International Journal of Pure and Applied Sciences and Technology

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1 Int. J. Pue Appl. Sc. Technol., 9( (, pp. -8 Intenatonal Jounal of Pue and Appled Scences and Technology ISSN 9-67 Avalable onlne at Reseach Pape Soluton of a Pobablstc Inventoy Model wth Chance Constants: A Geneal Fuzzy Pogammng and Intutonstc Fuzzy Optmzaton Appoach R Banejee and S Banejee,, Saoj Mohan Insttute of Technology, Hooghly-75 Coespondng autho, e-mal: (m.sb@edffmal.com (Receved: --; Accepted: -- Abstact: Ths pape consdes a pobablstc nventoy model wth unfom leadtme demand and fuzzy cost components unde pobablstc and mpecse constants. Fstly we solve the model by geneal fuzzy non-lnea pogammng technque. Then ntutonstc fuzzy optmzaton technque s appled and fnally, egadng the optmzaton of the objectve functon a compaatve study s pesented among fuzzy optmzaton technque, geneal fuzzy non-lnea pogammng technque and ntutonstc fuzzy optmzaton technque. All the esults and compaatve dscussons ae llustated numecally. Keywods: Geneal fuzzy non-lnea pogammng, Intutonstc fuzzy optmzaton, fuzzy cost components, stochastc nventoy, chance constant.. Intoducton The fst publcaton n fuzzy set theoy by Zadeh [] showed the ntenton to ommodate uncetanty n the non-stochastc sense athe than the pesence of andom vaables. Bellman and Zadeh [] fst ntoduced fuzzy set theoy n decson-mang pocesses. Late Zmmemann [, 4] showed that the classcal algothms could be used to solve a fuzzy lnea pogammng poblem. Fuzzy mathematcal pogammng has been appled to seveal felds le poject netwo, elablty optmzaton, tanspotaton, meda selecton fo advetsng; a polluton egulaton etc. poblems fomulated n fuzzy envonments. Detal lteatue on fuzzy lnea and non-lnea pogammng wth applcaton s avalable n two well-nown boos of Le and Hwang [5, 6].

2 Int. J. Pue Appl. Sc. Technol., 9( (, -8. In most of the exstng nventoy models, t s assumed that the nventoy paametes, objectve goals and constant goals ae detemnstc and fxed. But, f we thn of the pactcal meanng, they ae uncetan, ethe andom o mpecse. When some o all paametes of an optmzaton poblem ae descbed by andom vaables, the poblem s called stochastc o pobablstc pogammng poblem. In a stochastc pogammng poblem, the uncetantes n the paametes ae epesented by pobablty dstbutons. Ths dstbuton s estmated on the bass of the avalable obseved andom data. Abdel-Male, Montana, Moale [7] dscussed Exact, appoxmate, and genec teatve models fo the Newsboy poblem wth budget constant. Jagg and Vema [8] consdeed two-waehouse nventoy model fo deteoatng tems wth lnea tend n demand and shotages unde nflatonay condtons. Haseve and Moussouas [9] dscussed detemnaton of ode quanttes n multpoduct nventoy systems subject to multple constants and ncemental dscounts. Islam and Roy [] consdeed Fuzzy mult-tem economc poducton quantty model unde space constant. Al- Fawzan and Haga [] analyzed an ntegated nventoy-tagetng poblem wth tme-dependent pocess mean. Hala and M. E. EI-Sadan [] consdeed constaned sngle peod stochastc unfom nventoy model wth contnuous dstbutons of demand and vayng holdng cost. The objectve s to fnd the optmal puchase quantty whch mnmzes the expected total cost fo the peod unde a estcton on the expected vayng holdng cost when the demand dung the peod follows the unfom, the exponental dstbutons, usng Lagange multple appoach. L and Mao [] developed an nventoy model of peshable tem wth two types of etales. Ouyang and Chang [4] attempted to apply the fuzzy set concepts to deal wth the ambguous lost sales ate. They dscussed a mnmax dstbuton fee pocedue fo mxed nventoy models nvolvng vaable lead tme wth fuzzy lost sales. They beleved that f the fuzzy lost sales ate s expessed as fuzzy lost sales ate as the neghbohood of the fxed lost sales ate, and then t may match the eal stuaton bette. Tadtonally EOQ models dealng wth contnuous evew nventoy poblems often assume that demand dung the stoc out peod s ethe completely bacodeed o completely lost, and the lead tme s vewed as a pescbed constant o a andom vaable, whch theefoe s not subject to contol. Howeve thee ae not always tue: fo example, n eal maets, t can be obseved that, when the nventoy system s out of stoc, some of the customes may be wllng to wat fo the demands, whle othes may fll the demand fom anothe souce. Hence many eseaches extended the contnuous evew nventoy models to nclude the patal bacode stuaton. Then the total expected poft functon also becomes a fuzzy numbe. L, Kabad and Na [5] dscussed fuzzy models fo sngle- peod nventoy poblem. They consdeed the sngle peod nventoy poblem n the pesence of uncetantes. Two types of uncetantes, one asng fom andomness, whch can be ncopoated though a pobablty dstbuton and the othe fom fuzzness, whch can be chaactezed by fuzzy numbes, ae consdeed. Kao and Hsu [6] developed a sngle peod nventoy model wth fuzzy demand. Islam and Roy [7] dscussed a fuzzy mult-objectve nventoy model wth vaable deteoaton ate. Mandal and Roy [8] dscussed a mult-tem fuzzy nventoy poblem wth space constant va geometc pogammng method. Chou [9] developed a fuzzy bacode nventoy model and applcaton to detemnng the optmal empty-contane quantty at a pot. Chou [] also dscussed a sngle peod nventoy model wth fuzzy demand. Panda and Ka [] analyzed mult tem stochastc and fuzzy stochastc nventoy models unde mpecse goal and chance constants. The fuzzy-pobablstc pogammng poblem s fst educed to a coespondng equvalent fuzzy non-lnea pogammng poblem. Both the poblems ae solved by fuzzy non-lnea pogammng technques. Das, Roy and Mat [] developed a buye-selle fuzzy nventoy model fo a deteoatng tem wth dscount. They [] also dscussed a mult-tem stochastc and fuzzystochastc nventoy models unde two estctons. Hde and Hoa [4] solved some nventoy poblems wth fuzzy shotage cost. Lang-Yuh and Hung-Ch [5] descbed a mnmax dstbuton fee pocedue fo mxed nventoy models nvolvng vaable lead tme wth fuzzy lost sales. Consdeng the fuzzy demand a sngle peod nventoy model s dscussed by Chang and When-Ka [6]. Mahapata, G.S. and Roy, T.K. [7] used Geneal Fuzzy Pogammng technque on a elablty optmzaton model. Intutonstc Fuzzy Set (IFS was ntoduced by K. Atanassov [8] and seems to be applcable to eal wold poblems. The concept of IFS can be vewed as an altenatve appoach to defne a fuzzy set n case whee avalable nfomaton s not suffcent fo the defnton of an mpecse concept by means of a conventonal fuzzy set. Thus t s expected that, IFS can be used to smulate human decson-

