Fuzzy Optimization of Multi Item Inventory Model with Imprecise Production and Demand

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1 Internatonal Journal of Computatonal and ppled Mathematcs. ISSN Volume, Number 3 (7), pp esearch Inda Publcatons Fuzzy Optmzaton of Mult Item Inventory Model wth Imprecse Producton and Demand M. Maragathamˡ and.tahseen Jahan² ˡ PG and esearch Department of Mathematcs, Peryar EV College, Truchy, Inda. ² Department of mathematcs, Justce Basheer hmed Sayeed College for Women Chenna, Inda. bstract Fuzzy Inventory model s developed for producton of Mult-tems wth shortages beng allowed and fully backlogged. The preparaton tme whch s the tme gap between decson and actual producton of tems s consdered n a fuzzy envronment. Then t s reduced to crsp preparaton tme usng nearest nterval approxmaton and then reduced to mult- objectve optmzaton problem. The computaton s carred out by the method of defuzzfcaton usng Global Crtera Method. Keywords: Interval arthmetc, fuzzy nventory model, mult-tem, producton level, defuzzfcaton.. INTODUCTION In the conventonal nventory models dfferent types of vagueness n dfferent parameters and functons are experenced whch are very dffcult to be solved by the probablty nventory modellng approaches. To defne nventory optmzaton tasks n such envronment and to nterpret optmal solutons, fuzzy set theory rather than probablty theory s more convenent. Consderng the fuzzy set theory n nventory modellng renders an authentcty to the model formulated snce fuzzness s the closest possble approach to realty.

2 7 M. Maragatham and.tahseen Jahan In 97, Zadeh et al had proposed some strateges for decson makng n fuzzy envronment. Wde applcatons of fuzzy set theory can be found n zmmermann [9]. In the crsp nventory models, all the parameters n the total cost are known and have defnte values, but n the practcal stuatons t s not at all possble. Hence fuzzy set theory comes to the rescue but dfferent nventory models occur not only due to the varous fuzzy cost parameters but also due to the fuzzness n other varables and constrants. ead- tme s the tme gap between placement of order and the actual recept of the orders. It s a crucal factor to be consdered n the feld of busness. So far, most of the researches have dealt wth ether constant or stochastc lead-tme. In practce, t s dffcult to forecast the lead-tme n a defnte and precse manner and at tmes, the past records are also not avalable to form a probablty dstrbuton for the lead-tme. Hence the only choce left to the decson maker s to defne the lead-tme parameter mprecsely by a fuzzy number. Generally, lead tme s assocated wth EOQ model.e. nstantaneous procurement or purchase of the lot. But, n a producton system, the scenaro s dfferent. Here the tme gap between the decson producton and the actual commencement of producton known as preparaton tme plays a key role n the tme analyss of nventory control models. Ths preparaton tme means the tme to collect the raw-materals, to arrange sklled/unsklled labourers to get machne ready for producton, etc., whch n turn nfluences the set-up cost of the system. For the frst tme, Mahapatra and Mat [] formulated and solved producton nventory models for a deteroratng/breakable tem wth varyng preparaton tme. In ths paper we develop a fuzzy producton nventory model for mult tems n a warehouse or factory. The formulated EPQ model n crsp sense s solved by generalzed reduced gradent method whch s a sngle objectve optmzaton problem. The problem of mnmzng the average total cost n the fuzzy sense s solved by Global Crtera method.. PEIMINIES Basc defntons and theores Defnton. :et X denote the unverse of dscourse. Then the fuzzy subset of µ x : X, whch assgns a real X s defned by ts membershp functon µ xn the nterval,,to each element x X, where the value of µ x number at x shows the grade of membershp of x.

