FUZZY INVENTORY MODEL FOR DETERIORATION ITEMS THROUGH JUST IN TIME WITH SHORTAGES ALLOWED

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1 Inian J.i.e. : 6-7 IN: Pint IN: 5- Online FUZZY INVENTOY MODEL FO DETEIOTION ITEM THOUGH JUT IN TIME WITH HOTGE LLOWED J. JYNTHI a ND M. MGTHM b a Depatment of Mathemati Peiya Maniammai Univeity Thanjavu Inia b P.G. & eeah Depatment of Mathemati Peiya EV College Tiuhiappalli Inia BTCT Thi pape eive an inventoy moel fo eteioation item though JIT with hotage ha been oniee in a fuzzy envionment. To etemine the optimal total ot an the optimal oe uantity fo the popoe inventoy moel. In oe to aomplih thi pupoe we have been intoue the tapezoial fuzzy numbe. The woking out of eonomi oe uantity EOQ i onee out though efuzzifiation poe by uing igne itane metho. The igne itane metho i moe petinent than the othe metho of efuzzifiation. To illutate the eult of the popoe moel we have given two moel example an peente the omputational eult. enitivity fo thi moel i alo alulate whih how a linea elation among eman EOQ an total ot. Thi antiipate appoah i imple an give a bette eult in ompaatively le omputational wok. KEYWOD: Jut In Time; Quality uane; Deteioating Item; Tapezoial Fuzzy Numbe; Defuzzifiation; igne Ditane Metho. Inventoy play an extemely impotant ole in an intenational eonomy. n inventoy may be in the fom of aw mateial fulfille pat impatial pat emi finihe goo o finihe goo. Thee ae fou type of inventoy item namely Deteioation item Oboleene item Demolih o Pilfeage item No Oboleene / Deteioation item. Deteioation item efe to the item that beome eompoe mahe expie illogial epeiation o phyial haateiti of a mateial ue to flawe pakaging o abnomal toage onition. Oboleene item ae efee to the ahai pout loe thei value beaue of haty hange of euipment. Damage o Pilfeage item ae efee to the ime of tealing involving employee who teal item fom thei onign of evie in patiula in a manufatuing plant. No Oboleene / Deteioation item ae efee to the life yle of ome goo i impeie in natue. lak et al. efe Vo efinition of JIT: a iipline appoah to impoving oveall poutivity an eliminating wate. It povie fo the ot-effetive poution an elivey of only the neeay uantity of pat at the ight uality at the ight time an plae while uing a minimum amount of failitie euipment mateial an human eoue. JIT i epenent on the balane between the upplie flexibility an the ue flexibility. It i aomplihe though the appliation of element whih euie total employee involvement an team-wok. key philoophy of JIT i implifiation. The wiepea aoption of jut-in-time JIT inventoy piniple unoubtely make poution opeation moe effiient ot effetive an utome eponive. Companie effetively implementing JIT piniple have ubtantial ompetitive avantage ove ompetito that have not. The tik i figuing out how to apply JIT piniple to gain ompetitive avantage in you peifi inuty an buine ituation. The bai pemie of JIT i to have jut the ight amount of inventoy whethe aw mateial o finihe goo available to meet the eman of you poution poe an the eman of you en utome. No moe no le. Thi pape onie a imple an onvenient ituation an eive the minimum optimal olution with eteioating item whih integate inventoy an eminene affimation in a JIT. hotage ae allowe an lea time i zeo. Thi moel wa evelope by Hai. Wilon aoue inteet in the EOQ moel in aaemi an inutie. Late Haley et al analyze many inventoy ytem. In etain ituation unetaintie ae ue to fuzzine pimaily intoue by Zaeh i appliable. In 97 Zaeh et al popoe ome tategie fo eiion making in fuzzy envionment. Jain woke on eiion making in the peene of fuzzy vaiable. Kapyzk et al iue ome long-tem inventoy poliy-making though fuzzy-eiion making moel.wie appliation of fuzzy et theoy an be foun in Zimmeman an Pak. Ugeletti teate EOQ moel in fuzzy ene an ue tiangula fuzzy numbe. Chan an Wang ue tapezoial fuzzy numbe to fuzzify the oe ot inventoy ot an bakoe ot in the total ot of inventoy moel without bakoe. Vujoevi et al ue tapezoial fuzzy numbe to fuzzify the oe ot in the total ot of inventoy moel with bakoe. Futhe in a eie of pape Yao Coeponing autho

