A Fuzzy Production Inventory System. with Deterioration

Transcription

1 Alie Matematial Sienes, Vol. 5,, no. 5, - A Fuzzy Poution Inventoy System wit Deteioation A. Nagoo Gani PG & Resea Deatment of Matematis Jamal Moame College (Autonomous Tiuiaalli-, Tamilnau, Inia ganijm@yaoo.o.in P. Palaniammal PG & Resea Deatment of Matematis Seetalaksmi Ramaswami College (Autonomous Tiuiaalli-,Tamilnau, Inia. _s@yaoo.om Abstat A stuy is mae on a ontinuous fuzzy oution inventoy moel fo eteioating items witout sotages. Te fomulae fo total inventoy ost of te system e unit time an te otimal inventoy level une fuzzy envionment ae eive. A numeial examle is esente to illustate te oess of obtaining te fuzzy otimal oution lotsize an te fuzzy minimal total inventoy ost. Keywos: Fuzzy oution lotsize, fuzzy eteioation ate, fuzzy otimal ost. INTRODUCTION Reently, te fuzzy analysis of inventoy ontol moels as mu attention towas te maintenane of fuzzy oution inventoies of eteioating items like ugs, foo gains, fuits, vegetables, aio-ative substanes an so on. Stating fom Witin [5] many autos like Gae an Sae [7], Sa an Jaiswal [], Aggawal [], Dave an Patel [5], Goswami an Cauui [8], Misa [9], Deb an Cauui [6], Namias [] an Rafaa [], G.P. Samanta

2 A. Nagoo Gani an P. Palaniammal an A. Roy [] analyse is inventoy moels wit eteioation in iffeent aoaes. Again a two aamete an tee aamete Weibull istibutions wee anle by Covet an Pili [] an again by Pili [] esetively to eesent eteioation time. In te esent ae, we eal wit a ontinuous fuzzy oution inventoy ontol moel fo eteioating items witout sotages in wi te eman ate, oution ate, eteioation ate, oling ost e unit e unit time an eteioation ost ae taezoial fuzzy numbes an te time of eteioation of an item as an exonential istibution. Te main aim of te autos is to estimate te fuzzy otimal oution lotsize an tat of oution ost of an inventoy system une stuy. Fute, gae mean integation eesentation meto is alie fo efuzzifiation of tose fuzzy quantities wi ae involve in tis moel. Finally, a numeial examle to illustate te fuzzy oution inventoy moel wit eteioating items witout sotages is esente.. NOTATIONS, ASSUMPTIONS AND METHODOLOGY Te following notations an assumtions ae mae along wit metoology fo eveloing te fuzzy oution inventoy moel wit eteioating items... Notations % - fuzzy eman ate, % - fuzzy oution ate wee % > %, % - fuzzy oling ost e unit e unit time, % - fuzzy set u ost wi is onstant, s % - fuzzy ost fo eteioate item e unit wee %, %, %, % s an % ae all taezoial fuzzy numbes. T % i - fuzzy total inventoy ost (t % - fuzzy oution lotsize at time t(t.. Assumtions. Relenisment is instantaneous an lea time is zeo.. Te fuzzy oution yle time T % is fixe.. Sotages ae not emitte.. Relaement of te eteioate items is not emitte uing a given yle. 5. Te time of eteioation of an item follows an exonential istibution wose..f. is given by f( t% = -t % e, otewise wee is alle as te eteioation ate su tat < <. 6. Te oution oess of items is state at t% = % an stoe at t% = t%.

3 Fuzzy oution inventoy system 5 7. Te fuzzy oution lotsize attains a level % at time t% = t% 8. Te level of inventoy eeases gaually uing [ t%,t % ] ue to te fulfillment of eman an faing eteioation atly. 9. Te level of inventoy beomes zeo at time t% =T % at wi immeiately te oution oess of items is state gain an te oution yle is eeate afte time T %. Te oution inventoy moel une stuy is eesente by te iagam below. Inventoy O t T t T t T Fig. Continuous Poution Moel wit Deteioation Time.. Vaiation of Constant Fomula Te vaiation of onstant fomula fo solving te fist oe fuzzy iffeential equations is anle by Banabas [] et al an given by te following statement (i If α > ten ( t α ( t Θ t xθ ( t t = e ( α Θ e x t is iffeentiable an it is a solution of te oblem ( % t = - ( % t % %... ( (t % = % wee R, % R TRFN an % % : ( t%, t% R TRFN an (ii if α < an te Hukuaa iffeene t ( xθ t Θ Θ ( α Θ ( e t [ ] x is iffeentiable an it is a solution of te oblem given by (.