3 Int. J. Pue Appl. Sc. Technol., 9( (, -8. mang pocess and any actvttes equng human expetse and nowledge that ae nevtably mpecse o totally elable. Hee the degee of ecton and satsfacton ae consdeed so that the sum of both values s always less than unty. Atanossov also analyzed Intutonstc fuzzy sets n a moe explct way. Atanassov[9] dscussed an Open poblems n ntutonstc fuzzy sets theoy. An Inteval valued ntutonstc fuzzy sets was analyzed by Atanassov and Gagov[]. Atanassov and Kenovch[] mplemented Intutonstc fuzzy ntepetaton of nteval data. The tempoal ntutonstc fuzzy sets ae dscussed also by Atanossov[]. Intutonstc fuzzy soft sets ae consdeed by Maj Bswas and Roy[]. Angelov [4] mplemented the Optmzaton n an ntutonstc fuzzy envonment. He [5, 6] also contbuted n hs anothe two mpotant papes, based on Intutonstc fuzzy optmzaton. Paman and Roy [7] solved a vecto optmzaton poblem usng an Intutonstc Fuzzy goal pogammng. A tanspotaton model s solved by Jana and Roy [8] usng mult-objectve ntutonstc fuzzy lnea pogammng. Banejee and Roy [9] consdeed applcaton of the Intutonstc Fuzzy Optmzaton n the Constaned Mult-Objectve Stochastc Inventoy Model. Anothe mult-objectve stochastc nventoy model wth fuzzy cost components was solved by Intutonstc Fuzzy Optmzaton technque by Banejee and Roy [4].Banejee and Roy [4] also, dscussed a stochastc nventoy model solvng t fst tme by ntutonstc fuzzy geometc pogammng technque. In ths pape a fxed eode quantty system wth bacode and unfom lead-tme demand s solved by geneal fuzzy non-lnea pogammng technque [GFNLP]. The model wth puely pobablstc, puely fuzzy and the model wth mxed envonment s analyzed wth the pope assgnment of seveal weghts n fuzzy non-lnea pogammng. Then agan ths fxed eode quantty system wth lost sales s solved by geneal fuzzy non-lnea pogammng technque as well as ntutonstc fuzzy optmzaton technque [IFO]. Although, geneal fuzzy non-lnea pogammng technque optmzes the aveage annual cost moe than usual fuzzy non-lnea pogammng technque [FNLP] but IFO technque gves us the lowest value of the aveage annual cost n compason to FNLP and GFNLP technques.. Fxed Reode Quantty System wth Bacode Hee the polcy s to ode a lot sze Q when the nventoy level dops to a eode pont ant t s supposed that the nventoy poston of an tem s montoed afte evey tansacton. The demand n any gven nteval of tme s a andom vaable and the expected value of demand n a unt of tme, say a yea, s D. We let x denote the demand dung the lead tme and f(x denote ts pobablty dstbuton. The fxed pocuement cost s K and the unt vaable pocuement cost s C. The cost of cayng a unt of nventoy fo one unt of tme s h. All shotages ae bacodeed at a cost of π pe unt shot, egadless of the duaton of the shotage. Because of the pobablstc natue of demand, the numbe of cycles pe yea s a andom vaable that aveages D/Q. The pocuement cost pe cycle s KCQ. The shotages cost pe cycle s π b (, whee b ( s the expected numbe of shotages pe cycle and s a functon of eode pont. The amount of the shotage at the end of a cycle, when the eplenshment ode s eceved, s b(x, max [, x - ], whch has the expected value b ( ( x f ( x dx µ s the expected demand dung a lead tme. The quantty Q/ s often called the cycle stoc and - µ s efeed to as the safety stoc. Thus safety stoc fo the system s the amount by whch the eode pont exceeds the aveage usage dung a lead-tme. The aveage annual cost s: ( Q, KD Q πdb ( CD h( µ Q Q