2 JYNTHI ND MGTHM: FUZZY INVENTOY MODEL FO DETEIOTION ITEM THOUGH JUT et al. oniee the fuzzifie poblem fo the inventoy with o without bakoe moel. ye & ziz ue tapezoial fuzzy numbe. De an awat popoe an EOQ moel without hotage ot by uing tiangula fuzzy numbe. The total ot ha been ompute by uing igne itane metho. In Dutta an Pavan Kuma eive a fuzzy inventoy moel without hotage uing tapezoial fuzzy numbe with enitivity analyi. Hee we evelop a fuzzy inventoy moel with hotage uing tapezoial fuzzy numbe with enitivity analyi uing the igne itane metho fo efuzzifiation. The annual total ot i alulate a a funtion of five vaiable ie holing ot et up ot hotage ot eening ot an ewoking ot. METHODOLOGY Fuzzy Numbe ny fuzzy ubet of the eal line whoe membehip funtion µ atifie the following onition i a genealize fuzzy numbe. i µ i a ontinuou mapping fom to the loe inteval ii iii iv v vi [ ]. µ < x a µ Lx i titly ineaing on [a a ] µ w a x a µ x i titly eeaing on [a a ] µ x < a whee w an a a a an a ae < eal numbe. lo thi type of genealize fuzzy a a a a : ; w numbe be enote a L When w it an be implifie a a a a a L Tapezoial Fuzzy Numbe tapezoial fuzzy numbe epeente with membehip funtion a b i a: Figue : Tapezoial Fuzzy Numbe The Funtion Piniple uppoe ae two tapezoial fuzzy numbe then aithmetial opeation ae efine a: igne Ditane Metho Let be a fuzzy et efine on. Then the igne itane of i efine a: whee i ut of fuzzy et [a b a ] [ ] whih i a loe inteval. NOTTION ND UMPTION The mathematial moel of thi pape i evelope on the bai of the following notation an aumption. Notation : holing ot pe unit uantity pe unit time : et up o oeing ot pe oe : hotage ot o tok out ot pe unit uantity pe unit Inian J.i.e. : 6-7

3 JYNTHI ND MGTHM: FUZZY INVENTOY MODEL FO DETEIOTION ITEM THOUGH JUT time : oe uantity pe yle t heuling time peio z : oe level total eman ove the planning time peio [ t] : eening ot pe unit β : ewoking ot pe unit θ : peentage of efetive item TC total annual ot fo the peio[ t] : fuzzy holing ot pe unit uantity pe unit time : fuzzy hotage ot pe unit uantity pe unit time : fuzzy et up o oeing ot pe oe fuzzy total ot fo the peio [ t] F e-fuzzifye total ot fo [ t] F mimimum e-fuzzifye total ot fo [ t] : optimal oe uantity umption Deman ate i unifom an finite. Lea time i zeo. hotage ae allowe. Inventoy i ontinuouly eviewe. eening ot an ewoking ot i ontant. Holing ot etup ot an hotage ot ae taken a a tapezoial fuzzy numbe. Only a ingle oe will be poue at the beginning of eah yle an the entie lot i elivee in one bath. be the inventoy aying ot pe unit uantity pe unit time be the hotage ot pe unit uantity pe unit time & be the oeing ot pe oe known an ontant. i the lot-ize pe yle wheea z i the initial inventoy level afte fulfilling the bak-logge uantity of peviou yle an z be the maximum hotage level. t i the yle length o heuling peio. MODEL FOMULTION Popoe Inventoy Moel In Cip ene Fom the above notation an aumption we obtain the total annual ot fo the inventoy moel fo eteioation item though JIT with hotage in ip envionment. The total ot fo the peio t i given by TC aying ot et up ot hotage ot z TC t eening ot ewoking ot t z TC whee tz t z t z an t t t TC Patially iffeentiate euation with epet to t we get TC z t t t z t The optimum an an be obtaine by euating the fit patial eivative w..t to t of TC to zeo. T i.e give t t optimum peio Optimal oe uantity t Minimum total ot 5 6 TC [ ] 7 Diagammati epeentation: Inian J.i.e. : 6-7

