal B. and Pal A.K. 998 Ordr lvl invnory sysm wih ramp yp dmand ra for drioraing ims Journal of inrdisiplinary ahmais 49-. []. Wu K.S.. Ouyang L.Y. A rplnishmn poliy for drioraing ims wih ramp yp dmand ra for drioraing ims h Prodings of h Naional Sin ounil Par A: Physial Sin Enginring []. Singh S.R. and Singh.J. 7 AN EOQ invnory modl wih Wibull disribuion drioraion ramp yp dmand and parial baklogging ndian Journal of ahmais and ahmaial Sin 7-7. []. Giri B.. Jalan A.K. haudhuri K.S. Eonomi Ordr Quaniy modl wih Wibull drioraion disribuion shorag and ramp-yp dmand nrnaional Journal of Sysms Sin 4A 7-4. []. Goyal S.K.985. Eonomi ordr quaniy undr ondiions of prmissibl dlay in paymns Journal of h opraional Rsarh soiy Vol [4]. [4]. hung-yuan y A drioraing invnory modl wih sok dpndn dmand and parial baklogging undr ondiions of prmissibl dlay in paymns OPSEARH [5]. Udayakumar R. And Parvahi.P A drminisi invnory modl for drioraing ims wih parially bakloggd sok and im dpndn dmand. nrnaional Journal of mahmaial sins. []. Zimmrman H.J Fuzzy programming and linar programming wih svral objiv funions. Fuzzy Ss and Sysms doi:./ [7]. KaufmannA. and Gupa nroduion o fuzzy arihmi hory and appliaions. Nw York NY: Van Nosrand Rinhold. [8]. Kaprzyk J. and Saniwski P.98. Long-rm invnory poliy-making hrough fuzzy dision making modls. Fuzzy Ss and Sysms 87-.doi:./ [9]. Bllman R.E. and Zadh L.A. 97. ision-making in a fuzzy nvironmn. anagmn Sin doi:.87 /mns.7.4.b4. []. L H.. and YaoJ.S Eonomi ordr quaniy in fuzzy sns for nvnory wihou bakordr modl. Fuzzy Ss and Sysms 5-.doi:./S []. YaoJ.and Su.J.. Fuzzy invnory wih bakordr for fuzzy oal dmand basd on inrval-valud fuzzy s. Europan Journal of Opraional Rsarh doi:./S []. hn L.H. and Ouyang L.Y.. Fuzzy invnory modl for drioraing ims wih prmissibl dlay in paymn. Applid ahmais and ompuaion 87-7.doi:./j.am []. ahaag.. and GoswamiA. 9a. A fuzzy rplnishmn poliy for drioraing ims wih ramp yp dmand ra undr inflaion. nrnaional Journal of Opraional Rsarh doi:. 54/JOR.9.5. [4]. Vijayan. and Kumaran. 8. nvnory modl wih a mixur of bakordrs and los sals undr fuzzy os. Europan Journal of Opraional Rsarh doi:./j.jor [5]. Goyal S.K. and Giri B... h produioninvnory problm of a produ wih im varying dmand produion and drioraion ras. Europan Journal of Opraional Rsarh doi:./S []. Halim K.A. Giri B.. and haudhuri K.S. 8. Fuzzy onomi ordr quaniy modl for prishabl ims wih sohasi dmand parial baklogging and fuzzy drioraion ra. nrnaional Journal of Opraional Rsah doi:.54/ JOR [7]. L H.. and Yao J.S Eonomi produion quaniy for fuzzy dmand quaniy and fuzzy produion quaniy. Europan Journal of Opraional Rsarh 9 -.doi:./s [8]. Zimmrman H.J.99. Fuzzy s hory and is appliaions nd d.. Boson A: Kluwr Aadmi. SSN: 78 All Righs Rsrvd 5 JARE 558