3 Fuzzy Optmzaton of Mult Item Inventory Model wth Imprecse 7 Defnton.: fuzzy set on s convex f x x mn x, x x, x and,, where mn for all denotes mnmum operator. Defnton. 3: fuzzy set n the unverse of dscourse X s called a normal fuzzy set f there exsts at least one x X, such that µ x. Defnton.4: fuzzy set whch s both convex and normal s called as a fuzzy number n the unverse dscourse X. Fuzzy numbers are the fuzzy sets that are normalzed and convex. Defnton.5: fuzzy set defned on X and any number defned as / µ,, then cut s xx x. Fgure. shows a fuzzy number of unverse of dscourse X wth cuts. Defnton.6:The fuzzy number s sad to be Trangular fuzzy number f t s fully determned by a, a, a of crsp numbers such that a a a3 whose membershp 3 functon, representng trangle, can be denoted by The x a a a a x a x a 3 x a x a3 a3 a cuts of where, a a a otherwse s are the left and rght end ponts of. a a a, 3 3..,, where If a, a, a 3, then Fgure. Fuzzy number wth α-cuts.

4 7 M. Maragatham and.tahseen Jahan Fgure.Trangular Fuzzy Number. Defnton.7: Operatons on Fuzzy Numbers et X x, x x : x x x, x wherex, x left and rght lmts of X respectvely. et,,, / be a bnary operaton on the set of postve real numbers. If X and Y are closed ntervals then X Y x y : x X, y Y defnes a bnary operaton on the set of closed ntervals.in the case of dvson,t s assumed that Y and the formula s gven by X x x, Y y y,,, X Y x x y y x y x y kx where Y, x xand y y kx, kx, for k kx, kx, for k k s a real number Defnton.8: Preference elatons of Intervals: The decson maker s preference between nterval costs are defned for mnmzaton problems and let the uncertan costs be represented by ntervals X and Y respectvely. Then the preference relaton connectng n, by the left and rght lmts of ntervals s defned as below: The preference relaton between X x, x and Y y, y s X Y f x y and x y, X Y f X Yand x y. The order relaton represents the decson makers performance for the alternatve wth the least mnmum cost, that s f, X Y, then X s preferred to Y.

5 Fuzzy Optmzaton of Mult Item Inventory Model wth Imprecse 73 Defnton.9: The Nearest Interval pproxmaton Suppose X and Y are two fuzzy numbers wth α-cuts X, X Y, Y respectvely. Then the dstance between X and Y s, d X Y X Y d X Y d Gven X s a fuzzy number. We have to fnd a closed nterval Cd and X whch s nearest to X wth respect to the metrc d. We can do t snce each nterval s also a fuzzy number wth constant α-cut for all,. we have to mnmze wth respect to Hence Cd X C C, d d X C X X C d X C d C and. C In order to mnmze d mnmze the functon DC C d X Cd X D( C, C ) X ( ) d C C and,. Now d X, C X, t s suffcent to,,. The frst partal dervatves are D( C, C) X ( ) d C C when we solve D( C, C) and C D( C, C). C We get * C X ( ) d and * C X d ( ). lso D ( C, C ), * * C of the Hessan Matrx s D ( C, C ) * * C and the correspondng value,,, D C * C * D C * C * D C * C * * * H C, C. 4 C C C. C * * So DC, C.e,C d, Cd X X d X d d X X s the global mnmum. Therefore the nterval s the nearest nterval approxmaton of fuzzy number X wth respect to metrc. d et X x, x, x be a fuzzy number. The α- 3

6 74 M. Maragatham and.tahseen Jahan level nterval of X s defned as X X, X fuzzy number then X x x x when X s a trangular and X x x x. By nearest 3 3 nterval approxmaton method lower and upper lmts of the nterval are respectvely gven by C X d x x x d x x ( ) C X d x x x d x x 3 ( 3 ) 3. Therefore the nterval number of X as a trangular fuzzy number s gven by x x x x 3,. Defnton.: Mult- Objectve Non- near Problem: Mult Objectve Non-near Problem (MONP) wth nterval valued parameters can be stated as below: Mnmze k Z x C x n j a j j () Subject to k x j Bj, j, where C c, c, a, a x j,,, n, and Bj bj, b j. x x,,...,. x x n et us formulate the orgnal problem () as a Mult-Objectve Non near Problem. Snce the objectve functon Z x and the constrants contan some parameters represented by ntervals, t s natural that the soluton set of () should be defned by preference relatons between ntervals. So the nterval valued objectve functon and, Z x ts centre Z k n a j j j C x respectvely becomes: k n a j j j Z x C x, Z x C x Z x n terms of rght and left lmts Z. and Z x Z x Z x C x