4 JYNTHI ND MGTHM: FUZZY INVENTOY MODEL FO DETEIOTION ITEM THOUGH JUT Inian J.i.e. : 6-7 z t Figue : Vaiation of uantity with epet to time t Popoe Inventoy Moel In Fuzzy ene Hee we onie the moel in fuzzy envionment. ine the holing ot et up ot an hotage ot ae fuzzy in natue we epeent them by tapezoial fuzzy numbe. Let : fuzzy aying o holing ot pe unit uantity pe unit time : fuzzy et up o oeing ot pe oe : fuzzy hotage ot o tok out ot pe unit uantity pe unit time The total eman i onie a ontant. Now we fuzzify total ot given in the fuzzy total ot i given by: C T To petain igne itane metho to efuzzify the fuzzy total annual ot an then auie the optimal oe uantity by uing imple alulu tehniue. uppoe an ae fuzzy tapezoial numbe in L fom whee < < < an an ae known poitive numbe. Fom we have: C T θ β 9 θ β TC a b ay Now a b a L an inventoy tok

5 JYNTHI ND MGTHM: FUZZY INVENTOY MODEL FO DETEIOTION ITEM THOUGH JUT Inian J.i.e. : 6-7 Defuzzifying in by uing igne itane metho we have: [ ] TC L L [ ] [ ] F Computation of at whih F i minimum: F i minimum when F an whee > F Now F give the eonomi oe uantity a: lo at we have > F. Thi how that F i minimum at n fom euation we get: 5 NUMEICL EXMPLE Numeial Example in Cip ene The annual eman of an item i. unit / yea. nnual inventoy holing ot i.6 pe unit et up ot i. pe unit an hotage ot i. unit / yea. If thee i % efetive item then the upliate ot fo the efetive item i. / unit an the eening ot i. 5 / unit. Eonomi oe uantity an total annual ot ae etemine. ol: unit / yea. 6/ unit / yea. / unit / yea. / unit / yea θ %. / unit β. 5 / unit Eonomi oe uantity : 5.5 unit Total annual ot: ] [ TC..7

6 JYNTHI ND MGTHM: FUZZY INVENTOY MODEL FO DETEIOTION ITEM THOUGH JUT Numeial Example in Fuzzy ene Let unit / yea. 57 / unit / yea. 6 / unit / yea. 6 9 / unit / yea θ %. / unit β. 5 / unit Eonomi oe uantity : 5.6 unit Total annual ot:. No Deman.. Table : enitivity nalyi Fo Fo NUMEICL EXMPLE Numeial Example in Cip ene ompany ue unit of aw mateial whih ot 5 upee pe unit. Plaing eah oe ot upee aying ot % pe yea of the inventoy an hotage ot upee 5 pe yea. If thee i % efetive item then the upliate ot fo the efetive item i. 5 / unit an the eening ot i. / unit. Eonomi oe uantity an total annual ot ae etemine. ol: unit / yea. 5 X %. / unit / yea. 5 / unit / yea. / unit / yea θ %. 5 / unit β. / unit Eonomi oe uantity : 6.7 unit Total annual ot: TC [ ]. 6. Numeial Example in Fuzzy ene Let unit / yea. 6 9 / unit / yea. 6 9 / unit / yea. 6 9 / unit / yea θ %. 5 / unit β. / unit Eonomi oe uantity : 6.9 unit Total annual ot:. 6. Inian J.i.e. : 6-7