4 Int. J. Pue Appl. Sc. Technol., 9( (, -8.. Mult Objectve Stochastc Inventoy Model [MOSIM] wth Fuzzy Cost Components Tadtonal sngle objectve lnea o non-lnea pogammng poblem ams at optmzaton of the pefomance n tems of combnaton of esouces. In ealty, a manageal poblem of a esponsble oganzaton nvolves seveal conflctng objectves to be acheved smultaneously subject to a system of estctons (constants that efes to a stuaton on whch the DM has no contol. Fo ths pupose a latest tool s lnea o non-lnea pogammng poblem wth multple conflctng objectves. Hee B and F ae espectvely the total budget and total floo space aea. So the followng model may be consdeed:.. Model wth Puely Pobablstc Envonment Ths s a model wth pobablstc budget and floo space aea. Mnmze total annual cost of not only the fst objectve, but also the second objectve but also up to the nth objectve. It s a Mult- Objectve Stochastc Inventoy Model [MOSIM]. To solve the poblem (.. as a MOSIM, t can be efomulated as: K D Q π D Mn ( Q, ( CD h ( µ ( b (.(.. Q Q subject to the constants pˆ Q Bˆ fˆ Q Fˆ Q, > [,,., n] (Hee ^ ndcates andomzaton of the paametes. Hee, h ( h, h, h ; π ( π, π, π ; C ( C, C, C.. Model wth Puely Impecse Envonment ae tangula fuzzy numbes. Ths s a model wth fuzzy budget and floo space aea. Thus the model changes to: K D Q π D Mn ( Q, ( CD h ( µ ( b (.(.. Q Q subject to the constants Poss[ fq F ] α Poss p Q B α Q, [ ] >,(,,.., n < α, α < (Hee wavy ba ndcates fuzzfcaton of the paametes... Model wth Mxed Envonment Ths s a model wth pobablstc budget and fuzzy floo space aea.e. the model s analyzed n a mxed envonment and thus educes to: K D Q π D Mn ( Q, ( CD h ( µ ( b (.(.. Q Q subject to the constants pˆ Q Bˆ

5 Int. J. Pue Appl. Sc. Technol., 9( (, Poss [ f Q F] α Q, > [,,., n] < α <. Fxed Reode Quantty System wth Lost Sales Hee we consde the fxed eode quantty polcy when all shotages ae lost. We mae the same assumptons as n the bacodes case, except that the shotage cost π now ncludes the lost poft of an tem. The pocuement cost pe cycle s KCQ. The shotages cost pe cycle s π b (, whee b ( s the expected numbe of shotages pe cycle and s a functon of eode pont. In the lost sales case, the net nventoy and the on-hand nventoy ae the same. The on-hand nventoy at the end of a cycle, mmedately po to a ecept of a lot, s a (x, max (, x, whee x s the lead-tme demand. The expected on-hand nventoy at the end of a cycle s: a( ( x f ( x dx ( x f ( x dx ( x f ( x µ b ( The aveage annual cost s: KD ( Q, ( Q πd CD h µ ( h b ( Q Q. Mult Objectve Stochastc Inventoy Model [MOSIM] wth Fuzzy Cost Components.. Model wth Puely Pobablstc Envonment To solve the poblem (.. as a MOSIM, t can be efomulated as: K D Q π D Mn ( Q, ( CD h ( µ ( h b (.(.. Q Q subject to the constants pˆ Q Bˆ fˆ Q Fˆ Q, > [,,., n] (Hee ^ ndcates andomzaton of the paametes. Hee, h ( h, h, h ; π ( π, π, π ; C ( C, C, C.. Model wth Puely Impecse Envonment ae tangula fuzzy numbes. K D Q π D Mn ( Q, ( CD h ( µ ( h b (.(.. Q Q subject to the constants

6 Int. J. Pue Appl. Sc. Technol., 9( (, Poss Poss Q, [ fq F ] [ p Q B] α >,(,,.., n α < α, α < (Hee wavy ba ndcates fuzzfcaton of the paametes... Model wth Mxed Envonment K D Q π D Mn ( Q, ( CD h ( µ ( h b (.(.. Q Q subject to the constants pˆ Q Bˆ Poss f Q F α [ ] Q, > [,,., n] < α < 4. Few Stochastc Models 4. Demand follows Unfom Dstbuton We assume that demand fo the peod fo the th tem s a andom vaable whch follows unfom dstbuton and f the decson mae feels that demand values fo tem below a o above b ae hghly unlely and values between a and b ae equally lely, then the pobablty densty functon f (x ae gven by: f a x b b a f (x fo,,, n. othewse ( b So, b ( fo,,, n ( b a Whee, b ( ae the expected numbe of shotages pe cycle and all these values of b ( affects all the fou models of fxed eode quantty system wth bacode and lost sales espectvely. 5. Mathematcal Analyss 5. Geneal Fuzzy Non-lnea Pogammng (GFNLP Technque to Solve Mult-Objectve Non-Lnea Pogammng Poblem (MONLP A Mult-Objectve Non-Lnea Pogammng (MONLP o Vecto Mnmzaton poblem (VMP may be taen n the followng fom: Mn f(x [f (x, f (x, f (x,., f (x] T Subject to x є X {x є R n : g j (x o o b j fo j,.., m }.(5.. and l x u (,,., n. Zmmemann [4] showed that fuzzy pogammng technque can be used to solve the mult-objectve pogammng poblem. To solve MONLP poblem, followng steps ae used:

7 Int. J. Pue Appl. Sc. Technol., 9( (, STEP : Solve the MONLP (5.. as a sngle objectve non-lnea pogammng poblem usng only one objectve at a tme and gnong the othes, these solutons ae nown as deal soluton. STEP : Fom the esult of step, detemne the coespondng values fo evey objectve at each soluton deved. Wth the values of all objectves at each deal soluton, pay-off matx can be fomulated as follows: x x x f ( x f ( x f ( x f ( x f( x f( x f ( x f ( x f ( x f ( x f ( x f ( x Hee x, x,., x ae the deal solutons of the objectve functons f (x, f (x,..,f (x espectvely. So U max {f (x, f (x,.,f (x } and L mn {f (x, f (x,.,f (x } [L and U be lowe and uppe bounds of the th objectve functons f (x fo,,,]. STEP : Usng aspaton level of each objectve of the MONLP (5.. may be wtten as follows: Fnd x so as to satsfy f (x L ( fo,,., x ε X Hee objectve functons of (5.. ae consdeed as fuzzy constants. These type of fuzzy constants can be quantfed by elctng a coespondng membeshp functon: w µ [f (x] o f f (x U w µ (f (x f L f (x U (,,,.(5.. w f f (x L Hee w ae the weghts µ (f (x s a stctly monotonc deceasng functon wth espect to f (x (,,,. w Havng elcted the membeshp functons (as n (5.. µ [f (x] fo,,,, ntoduce a geneal aggegaton functon w w w ( D ( µ x G µ ( f ( x, µ ( f ( x,..., µ ( f ( x. So a fuzzy mult-objectve decson mang poblem can be defned as w Max µ ( x D subject to x є X. w.(5.. Hee we adopt Fuzzy decson based on mnmum opeato (le Zmmemann s appoach []. In ths case (5.. s nown as FNLP M. Then the poblem (5.., usng the membeshp functon as n (5.., odng to mn-opeato s educed to: Max α.(5..4 Subject to wµ [f (x] α fo,,.., x є X, α ε [, w], w ε (,] w mn (w, w,, w STEP 4: Solve (5..4 to get optmal soluton.

8 Int. J. Pue Appl. Sc. Technol., 9( (, Fomulaton of Intutonstc Fuzzy Optmzaton [IFO] When the degee of ecton (non-membeshp s defned smultaneously wth degee of eptance (membeshp of the objectves and when both of these degees ae not complementay to each othe, then IF sets can be used as a moe geneal tool fo descbng uncetanty. To maxmze the degee of eptance of IF objectves and constants and to mnmze the degee of ecton of IF objectves and constants, we can wte: max µ ( X, XεR,,,..,K n mn υ ( X, XεR,,,.., K n Subject to υ ( X, µ ( X υ ( X µ ( X υ ( X < X Whee µ (X denotes the degee of membeshp functon of (X to the denotes the degee of non-membeshp (ecton of (X fom the th IF sets. 5. Stochastc Model wth Pobablstc and Fuzzy Constants A stochastc non-lnea pogammng poblem s consdeed as: Mn f (X Subject to f j(x c j (j,,..,m X. th IF sets and ν (X.e Mn f (X.(5.. Subject to f j (X (j,,..,m X. Whee, f j (X f j(x - c j Hee X s a vecto of N andom vaables y, y,.,y n and t ncludes the decson vaables x, x,.,x n. Expandng the objectve functon f (X about the mean value y of y and neglectng the hghe ode tem: N f f X f X X ( ( ( y y y ξ (X (say.(5.. If y (,,.,n follow nomal dstbuton then so does ξ (X. The mean and vaance of ξ (X ae gven by: ξ ξ (X.(5.. N f σ ξ X σ y.(5..4 y When some of the paametes of the constants ae andom n natue then the constants wll be pobablstc and thus, the constants of (5. can be wtten as: P( f (j,,,m.(5..5 j j

9 Int. J. Pue Appl. Sc. Technol., 9( (, Then n the lght of the theoetcal conventon gven above, equvalent detemnstc constants ae: / N f j f ( j φ j j X σ y (j,,,m.(5..6 y whee, φ s the value of the standad nomal vaate coespondng to the pobablty j. j ( j When some of the paametes of the constants ae fuzzy then the constants wll be mpecse and thus, we ae to consde the followng theoem: Theoem Let X : Γ R be a nomal fuzzy vaable wth paametes (a, b. Fo a chosen confdence level α, ε α f [Poss(X x] α then, x ε [ X L α, X U α ] Whee, X L α a - b logα, X U α a b logα. Poof: Fom defnton, µ X (x σ[ X - (x], x ε R. Now, [Poss(X x] α µ X (x α when, X N (a, b µ X (x exp(-((x-a/b, - < X <. Theefoe, x a b log α x a logα logα b a b logα x a b logα If the fuzzy constant s of the fom: n Poss A x b j j α, j,,..,j...(5..7 j Then, we defne J nomal fuzzy vaables as follows: Y n j A x j b j, j,,..,j whee A j and b j ae mutually mn-elated nomal fuzzy vaables and ( Y, Y N m j d j. Y j So, the fuzzy constant (5..7 changes to: α, j. Y Poss [ j ] j Hence, fom the above Theoem, we have J pas of equvalent csp constants as follows: Y m Y j d j logα, j Y m Y j d j logα, j.(5..8 j