4 6 A. Nagoo Gani an P. Palaniammal Tis metoology is anle to solve te fuzzy oution inventoy system, in wi te fuzzy taezoial eal numbes ae use... Fuzzy aitmetial oeations une funtion inile an squae oot Some fuzzy aitmetial oeations une funtion inile ae omonent wise efine as below: If A % = (a, a, a, a an B % = (b, b, b, b ae two taezoial fuzzy numbes, ten ( A % B% = (a + b, a + b, a + b, a + b wee a, a, a, a, b, b, b, b ae any eal numbes. ( A % B% = (,,,, wee T = (a b, a b, a b, a b, T = (a b, a b, a b, a b, = min T, = min T, = max T, = max T If a, a, a, a, b, b, b an b ae all non-zeo ositive eal numbes, ten A % B % = (a b, a b, a b, a b ( B % = (-b, -b, -b, -b an A % B % = (a -b, a -b, a -b, a -b wee a, a, a, a, b, b, b, b ae any eal numbes. - ( = B % =,,, wee b B%, b, b, b ae all ositive eal b b b b numbes. (5 A % B % a a a a =,,, b b b b wee a, a, a, a, b, b, b, b ae all non-zeo ositive eal numbes. (6 If α R, ten (i α Ã = (αα, αα, αα, αα if α (ii α Ã = (αα, αα, αα, αα if α < (7 A % = (a, a, a, a = ( a, a, a, a wee a, a, a, a ae all non-zeo ositive eal numbes..5. Gae Mean Integation Reesentation of a Genealize Fuzzy Numbe Te meto of efuzzifiation of a genealize taezoial fuzzy numbe A = (a, a, a, a by its gae mean integation eesentation was oose by Cen an Hsie [, 5] an is efine by GMI( A = [ (a+ a + (a- a- a + a ] a + a + a + a = 6. FORMULATION AND ANALYSIS OF THE MODEL If (t % enotes te available inventoy at time t% ( % t% T % ten te linea fist oe fuzzy iffeential equations, govening a oution inventoy system

5 Fuzzy oution inventoy system 7 wit eteioating items an witout sotages, ae given by (t % (t % = % t % % if % % t % t (t % = %..... ( (t % (t % = t % % if % t % t T %..... ( wit te bounay onitions ( % = %, (t % = % an (t % = T % = % an te solution is foun as ( Θ [ t e ] ( Θ, if t t (t % = Θ ( t t Θ ( e if t t T Teefoe une fuzzy aitmeti oeations % = (t % leas to te following (i % = ( ( t Θ Θe (ii % t = ø ( Θ ( ø ( Θ (iii T % = ø ( ( [ ( Θ Θ ( ø ( ( Θ Θ ] (iv Te fuzzy total eteioation uing [ %,T % ] is foun an is given by T % = [( ø ( Θ ] (v Te fuzzy eteioation ost ove te eio [ %,T % ] is D % = [( ø ( ø ( Θ + ] wee % enotes te unit fuzzy ost of eteioation. (vi Te fuzzy inventoy aying ost (oling ost ove te yle [ %, T % ] is I% = [( ø( ( Θ + ( ø ( ( Θ ] (vii Teefoe, te total fuzzy inventoy ost of te system e unit time is foun As T % i = ( ø T [( ø ( ( Θ ] ( ø T [( ø ( ( Θ ( ø ø ( ( Θ ] s wi is te sum of te fuzzy eteioating ost e unit time, te fuzzy inventoy oling ost e unit time an te onstant fuzzy setu ost. (viii Te fuzzy otimal inventoy level % wi minimizes te total inventoy ost, T % i = C( %, is obtaine as te solution of te fist oe fuzzy iffeential equation ( T % i = % an it is foun as % % = ( Θ ø ( [ ( Θ ± ( Θ8 ]

6 8 A. Nagoo Gani an P. Palaniammal (ix Hene, te esete fuzzy total minimal ost of te system e unit time is obtaine as T % i = T( % = i øt [( ø ( Θ + ] ( ø T [( ø ( ( Θ ø ( ( Θ ] wee % is ovie by (viii. %, %, %, % enote te taezoial fuzzy numbes efine by If % =( % = (,,,,,, ; (i otimal oution lotsize (ii otimal oution time % t % =(,,, % =(,,, ten ove a yle te fuzzy % (iii yle time T % (iv total otimal eteioation T % (v total otimal inventoy aying ost T % (vi total otimal eteioating ost D % an (vii total otimal oution inventoy ost T % i ae alulate une aitmeti oeations of funtion inile an tei esetive fomulae ae exibite below: (i % = [(,,, ø (,,, an ( ± (,,, ( (( ( +, + +, ( C P +C P C P + C P 8C C, + 8 (( ( ( (ii t% = + + -,, ( ( ( - +, + ( ( ( (iii T % = ( + ( ( -, (