7 Fuzzy Optmzaton of Mult Item Inventory Model wth Imprecse 75 Thus the problem () s transformed nto Mnmze Z, Z Subject to C k a x j bj, k a x j bj, j, x j,,, n, x x x x,,...., n Defnton.: mult objectve Non- lnear optmzaton problem (MONP) s convex f all the objectve functons and the feasble regon are convex. Defnton.: The Best choce of the objectve vector ', Z t t whch are functons of the varable t', t are sad to be pareto-optmal soluton to the MONP f and only f there exsts unque solutont', * t * n the feasble regon such that ', * * ', Z t t Z t t for all. If the mult-objectve optmzaton problem s convex, then every locally pareto-optmal soluton s also globally paretooptmum[3]. 3. NOTTIONS ND SSUMPTIONS To develop our nventory model we need the followng notatons and assumptons:-. Demand s dependent on unt producton cost and the current stock level.. Shortages are allowed & backlogged fully. 3. Preparaton tme n producton of new tems s allowed and fuzzy n nature. 4. Set up cost s dependent on preparaton tme 5. Producton cost, set up cost and shortage cost are all known constants. Shortages are allowed and fully backlogged. 6. Tme of plan s nfnte 7. ndex of tems where our nventory system nvolves mult tems (non deteroratng),say where ranges from to n. 8. I t Inventory level at any tme t for th term 9. Imax Maxmum nventory level n a tme cycle t. Is Maxmum shortage allowed n a tme cycle t. T Fuzzy preparaton tme for the next producton cycle.

8 76 M. Maragatham and.tahseen Jahan. t Cycle length of a cycle. 3. t ' e producton tme.e. when the next producton starts whch depends on decson maker. 4. u Set up cost whch depends on the fuzzy preparaton tme and hence t has the form u u ut where, u and u are constants so chosen to best ft the set up cost. 5. h Holdng cost per unt per unt tem 6. s Shortage cost per unt per unt tem 7. p Producton cost per unt per unt tem 8. Total producton cost for all tems p 9. D = Demand ate for tem whch depends on producton prce and stock and so t has the form D I t n I I t (t) f ( ) f I ( t) where and are postve real constants.. K ate of producton for th tem whch has the form K D where. 4. MODE FOMUTION 4. CISP MODE : The Inventory producton cycle starts wth shortages at tme t and at tme t, t reaches the maxmum shortage level I s and these shortages are backlogged fully and after tme t the shortages go to the level zero and then the nventory agan shoots up to the level I n tme t. 3 fter ths the producton s max halted, the nventory on hand agan declnes due to demand and reaches the level zero at tme t. Ths s represented n the followng Fgure(3)

9 Fuzzy Optmzaton of Mult Item Inventory Model wth Imprecse 77 Fgure (3): Inventory Begns wth backlogged shortages and end wth no nventory The change of nventory level can be descrbed by the followng dfferental equatons: D f t t di (t) K D f t t t dt K D f tt t3 D f t3 t t Wth the boundary condtons I I t I t can be rewrtten as follows: (). These dfferental equatons f t t (t) f t t t di dt I (t) f t t t3 I (t) f t3 t t Solvng 3() and 3() wth boundary condtons n the tme ntervals and also puttng we have the followng equatons: (3),t and t, t t I (t) t t (4) So at tme t t, the above equatons become t t t t t (5)

10 78 M. Maragatham and.tahseen Jahan In a Smlar manner solvng 3() and 3(v) wth the boundary condtons I t I t n the tme ntervalst, t and t, t respectvely we have the followng equatons: where ( ). I t e e t tme t t, 3 the above equatons become 3 tt ( t t) 3 3 t t t t t e e t3 t, usng (5). (7) Now calculatng the dfferent total costs as follows: Total shortage cost durng tme ntervals 3,t and t, t t t Cs s I t dt I t t dt s t s t t t where t T t ', t t and t t 3. Total holdng cost durng the tme ntervals t3 t Ch h I tdt I tdt t t3 t t and t, t, 3 t3 t h t t3 s calculated as below: 3 h e t e t Total producton cost durng tme ntervals p t t 3 t t t t and t, t, 3 C dt I t dt t3 t t t e Hence thetotalaverage cost ', (6) (8) s calculated as follows: s calculated as follows: Z t t s gven by the followng equatons: (9) ()