10 Int. J. Pue Appl. Sc. Technol., 9( (, Stochastc Models wth Fuzzy Cost Components Stochastc non-lnea pogammng poblem wth fuzzy objectve coeffcent consdes as: MnZ CX...(5.4. X Hee C epesents a vecto of fuzzy paametes nvolved n the objectve functon Z. We assume C ( c, c, c, whch s a tangula fuzzy numbe wth membeshp functon: µ ( t C t c c c c t c c fo c fo c fo t > c t c t c o t < c So equaton (5.4. becomes: MnZ ( c X, c X, c X X Whee, c c, c,, c c ( c (, c,, n c n -...( (5.4. c ( c, c,, c n Accodng to Kaufman and Gupta [4] by combnng thee objectves nto a sngle objectve functon, equaton (5.4. can be educed to a LPP by most lely ctea as: c 4c c Mn X ( Weghts n FNLP Hee, postve weghts W eflect the decson mae s pefeences egadng the elatve mpotance of each objectve goal f (x fo,,..,. These weghts can be nomalzed by tang W. To acheve the same objectve, sutable nvese weghts ae assgned to dffeent membeshp functons n the fuzzy non- lnea pogammng FNLP method. As thee s no one-to-one coespondence between the dect and nvese weghts untl they ae equal. So ntoducng nomalzed weghts n FNLP, ((5.4. become, Max α subject to W µ w (f (x α fo,,, ; x є X, α ε [, w], w ε (,] w mn (w, w,, w whee W, < W < (fo,,,

11 Int. J. Pue Appl. Sc. Technol., 9( (, Soluton of the Stochastc Model 6. Soluton of the Stochastc Inventoy Model by Geneal Fuzzy Non- Lnea Pogammng Technque To solve MOSIM of secton. o secton., step of secton 5. s used. Afte that odng to step Pay-off matx s fomulated as follows: ( Q, ( Q, ( Q, Hee,, and ae the functons of ( Q, Q, Q,,, Now, U, L (whee L deal solutons of the objectve functons (, Q, Q,,, (, Q, Q,, Q U ae dentfed and Q [fo,, ], ( Q, ae the Hee, fo smplcty, fuzzy lnea membeshp functons w µ (, Q, Q,,, Q fo the objectve functons ae dentfed as follows: w µ (, Q, Q,,, Q U w ( w [Fo,, ] - ( Q, Q, Q U L,,, fo fo L ( Q, Q fo ( Q, Q, Q,,, L ( Q, Q, Q,,, U, Q,,, U Accodng to step of atcle 5. havng elcted the above membeshp functons csp lnea pogammng poblems ae fomulated as follows: (followng to (5..4 Max α (6.. subject to w µ (, Q, Q,,, Q > α α ε [, w ], w ε (,] w mn ( w, w, w Q, >. [,,] [Constants of the espectve models ae also to be consdeed]

12 Int. J. Pue Appl. Sc. Technol., 9( (, -8. O, usng postve weghts W fo the th objectve f (x : Max α subject to (6.. w W µ (, Q, Q,, Q > α, α ε [, w ], w ε (,] w mn ( w, w, w Q, >. [,,] Whee, W, < W < (fo,, [Constants of the espectve models ae also to be consdeed] Solve the csp non-lnea pogammng poblem (6.. by an appopate mathematcal pogammng algothm. 6. An Intutonstc Fuzzy Appoach fo Solvng MOSIM wth Lnea Membeshp and Non-Membeshp Functons To defne the membeshp functon of MOIM poblem, let L and U be the lowe and uppe th bounds of the objectve functon. These values ae detemned as follows: Calculate the ndvdual mnmum value of each objectve functon as a sngle objectve IP subject to the gven set of constants. Let X, X,.. X be the espectve optmal soluton fo the dffeent objectve and evaluate each objectve functon at all these optmal soluton. It s assumed hee that at least two of th these solutons ae dffeent fo whch the objectve functon has dffeent bounded values. Fo each objectve, fnd lowe bound (mnmum value L and the uppe bound (maxmum value U. But n ntutonstc fuzzy optmzaton (IFO, the degee of ecton (non-membeshp and degee of eptance (membeshp ae consdeed so that the sum of both values s less than one. To defne membeshp functon of MOIM poblem, let L and be the lowe and uppe bound of L L U the objectve functon Z (X whee U U. These values ae defned as follows: The lnea membeshp functon fo the objectve Z (X s defned as: f Z ( X L U Z ( X µ ( Z ( X f L Z ( X U. (6.. U L f Z ( X U f Z ( X U Z ( X L ν ( Z ( X f L Z ( X U. (6.. U L f Z ( X L

