International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August ISSN
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1 Interntionl Journl of Scientific & Engineering Reerc Volume Iue 8 ugut- 68 ISSN n Inventory Moel wit llowble Sortge Uing rpezoil Fuzzy Number P. Prvti He & ocite Profeor eprtment of Mtemtic ui- E -Millt Government College for Women utonomou Cenni-6 prvtigl@gmil.com S. Gjlkmi Reerc Scolr eprtment of Mtemtic ui- E -Millt Government College for Women utonomou Cenni-6 gjlkmi@gmil.com btrct: In ti pper we invetigte eterminitic inventory moel wit llowble ortge n it i completely bcklogge. Our gol i to etermine te optiml totl cot n optiml orer quntity for te propoe inventory moel. Orering cot oling cot ortge cot re tken trpezoil fuzzy number. Signe itnce meto i ue for efuzzifiction. Senitivity nlyi i crrie out troug te reult of te numericl exmple. Keywor: rpezoil fuzzy number llowble ortge crip fuzzy inventory moel efuzzifiction. Introuction: Inventory control i very importnt fiel for bot rel worl ppliction n lo reerc purpoe. In erlier tge te uncertintie of inventory moel re trete rnomne n re nle by uing probbility teory. e firt quntittive tretment of inventory w te imple EO moel. i moel w evelope by [] [] n tey re invetigte in cemic n inutrie. Lter [5] nlyze mny inventory ytem. In ome itution uncertintie re ue to fuzzine primrily introuce by [] i pplicble. Lter on o mny reercer worke on tee re. Mny ppliction of fuzzy et teory cn be foun in [6]. In EO moel we ve ientifie te orer ize tt minimize te um of nnul totl cot of inventory. u EO moel i ueful pproximtion to mny rel life problem. Urgeletti [7] evelope EO moel in fuzzy nture n ue tringulr fuzzy number. [8] ue trpezoil fuzzy number to fin te totl cot witout bckorer. [9] ue trpezoil fuzzy number to fin te totl cot wit bckorer. en tey foun fuzzy totl cot. In our pper we re eveloping n inventory moel uing trpezoil fuzzy number for oling cot orering cot n ortge cot. Signe itnce meto i ue for efuzzifiction. ue to irregulritie or pyicl propertie of te mteril ll te time we cnnot tke prmeter vrible. For ti itution we pply fuzzy concept. Sortge i llowe n it i completely bcklogge. n lgoritm i evelope to fin te optiml orer quntity n lo for minimizing te totl cot. Senitivity nlyi i crrie out troug te numericl exmple. efinition n Preliminrie: fuzzy et on te given univerl et X i et of orere pir { x µ x : x X } were µ : X [ ] i clle memberip function. e - cut of i efine by { x; µ x }. If R i te rel line ten fuzzy number i fuzzy et wit memberip : X µ ving te following propertie: function [ ] µ i i norml i.e. tere exit x ii i piece-wie continuou x R uc tt iii upp cl x R : µ x > were cl repreent te cloure of { } et iv i convex fuzzy et. rpezoil fuzzy number trpezoil fuzzy number b c i repreente wit memberip function µ : ttp://
2 Interntionl Journl of Scientific & Engineering Reerc Volume Iue 8 ugut- 69 ISSN µ - x L x x b; b I b x c; x x R x c x ; c oterwie n Suppoe B b b b b en i e ition of re two trpezoil fuzzy number. n B i B b b b b n B i B b b b b B b b b b B b b b b K K K K K ifk > K K K K K ifk < ii e multipliction of iii iv v For ny rel number K Signe itnce Meto LR form efuzzifiction of cn be foun by igne fuzzy et i clle in LR- form if tere exit reference function L for left R for rigt n clr m > itnce meto. If i trpezoil fuzzy number ten n n > wit memberip function te igne itnce from to i efine σ x L x σ ; m [ ] µ x σ x γ ; L R x γ were R x γ n were σ i rel number clle te men vlue of [ L R ] m [ b c ] [ ] n n re clle te left n rigt pre repectively. e function L n R mp R [ ] n re ecreing. Nottion LR- type fuzzy number cn be repreente Orering cot per orer H Holing cot per unit quntity per unit time σ γ m n. LR e Function Principle i principle i ue for te opertion for ition ubtrction multipliction n iviion of fuzzy number. Sortge cot per unit quntity S Mximum orer level Lengt of te pln emn wit time perio [ ] Orer quntity per cycle C otl cot for te perio [ ] C - Fuzzy totl cot for te perio[ ] F e-fuzzifie totl cot for [ ] F Minimum e-fuzzifie totl cot for [ ] - Optiml orer quntity umption In ti pper te following umption re coniere: i otl emn i coniere contnt. ii ime of pln i contnt. ttp://