7 JYNTHI ND MGTHM: FUZZY INVENTOY MODEL FO DETEIOTION ITEM THOUGH JUT. No Deman Table : enitivity nalyi Fo Fo CONCLUION In thi pape we evelop a eonomi oe uantity an total annual inventoy ot in the ip ene a well a fuzzy ene. Caying ot et up ot an hotage ot ae taken a a tapezoial fuzzy numbe. Hee we auie the efetive item in tem of peentage; eening an ewoking ot ae taken a ontant. Thi moel i olve analytially though JIT by minimizing the total inventoy ot. Finally the popoe moel ha been veifie by the two numeial example along with the enitivity analyi. In the futue tuy we apply the fuzzy onept fo all poviion in thi antiipate moel. EFEENCE Chan W Bakoe fuzzy inventoy moel une funtion piniple Infomation iene 95- :7-79. Dutta D. an Kuma P.. Fuzzy inventoy moel without hotage uing tapezoial fuzzy numbe with enitivity analyi IO Jounal of Mathemati :-7 IN: Haley G. an Whitin T.M. 96. nalyi of inventoy ytem Pentie-Hall Englewoo lipp NJ 96. Hai F. 95. Opeation an ot W haw Co. Chiago. Hieh C.H.. Optimization of Fuzzy Poution Inventoy Moel Infomation iene 6-:9-. Jain Deiion making in the peene of fuzzy vaiable IIIE Tanation on ytem Man an Cybeneti 7:69-7. Kapyzk J. an taniewki P. 9. Long-tem inventoy poliy-making though fuzzyeiion making moel Fuzzy et an ytem :7-. Kao C.K. an Hu W.K.. ingle-peio inventoy moel with fuzzy eman Compute an Mathemati with ppliation :-. Pak K Fuzzy et Theoetial Intepetation of eonomi oe uantity IEEE Tan. ytem Man. Cybenet MC 7:-. De P.K. an awat.. fuzzy inventoy moel without hotage uing tiangula fuzzy numbe Fuzzy Infomation & Engineeing :59-6.DOI:./ajap.. lak N. Chambe. Halan C. Haion. an Johnton Opeation Management Pitman Publihing Lonon. ye J.K. an ziz L.. 7. Fuzzy inventoy moel without hotage uing igne itane metho pplie Mathemati &Infomation iene :-9. Ugeletti Tinaelli G. 9. Inventoy ontol moel an poblem Euopean Jounal of Opeational eeah :-. Vujoevi M. Petovi D. an Petovi EOQ Fomula when Inventoy Cot i Fuzzy Intenational Jounal of Poution Eonomi 5:99-5. Wilon. 9. ientifi outine fo tok ontol. Hava Buine eview :6. Yao J.. an Chiang J.. Inventoy without bak oe with fuzzy total ot an fuzzy toing ot efuzzifie by entoi an igne itane Euopean Jounal of Opeational eeah :-9. Yao J.. an Lee H.M Fuzzy Inventoy with o without bakoe fo fuzzy oe uantity with tapezoial fuzzy numbe Fuzzy et an ytem 5:- 7. Yao J.. an Lee H.M Eonomi oe uantity in fuzzy ene fo inventoy without bakoe moel Fuzzy et an ytem 5:-. Zaeh L Fuzzy et Infomation Contol :-5. Inian J.i.e. : 6-7

8 JYNTHI ND MGTHM: FUZZY INVENTOY MODEL FO DETEIOTION ITEM THOUGH JUT Zaeh L.. an Bellman.E. 97. Deiion Making in a Fuzzy Envionment Management iene 7:-6. Zimmeman H.J. 9. Uing fuzzy et in opeational eeah Euopean Jounal of Opeational eeah :-6. Inian J.i.e. : 6-7