13 Int. J. Pue Appl. Sc. Technol., 9( (, -8. µ,ν µ ( Z (X ν ( Z (X L L U U Z (X Membeshp and non-membeshp functons of the objectve goal Fgue- Lemma: In case of mnmzaton poblem, the lowe bound fo non-membeshp functon (ecton s always geate than that of the membeshp functon (eptance. Now, we tae new lowe and uppe bound fo the non-membeshp functon as follows: L L t( U L whee < t < U U t( U L fo t Followng the fuzzy decson of Bellman-Zadeh [] togethe wth lnea membeshp functon and non-membeshp functons of (6.. and (6.., an ntutonstc fuzzy optmzaton model of MOIM poblem can be wtten as: max µ ( X, XεR,,,..,K mn υ ( X, XεR,,,..,K (6.. Subject to υ ( X, µ ( X υ ( X µ ( X υ ( X < X The poblem of equaton (6.. can be educed followng Angelov [4] to the followng fom: Max α β.(6..4 Subject to Z ( X U α ( U L Z ( X L β ( U β α β α β < L X Then the soluton of the MOIM poblem s summazed n the followng steps:

14 Int. J. Pue Appl. Sc. Technol., 9( (, -8. Step. Pc the fst objectve functon and solve t as a sngle objectve IP subject to the constant, contnue the pocess K-tmes fo K dffeent objectve functons. If all the solutons (.e. X X.. X (,,., m; j,,..,n same, then one of them s the optmal compomse soluton and go to step 6. Othewse go to step. Howeve, ths aely happens due to the conflctng objectve functons. Then the ntutonstc fuzzy goals tae the fom (X L ( X,,..., K., Z Step. To buld membeshp functon, goals and toleances should be detemned at fst. Usng the deal solutons, obtaned n step, we fnd the values of all the objectve functons at each deal soluton and constuct pay off matx as follows: Z( X Z( X Z( X Z ( X Z ( X Z ( X Z Z Z ( X ( X ( X Step. Fom Step, we fnd the uppe and lowe bounds of each objectve fo the degee of eptance and ecton coespondng to the set of solutons as follows: U max( Z ( X and L mn( Z ( X Fo lnea membeshp functons, L L t( U L whee < t < U U t( U L fo t Step 4. Constuct the fuzzy pogammng poblem of equaton (6.. and fnd ts equvalent LP poblem of equaton (6..4. Step 5. Solve equaton (6..4 by usng appopate mathematcal pogammng algothm to get an optmal soluton and evaluate the K objectve functons at these optmal compomse solutons Step 6. STOP. 7. Numecal Examples To solve the model (.., (.. and (.. by Geneal Fuzzy Non-Lnea Pogammng [GFNLP] Technque when pobablty densty functon of demand dung lead-tme follows unfom dstbuton, we consde the followng data: K $;D 5; C $(,5,8; h $(,,4;µ (a b /; π $(,4,6;b 7;a 5; K $;D 6; C $(,6,9; h $(,4,5; µ (a b /; π $(4,6,8;b 8;a 6; K $;D 6; C $(,5,8; h $(,4,6; µ (a b /; π $(,4,6;b 7;a 6; Fˆ (,; F (,; Bˆ $(5,5; B $(5,5;α.5;α.5; w.6; w.7; ˆp $(,.; ˆp $(,.; ˆp $(4,.4; p $(,.;

15 Int. J. Pue Appl. Sc. Technol., 9( (, p $(,.; p $(,.; ˆf (,.; ˆf (4,.4; ˆf (5,.5;.95; f (,.; f (4,.4; f (,., w.6,.65,.7. w w Hee, Fˆ, Bˆ, pˆ, fˆ [,, ] ae stochastc nomal vaables and F, B, p, f [,, ] ae fuzzy nomal vaables whee the fst and second components of the vaables ndcate espectvely the mean and standad devaton of the nomal vaables. Now, we apply secton 5., 5., 5.4, 5.5 and secton 6. and thus the followng esults ae obtaned n TABLE-, TABLE- and n TABLE-: TABLE-: GFNLP technque to solve the model (.. wth equal and unequal weghts Q Q Q ASPIRATION Weghts ($ ($ ($ LEVEL (W,W,W α.7 (/,/,/ α.65 (.6,., α.66 (.,.6, α.6 (.,.,.6 TABLE-: GFNLP technque to solve the model (.. wth equal and unequal weghts Q Q Q ASPIRATION Weghts ($ ($ ($ LEVEL (W,W,W α.7 (/,/,/ α.69 (.6,., α.6 (.,.6, α.67 (.,.,.6 TABLE-: GFNLP technque to solve the model (.. wth equal and unequal weghts Q Q Q ASPIRATION Weghts ($ ($ ($ LEVEL (W,W,W α.7 (/,/,/ α.65 (.6,., α.66 (.,.6, α.6 (.,.,.6 Now, to solve the model (.., (.. and (.., we apply secton 5 and secton 6 and thus usng also the deved set of data gven below, the followng esults ae obtaned n TABLE-4, TABLE-5 and n TABLE-6: U U $5987.4, L $587.87, L $5, U U $664.98, L $6.67, L $6, U U $744.6, L $8.4, L $8. It s to be noted that Fuzzy Non-Lnea Pogammng [FNLP] Technque can be consdeed as a patcula case of Geneal Fuzzy Non-Lnea Pogammng [GFNLP] Technque when w w w.