7 Fuzzy oution inventoy system 9 (iv T % = ( + ( ( - ( -, ( ( ( + ( ( - ( -, ( ( ( ( + ( ( - ( - ( ( ( +, +, +, + (v D % = +, + ( - ( - +, + ( ( (vi (vii,, I% = + + ( - ( ( ( ( , + + ( - ( ( ( ( - T % i = + + +, T ( - T ( - - ( + + +, T ( - T ( ( ( , T ( - T ( ( ( - - -

8 A. Nagoo Gani an P. Palaniammal T ( - T ( - - (. EXAMPLE Let =.5, % = (, 5, 5, 6, % = (9, 5, 5, 5, % = (9,,, an % = (9,,, Ten te fuzzy an effuzzifie otimal quantities ae foun as (i % = (6.8, 67., 67., GMIR( % = 67. units (ii t% = (5.985, , , GMIR( % t = units of time. (iii T % = (.6667, , ,.798 GMIR( T % = (iv T % = (8.8599, , , GMIR( T % = Hene te aveage eteioation e yle =.9558 units an te eentage of eteioation of items e yle = % (v D % = (6., , , GMIR( D % = units of ost. (vi I% = ( , 7.66, 7.66,.88 GMIR( I% = units of ost. (vii T % i = ( , , , 8.56 GMIR( T % = i 5. CONCLUSION In tis ae, a ontinuous oution inventoy system wit eteioation is analyse une fuzzy envionment to estimate vaious fuzzy otimal quantities along wit te esetive efuzzifie esults by assuming te eman an oution ates along wit te oling ost an eteioating ost as taezoial fuzzy numbes an te time of eteioation is exonential. It is obseve tat

9 Fuzzy oution inventoy system tese fuzzy estimates eflet bette tan te esetive is estimates of te eal systems. Refeenes [] S.P.Aggawal, A note on an oe-level inventoy moel fo a system wit onstant ate of eteioation. Osea, 5 ( [] S.H. Cen, C.Hsie, Gae mean integation eesentation of a genealize fuzzy numbe, Jounal of Cinese Fuzzy Systems 5( ( [] Ci Hsun Hsie, Otimization of fuzzy oution inventoy moels, Jounal of Infomation Sienes 6 ( 9. [] R.P. Covet, an G.C. Pili, An EO moel fo items wit Weibull istibution eteioation, AilE Tansation, 5 (97-6. [5] U. Dave, an L.K. Patel, (T.S i oliy inventoy moel fo eteioating items wit time ootional eman. Jounal of te Oeational Resea Soiety, ( [6] M. Deb an K.S. Cauui, An EO Moel fo items wit finite ate of oution an vaiable ate of eteioation, Osea, ( [7] P.M. Gae an G.P. Sae, A moel fo exonentially eaying inventoies. Jounal of Inustial Engineeing, ( [8] A. Goswami an K.S. Cauui, An EO moel fo eteioating items wit sotages an a linea ten in eman, Jounal of te Oeational Resea Soiety, ( [9] R.B. Misa, Otimum oution lot-size moel fo a system wit eteioating inventoy, Intenational Jounal of Poution Resea, ( [] S. Namias, Peisable inventoy teoy: A eview, Oeations Resea, ( [] G.C. Pili, A genealize EO moel fo items wit Weibull istibution eteioation, AUE Tansation. 6 ( [] F. Rafaa, Suvey of liteatue on ontinuously eteioating inventoy moel, Jounal of te Oeational Resea Soiety, ( [] Y.K. Sa an M.C. Jaiswal, An oe-level inventoy moel fo a system wit onstant ate of eteioation, Osea, ( [] G.P. Samanta an A. Ajanta Roy,. Poution Inventoy Moel wit eteioating items an sotages, Yugoslav Jounal of Oeations Resea, (, Numbe, 9-. [5] T.M. Witin, Teoy of Inventoy Management, Pineton Univesity Pess, Pineton, NJ,, 957. [6] Zae L.A., Fuzzy sets, Infomation an ontol, 965, 8: Reeive: August,