Multi-Objective Inventory Model of Deteriorating Items with Shortages in Fuzzy Environment Omprakash Jadhav 1, V.H. Bajaj 2

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1 Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6, Issue 1, 21 pp 45 MultObjectve Ivetory Model of Deteroratg Items wth Shortages uzzy Evromet Omprakash Jadhav 1, V.H. Bajaj 2 Departmet of Statstcs, Dr. B. A. M. Uversty, Auragabad, Maharashtra414, INDIA. Abstract: I ths paper multobjectve vetory model uder lmted storage area, deteroratg. Items wth stockdepedet demad are developed a fuzzy evromet. Here, objectves are to maxmze the proft ad to mmze the total average cost ad wastage cost, where urchasg prce, setup cost, holdg cost, storage area, Ivetory cost, shortage cost, rate of deterorato, total storage area, ad objectve goals are fuzzy ature. I ths model, fuzzy parameters are represeted by lear membershp fuctos ad after the fuzzfcato, t s solved by fuzzy olear programmg (NL) ad weght NL (WNL), fuzzy product goal programmg, (G) ad weght G (WG) method are preseted. The model s llustrated umercally ad the results obtaed from both WNL ad WG methods. Solvg ths problem also for some umercal values by both WNL ad WG methods, optmum results are preseted tabular form for dfferet weghts. Key words: Mult objectve, Deteroratg tems, Shortages, uzzy Goal rogrammg, Lear membershp fuctos. 1. Itroducto Though MultObjectve Decso Makg (MODM) problems have bee formulated may areas lke ar polluto, trasportato, structural aalyss etc. It may deped o tme, o had vetory level or tal stock level, etc. After that, the models wth tme depedet demad rate have bee studed by several researchers Datta ad pal [6], Che ad Wag [5], Bakh [1] ad others. I ths area, a lot of research papers have bee publshed by several researches such as apachrstos ad Skour [8], Chag et. al. [], Balkh [1], Chag [4] ad others. However, Goyal ad Gr [7] preseted a revew artcle o deteroratg tems cludg the publcatos up to 21.Over the past two decades, o extesve research work has bee doe to deal wth more tha oe objectve vetory maagemet system. Bookbder ad Che [2] developed a olear mxed tegerprogrammg model wth two objectves for the warehouseretaler system uder determstc demad. The objectves ths model are mmzato of aual vetory ad trasportato costs. They also cosdered two probablstc models Correspodg Addresses: 1 drjadhav@gmal.com, 2 vhbajaj@gmal.com Research Artcle wth customer s servce as aother objectve. Roy ad Mat [9] formulated a vetory problem of deteroratg tems wth two costrats, amely, storage space costrat ad total average cost costrat ad two objectves, amely, maxmzg total average proft ad mmzg total wastage cost fuzzy evromet. Very few researchers lke Omprakash Jadhav ad V.H. Bajaj [1,11] have formulated t the feld of vetory. They formulated a vetory problem of deteroratg tems wth two objectvesmmzato of total average cost ad wastage cost crsp evromet. ad solved usg olear goal programmg method. But, there are lots of reallfe vetory problems, whch ca be better represeted by the MODM formulato. I the vetory problem of a wastagable /damageable tem say gold, damod, etc., the tem may be so costly or so scarce that oe ca t have ulmted wastage of the materals for the sake of maxmum proft. I ths case, wastage has to be mmzed eve f t brgs dow the proft level. Such a stuato s truthfully represeted by takg two objectves ) Maxmzato of proft ad ) Mmzato of wastage ad the a compromse soluto s foud out to satsfy both the objectves a best possble way. Smlarly, for the sake of proft as much as possble, oe retaler caot vest the ulmted amout for hs busess, f requred. Retalers/busessme always try to vest the amout as less as possble ad to make proft as much as possble agast that vestmet. Hece, here aga, mmzato of the average vestmet cost may be a addtoal objectve addto wth the usual objectves of proft maxmzato ad wastage mmzato for damageable tems. I most of the earler vetory models, lfetme of a tem s assumed to be fte whle t s storage. But, realty, may physcal goods deterorate due to dryess, spolage, vaporzato etc. ad are damaged due to hoardg loger the ther ormal storage perod. The deterorato also depeds o preservg facltes ad evrometal codto of warehousestorage. So, due to deterorato effect, a certa fracto of the tems are Copyrght 21, Statperso ublcatos, Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6 Issue 1 21