16 Int. J. Pue Appl. Sc. Technol., 9( (, TABLE-4: Soluton of the model (.. Methods α β ($ ($ ($ FNLP GFNLP IFO In addton to the above set of data we also have to consde the followng deved data to obtan TABLE-5 U U $584.56, L $5.87, L $5, U U $674.78, L $64.47, L $65, U U $94.75, L $79., L $8. TABLE-5: Soluton of the model (.. Methods α β ($ ($ ($ FNLP GFNLP IFO Agan, n addton to the above set of data we also have to consde the followng deved data to obtan TABLE-6 U U $59.4, L $576.6, L $58, U U $685., L $6., L $64, U U $878.4, L $784.9, L $9. TABLE-6: Soluton of the model (.. Methods α β ($ ($ ($ FNLP GFNLP IFO If we consde TABLE- sepaately, then t s obseved that when W.6.e. moe mpotance s gven to the objectve functon lowest value $4855 s obtaned n ths case n compason to othe thee cases and smla esults ae also obtaned fo othe two objectve functons and. Natues of esults eman unalteed fo the models (.. and (.. n TABLE- and TABLE-.. In TABLE-4, TABLE-5 and TABLE-6, GFNLP technque mnmzes the aveage annual cost [,, ] moe than FNLP Technque. Wheeas, applyng IFO technque, the moe mnmzed value of the aveage annual cost [,, ] s obtaned n compason to GFNLP technque. Hence, IFO technque s the best one to optmze an objectve functon.

17 Int. J. Pue Appl. Sc. Technol., 9( (, Conclusons and Futue Scope of Reseach Objectve of ths pape s to analyze a stochastc nventoy model by GFNLP technque athe than usual FNLP technque. We also establsh the fact that although GFNLP optmze the objectve functon moe than the FNLP technque but f we consde the pefomance egadng the optmzaton of the objectve functon (Hee, aveage annual cost, IFO technque s the best one. Instead of unfom lead-tme demand, the model can be solved also when the lead-tme demand follows nomal o exponental dstbutons. The concept of fuzzy pobablty can also be used n ths model as t was done n the dffeent stochastc nventoy model by Banejee and Roy [4]. The moe mpoved soluton may be obtaned f fuzzy geometc pogammng technque s appled as Banejee and Roy [44] appled t n the dffeent model. Refeences [] L.A. Zadeh, Fuzzy sets, Infomaton and Contol, 8(965, [] R.E. Bellman and L.A. Zadeh, Decson-mang n a fuzzy envonment, Management Scence, (97, 7B4-B64. [] H.J. Zmmeman, Descpton and optmzaton of fuzzy system, Intenatonal Jounal of Geneal Systems, (976, 9-5. [4] H.Z. Zmmemann, Fuzzy lnea pogammng wth seveal objectve functons, Fuzzy Sets and Systems, (978, [5] Y.J. La and C.L. Hwang, Fuzzy Mathematcal Pogammng: Methods and Applcatons, Spenge-Velag, Hedelbeg, 99. [6] Y.J. La and C.L. Hwang, Fuzzy Multple Objectve Decson Mang, Spenge-Velag, Hedelbeg, 994. [7] L. Abdel-Male, R. Montana and L.C. Moales, Exact, appoxmate, and genec teatve models fo the Newsboy poblem wth budget constant, Intenatonal Jounal of Poducton Economcs, 9 (4, [8] C.K. Jagg and P. Vema, Two-waehouse nventoy model fo deteoatng tems wth lnea tend n demand and shotages unde nflatonay condtons, Intenatonal Jounal of Pocuement Management, ( (, [9] C. Haseve and J. Moussouas, Detemnng ode quanttes n mult-poduct nventoy systems subject to multple constants and ncemental dscounts, Euopean Jounal of Opeatonal Reseach, 84(8, [] S. Islam and T.K. Roy, Fuzzy mult-tem economc poducton quantty model unde space constant: A geometc pogammng appoach, Appled Mathematcs and Computaton, 84(7, 6-5. [] M.A. Al-Fawzan and M. Haga, An ntegated nventoy-tagetng poblem wth tmedependent pocess mean, Poducton Plannng and Contol, ( (, -6. [] A.F. Hala and M.E. EI-Saadan, Constaned sngle peod stochastc unfom nventoy model wth contnuous dstbutons of demand and vayng holdng cost, Jounal of Mathematcs and Statstcs, ( (6, 4-8. [] J. L and J. Mao, An nventoy model of peshable tem wth two types of etales, Jounal of the Chnese Insttute of Industal Engnees, 6( (9, [4] O. Lang- Yuh and C. Hung-Ch, A mnmax dstbuton fee pocedue fo mxed nventoy models nvolvng vaable lead tme wth fuzzy lost sales, Int. J. Poducton Economcs, 76(, -. [5] L. L, S.N. Kabad and K.P.K. Na, Fuzzy models fo sngle-peod nventoy poblem, Fuzzy Sets and Systems, (, [6] C. Kao and Wen-Ka Hsu, A sngle-peod nventoy model wth fuzzy demand, Computes and Mathematcs wth Applcatons, 4(, [7] S. Islam and T.K. Roy, A fuzzy mult-objectve nventoy model wth vaable deteoaton ate, J. Tech., XXXVIII( (6, 7-8.