11 Fuzzy Optmzaton of Mult Item Inventory Model wth Imprecse 79 Z TC t ', t, where TC Cs Ch Cp u. t Hence the proposed model n the crsp sense s formulated as follows: Mnmze Z t ', t s t s t t t h ( t3 t) h tt3 e t e t t3 t t t e u u T. () 4. Fuzzy Model: Due to uncertanty n the envronment preparaton tme T s a fuzzy number represented by an apt nterval arthmetc number n the formt T T substtutngt T T,,. Hence t ' t, t (say) and also for the rest of the tme factors by ther correspondng fuzzy parameters n the nterval arthmetc number we have the followng equatons: t t t t t t t t t 3 3 3, say, say The correspondng dfferent costs assocated wth the proposed nventory model wth ther left & rght fuzzfed -cuts are deduced as follows : Total shortage cost C C, S where ( ). S S S Total holdng cost C C, C say, where Cs s t s t s tt s t () h h h say, where h t3t h h tt3 h Ch e t e t (3)

12 7 M. Maragatham and.tahseen Jahan Total producton cost C C, C p P P The set up cost u can be wrtten as say, where ( ) t3t CP t t e u u u T u u T u u,, (4) say (5) Smlarly the rght -cuts of the fuzzfed average total cost can be got by substtutng nstead of and nstead of n the above formulae. Hence the total average cost now equals Z TC Cs, C C,,,, s h C h C p C p u u Z Z t t where Z Cs C h C p u t, Z Cs C h C p u t and Z C Z Z (6) 5. SOUTION METHODOOGY 5. Crsp model: To get the optmum value of average total cost we have to solve t ' Z t ', t t ' ', and Z t t Z t t ', t TC h h h Z t ', t e e t ' t t ' t t & Z t t ', t T t' t T t ' t T t' T t' e s T t ' s

13 Fuzzy Optmzaton of Mult Item Inventory Model wth Imprecse 7 t t t t t t T t' TC h ' h ( ) t T t Z t ', t TC e e t T t' h e TC t (7) TC whch mples TC. t t Hence the average total cost s strctly convex snce all the postve mnors of ts Hessan matrx are strctly postve. Snce we have Z t ' t ', t Z, t t ', t and ', ', ', Z t t Z t t Z t t. t ' t tt ' 5. Fuzzy Model: The objectve functons of the fuzzfed optmzaton problem.e. Z, Z C, Z are all functons of t ' & t as t, t and t 3 are dependent varables whch depend on t ' and t. Ths total average cost mnmzaton problem wth nterval objectve functon s converted to mult objectve non-lnear optmzaton problem, whose objectves are to mnmze the centre Z and rght lmt C Z of the nterval objectve functon. s seen n model I, we can also show and hence Z C are strctly convex and can also be seen that an objectve vector * * Z C, Z C s pareto-optmal f there does not exst another objectve vector ( Z, Z ) such that ', * * ', Z t t Z t t for all C, ndex j C, * *. Hence C, and Z * * t ', t Z t ', t j j C for at least one Z Z s pareto-optmal f the decson vector correspondng to t s pareto-optmal and also snce our mult objectve optmzaton problem s convex, we know every locally pareto-optmal soluton s also globally pareto-optmum [3]. Hence the mult objectve problem s solved by Global Crtera Method whose solvng steps are gven below: Step-: The mult-objectve programmng problem s converted to sngle objectve problem usng only one objectve at a tme. Step-: pay-off matrx s formed from the above last table as follows:

14 7 M. Maragatham and.tahseen Jahan Z Z mn C max C Step-3: Fnd the deal objectve vector from the above pay-off matrx of step-, for example mn mn max max ( Z Z ) say and ts correspondng value of ( Z Z ).Then ts auxlary C problem s solved as below: Z Z max mn C mn mn ', ', Z t t Z C C Z t t Z C Global Crtera Z GC Mnmze max mn max mn Z Z Z C Z where and for calculaton sake we take the value of as. Hence the soluton set gven above s referred to as Global Crtera. 6. NUMEIC EXMPE Consder an nventory model wth the followng parametrc values: 6. Crsp Model: 3,,.7,.5, h s.5/ unt / year, u s / unt / year, u s3/ unt / year, s s5/ unt / year,.8, T.6 year By Graded Based Non lnear optmzaton Method we get the soluton of the Crsp Model as: Z t ', t , t '.669, t , I , I s max 6. Fuzzy Model:.8, 3,,.7,.5, h s..5/ unt / year, s.5/ unt / year, u s./ unt / year, u s.3/ unt / year, s s.5/ unt / year, T.5, T.8 Z C Z Usng the above data, we get the followng pay off matrx By Global Crtera Method we get the soluton of the fuzzy model