3 Interntionl Journl of Scientific & Engineering Reerc Volume Iue 8 ugut- 7 ISSN iii Sortge i llowe n it i completely bcklogge. iv Only oling cot orering cot n ortge cot re fuzzy in nture. Inventory moel in Crip ene Firt we el n inventory moel wit ortge in crip environment. e economic lot ize i obtine by te following eqution: Mximum orer level i S H H H e totl emn n time of pln re coniere contnt. Now we fuzzify totl cot given in 8 te fuzzy totl cot i given by H C H H Our im i to pply igne itnce meto to efuzzify te totl cot n ten obtin te optiml orer quntity by uing imple clculu tecnique. Suppoe H re trpezoil fuzzy number in LR form were < < H n ll re poitive number ten from we ve e totl cot for te perio [ ] i HS C S H Subtituting 6 in 7 n implifying we get H C H e optimum n C cn be obtine by equting te firt orer prtil erivtive of C to zero n olving te reulting eqution Optiml orer quntity H H C --- By uing ritmetic function principle in n implifying we get C Minimum totl cot H H Inventory moel in Fuzzy ene We conier te moel in fuzzy environment. Since te orering cot oling cot n ortge cot re fuzzy in nture we repreent tem by trpezoil fuzzy number. Let : fuzzy orering cot per orer H : fuzzy crrying or oling cot per unit quntity per unit time : fuzzy ortge cot per unit quntity per unit time ttp://
4 Interntionl Journl of Scientific & Engineering Reerc Volume Iue 8 ugut- 7 ISSN ttp:// C [ b c ] y Now L b n R - c efuzzifying C by uing igne itnce meto we get ] [ H C R L
5 Interntionl Journl of Scientific & Engineering Reerc Volume Iue 8 ugut- 7 ISSN ttp:// Integrting n implifying we get [ ] 8 8 F y Computtion of t wic F i minimum: F i minimum wen F n were > F Now F give te economic orer quntity : [ ] lo t we ve > F i ow tt F i minimum t. n from 6
6 Interntionl Journl of Scientific & Engineering Reerc Volume Iue 8 ugut- 7 ISSN F [ ] lgoritm for fining fuzzy totl cot n fuzzy optiml orer quntity: Step : Clculte totl cot for te crip moel for te given crip vlue of H n. Step : Now etermine fuzzy totl cot uing fuzzy ritmetic opertion on fuzzy oling cot fuzzy orering cot n fuzzy ortge cot tken trpezoil fuzzy number. Step : Ue igne itnce meto for efuzzifiction. Reference: en fin fuzzy optiml orer quntity wic cn be obtine []. Inventory ytem by Eliezer Nor by putting te firt erivtive of F equl to zero n were te econ erivtive i poitive t. Numericl exmple: Exmple : Crip moel: Let R. /- per unit H R. /- per unit 5 unit 6 y R. 6/- per unit. en unit C R..8. Fuzzy moel: Let 5 unit 6 y 5 9 H en unit F R S. N o emn For 5 9 For 5 8 F F From te bove tble we oberve tt: i e economic orer quntity obtine by igne itnce meto i cloer to crip economic orer quntity. ii otl cot obtine by igne itnce meto i le tn crip totl cot. iii For ifferent vlue of orering quntity by cnging mile two pre te economic orer quntity remin fixe. Sme i true for totl cot. Concluion: In ti pper we ve ue igne itnce meto for efuzzifying te oling cot orering cot n ortge cot. ee cot re tken trpezoil fuzzy number. We conclue tt for n EO moel if oling cot orering cot n ortge cot re expree trpezoil fuzzy number ten te reult obtine re muc better tn te ce of tringulr fuzzy number. Finlly we conclue tt even toug we re llowing ortge for n EO moel te totl cot i muc leer tn te moel propoe by []. Numericl exmple re given to illutrte ti moel. [] Fuzzy et n logic by Ze []. Hrri F. Opertion n cot W Sw Co. Cicgo 95. []. Wilon R. cientific routine for tock control. Hrvr Buine review [5]. Hley G. Witin. M. nlyi of inventory ytem Prentice Hll Englewoo ipp NJ 96. [6]. Zimmermn H. J. Uing fuzzy et in opertionl reerc Europen journl of opertionl reerc [7]. Urgeletti inrelli G. Inventory control moel n problem Europen Journl of opertion reerc 98. [8]. Cn Wng Bckorer fuzzy inventory moel uner function principle Informtion cience [9]. Vujoevic M. Petrovic. Petrovic R. EO formul wen inventory cot i Fuzzy Interntionl Journl of prouction Economic [].. utt Pvn kumr Fuzzy inventory moel witout ortge uing rpezoil fuzzy number wit enitivity nlyi. Senitivity nlyi ttp://
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