18 Int. J. Pue Appl. Sc. Technol., 9( (, [8] N.K. Mandal and T.K. Roy, Mult-tem fuzzy nventoy poblem wth space constant va geometc pogammng method, Yugoslav Jounal of Opeaton Reseach, 6( (6, [9] Chen-Chang Chou, A fuzzy bacode nventoy model and applcaton to detemnng the optmal empty-contane quantty at a pot, Intenatonal Jounal of Innovatve Computng, Infomaton and Contol, 5((B (9, [] Chen-Chang Chou, Fuzzy economc ode quantty nventoy model, Intenatonal Jounal of Innovatve Computng, Infomaton and Contol, 5(9 (9, [] D. Panda and S. Ka, Mult tem stochastc and fuzzy stochastc nventoy models unde mpecse goal and chance constants, Advanced Modelng and Optmzaton, 7(, [] K. Das, T.K. Roy and M. Mat, Buye-selle fuzzy nventoy model fo a deteoatng tem wth dscount, Intenatonal Jounal of System Scence, 5(8 (4, [] K. Das, T.K. Roy and M. Mat, Mult-tem stochastc and fuzzy-stochastc nventoy models unde two estctons, Computes and Opeatons Reseach, (4, [4] H. Katag and H. Ish, Some nventoy poblems wth fuzzy shotage cost, Fuzzy Sets and Systems, (, [5] Lang-Yuh Ouyang and Hung-Ch Chang, A mnmax dstbuton fee pocedue fo mxed nventoy models nvovng vaable lead tme wth fuzzy lost sales, Intenatonal Jounal of Poducton Economcs, 76(, -. [6] C. Kao and H. Wen-Ka, A sngle peod nventoy model wth fuzzy demand, Computes and Mathematcs wth Applcatons, (, [7] G.S. Mahapata and T.K. Roy, Fuzzy mult-objectve mathematcal pogammng on elablty optmzaton model, Appled Mathematcs and Computaton, 74 (6, [8] K. Atanassov, Intutonstc Fuzzy Sets and System, (986, [9] K. Atanassov and G. Gagov, Inteval valued ntutonstc fuzzy sets, Fuzzy Sets and Systems, ( (989, [] K. Atanassov, Open poblems n ntutonstc fuzzy sets theoy, Poceedngs of 6-th Jont Conf. on Infomaton Scences, Reseach Tangle Pa (Noth Caolna, USA, 989. [] K. Atanassov and V. Kenovch, Intutonstc fuzzy ntepetaton of ntetval data, Notes on Intutonstc Fuzzy Sets, 5( (999, -8. [] K. Atanassov, Intutonstc Fuzzy Sets, Spnge Physca-Velag, Beln, 999. [] P.K. Maj, R. Bswas and A.R. Roy, Intutonstc fuzzy soft sets, The Jounal of Fuzzy Mathematcs, 9( (, [4] P.P. Angelov, Optmzaton n an ntutonstc fuzzy envonment, Fuzzy Sets and Systems, 86(997, [5] P.P. Angelov, Intutonstc fuzzy optmzaton, Notes on Intutonstc Fuzzy Sets, (995, 7-. [6] P.P. Angelov, Intutonstc fuzzy optmzaton, Notes on Intutonstc Fuzzy Sets, ( (995, -9. [7] S. Paman and T.K. Roy, An ntutonstc fuzzy goal pogammng appoach to vecto optmzaton poblem, Notes on Intutonstc Fuzzy Sets, ( (5, -4. [8] B. Jana and T.K. Roy, Mult-objectve ntutonstc fuzzy lnea pogammng and ts applcaton n tanspotaton model, Notes on Intutonstc Fuzzy Sets, ( (7, 4-5. [9] S. Banejee and T.K. Roy, Applcaton of the ntutonstc fuzzy optmzaton n the constaned mult-objectve stochastc nventoy model, J. Tech, XXXXI(Septembe (9, [4] S. Banejee and T.K. Roy, Soluton of sngle and mult-objectve stochastc nventoy models wth fuzzy cost components by ntutonstc fuzzy optmzaton technque, Advances n Opeatons Reseach, ( (. [4] S. Banejee and T.K. Roy, Applcaton of fuzzy geometc and ntutonstc fuzzy geometc pogammng technque n the stochastc nventoy model wth fuzzy cost components, Advances n Fuzzy Sets and Systems, 6( (, -5. [4] A. Kaufmann and M. Gupta, Fuzzy Mathematcal Models n Engneeng and Management Scence, Noth Holland, 988.

19 Int. J. Pue Appl. Sc. Technol., 9( (, [4] S. Banejee and T.K. Roy, Pobablstc nventoy model wth fuzzy cost components and fuzzy andom vaable, Intenatonal Jounal of Computatonal and Appled Mathematcs, 5(4 (, [44] S. Banejee and T.K. Roy, Soluton of a stochastc nventoy model wth fuzzy cost components by geometc pogammng and fuzzy geometc pogammng technque, Advances n Theoetcal and Appled Mathematcs, 5( (, -5.