15 Fuzzy Optmzaton of Mult Item Inventory Model wth Imprecse 73 Z 369.4, Z 73.3, Z 8.6, t ' , t , GC C SENSITIVITY NYSIS Senstvty analyss of the varables t'& t for the crsp model has been made where the parameters,, T, are changed from -5% to +5%. The correspondng change * of Z t t * * ',, I, I are enumerated n Table I. s max Table: I. SENSITIVITY NYSIS Parameter %changes % of Z (t t ) t ' t I I ' * *, S max T No soluton CONCUTIONS The generalzed reduced gradent method has been appled to get the soluton of crsp model whch s a sngle objectve optmzaton problem, whereas fuzzy model s a mult-objectve optmzaton problem whose objectves are to mnmze the centre Z C and rght lmt Z of the nterval objectve functon. These two objectve functons take care of both average case and the worst case. In ths fuzzy model the

16 74 M. Maragatham and.tahseen Jahan pareto-optmal solutons obtaned usng Global Crtera Method gve the best possble soluton n terms of best choce to be made by decson maker as the preparaton tme s fuzzy n nature. The preparaton tme for the next producton cycle s hghly sgnfcant as t mpacts several costs lke set up cost, producton costs, shortage costs, etc. It s also seen that the demand of a commodty decreases wth the ncrease n producton costs but ncreases wth the ncrease of stock put up on dsplay and vce versa s true. The proposed model s generally a producton model wth demand dependent producton rate and the unt producton cost whch s assumed to be constant for each of the tems under producton whch s not so n real terms as t vares wth the preparaton tme and quantty produced. Ths model can also be mprovsed as tme dependent producton rate, partally lost sales, nflaton, quantty dscounts on the dsplayed stock are the dfferent factors whch can be consdered to make ths model more pragmatc and dynamc n the future scope of research. EFEENCES [] Bhuna. K and Mat M, n Inventory Model for Decayng Items wth Sellng Prce, Frequency of dver-tsement and nearly Tme Dependent Demand wth Shortages IPQ Transactons, Vol., 997, pp [] Chandra K. Jagg, Sarla Pareek, nuj Sharma, Fuzzy Inventory Model for Deteroratng Items wth Tme varyng Demand and Shortages mercan Journal of operatonal esearch, [3] Foote B, Kebrac N and Kumn H, Heurstc Polces for Inventory Orderng Problems wth ong and andom Varyng ead Tmes Journal of Operatons Manage-ment, Vol. 7, No. 3-4, 988, pp do:.6/7-6963(8)98-5 [4] qun, J, Determnstc EOQ nventory model for Non-Instantaneous deteroratng tems startng wth shortages and endng wthout shortages IEEE, Internatonal conference on servce operatonal and ogstcs and Informatcs Begng 95-99, 8. [5] Magson D, Stock Control When ead-tme Can Not Be Consdered Constant Journal of the Operatonal e-search Socety, Vol. 3, 979, pp [6] Mahapatra N. K, oy T. K. and Mat M, Mult-Ob-jectve Mult-Item Inventory Problem, Proceedngs of the Semnar on ecent Trends and Developments n p-pled Mathematcs, Howrah, 3 March, pp [7] Mahapatra N.K, Das K, Bhuna.K and Mat M, Mult objectve Inventory Model of Deteroratng Items wth ramp Type Demand Dependent Producton, Setup and Unt Costs, Proceedngs of the Natonal Symposum on ecent dvances of Mathematcs and ts pplcatons n Scence and Socety, Unversty of Kalyan, - November, pp. 89-.

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18 76 M. Maragatham and.tahseen Jahan