2 Omprakash Jadhav, V.H. Bajaj ether damaged or decayed ad are ot perfect codto to satsfy the future demad of customers as good tems. Deterorato for such tems s cotuous ad costat or tme depedet ad /or depedet o the ohad vetory. Normally, marketg durato of seasoal products s costat ad these are avalable the market every year at some fxed terval of tme. Hece the tme perod for the busess of seasoal goods s fte. Several researchers have developed ths type of vetory models. I ths paper, uder lmted storage area, a multobjectve vetory model of deteroratg tems wth stockdepedet demad s formulated crsp ad fuzzy evromet. Here, objectves are to maxmze the proft ad to mmze the total average cost ad the wastage cost. The problem s solved by ) uzzy rogrammg Techque (T) based o Zmmerma s approach. ) Weghted uzzy rogrammg Techque (WT). ) v) Goal rogrammg (G) ad Weghted Goal rogrammg (WG), methods crsp evromets. The model s llustrated umercally ad the results obtaed from dfferet methods are compared. I fuzzy evromet, vetory costs, prces, proft goal, total cost, wastage cost ad storage area are assumed to be mprecse ature. I ths model, fuzzy parameters are represeted by lear membershp fuctos ad after the fuzzfcato, t s solved by uzzy No Lear rogrammg (NL), Weghted NL (WNL), uzzy roduct Goal rogrammg (G) ad weghted G (WG) methods. The model s llustrated umercally ad the results obtaed from both Weghted uzzy No Lear rogrammg (WNL) ad WG methods are preseted. A sestvty aalyss o fuzzy goals s preseted wth dfferet toleraces o proft, total cost, storage area, ad wastage cost. 2. Notatos Assumptos D (S ) = Demad at tme t, gve by D (S ) = α I Q β (t), for ( Q Q 1 ) s (t) = α Q β (t), for < s (t) < Q = α s β (t), for Q s (t ) Q 1 Where α ad z are costats; α, β, >, Q o = stock level (whch s costat), whe the vetory level s less the ths quatty, the cosumpto rate becomes costat, Q 1 = Hghest stock level, T = Tme perod of each cycle, S (t) = Ivetory level at tme t, = te rate of producto, β = Deterorato rate, 2 = ellg prce, 1 = urchasg rce, f = Space requred for oe ut of th tem, = loor space or shelfspace avalable, C 1 = Ivetory holdg cost per ut tem per ut tme, C = Setup cost per perod, ϕ = Total shortage uts, θ = Total deteroratg uts, (Q,Q) = Sum of average costs of the system, (Q,Q) = Sum of average profts of the system, C(Q,Q) = Sum of Wastage costs of the system, (Where Q ad Q are the vector of decso varables Q ad Q 1, = 1, 2, respectvely). A deteroratg multtem vetory model wth fte rate of repleshmet, purchasg prce depedet sellg prce, stockdepedet polyomal form of demad, partally backlogged shortages wth lmted storage space costrat s developed uder the followg assumptos. ) roducto rate s fte, ) Shortages are allowed ad partally back logged, ) Lead tme s zero, v) The tme horzo of the vetory system s fte.. Mathematcal ormulato d s d t = α (Q β ), ( Q Q 1 ) s (t) = α (Q β ) a s, s (t) Q (1) = α (s β ) a s, Q s (t) Q 1 (2) Where α >, < a, <1. So, the legth of the cycle T, for th tem, holdg cost, total umber of deteroratg utes, shortage cost, reveue of the oe cycle respectvely are gve by Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6 Issue 1 age 44

3 Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6, Issue 1, 21 pp 45 T = t 1 t 2 t () Q1 s d s c G (Q ) = c D (s ) a s (4) Q 1 asds = a θ (Q ) = G (Q 1 ) 1 D (s ) as (5) ( Q Q ) ( ) 1 s d s c φ (Q, Q ) = c 1 D (s ) (6) N (Q,Q ) = (p p )(Q θ (Q ) )p θ (Q ) (7) The sum of average proft, cost ad wastage cost are respectvely gve by [ ] ( Q, Q ) = N ( Q, Q ) C C G ( Q ) C φ ( Q, Q ) / T = 1 (8) (Q,Q ) = [c c G (Q ) c φ (Q,Q ) p Q ] /T = (9) C (Q,Q ) = θ (Q ) / T = 1 1 (1) ) Crsp Model: Hece our problem s to maxmze the total average proft ad to mmze both the total average ad wastage costs uder the lmtato of total storage area. Maxmze (Q,Q ) = (Q Q 1 =1, ) Mmze (Q,Q ) = (Q,Q ) 1 =1 Mmze C (Q,Q ) = C(Q,Q ) 1 =1 f Q, Q > Q, Q, Q >, = 1 2,,.., ) 1 1 (11) = 1, uzzy Model: Whe the vetory parameters such as purchasg prce, setup cost, holdg cost, storage area, the vestmet cost, shortage cost, back logged coeffcet, rate of deterorato ad total storage area ad objectve goals are fuzzy, the sad crsp model (9) s trasformed to a fuzzy model ad s represeted as M a x m z e (Q,Q ) = (m p% Q a% G (Q )) c% G (Q )c% φ (Q Q ) 1 1 2, 1 = 1 c% p% Q /T M m z e (Q,Q ) = c% G (Q ) c% φ (Q, Q ) c % p% Q /T = M m z e C (Q,Q ) = p% a% G (Q )/T = (12) f Q, Q > Q, Q, Q >, = = 1 1, 2,,.., Mathematcal Aalyss ) uzzy rogrammg Techque: To solve the above multobjectve programmg problem (9) by T. The frst step s to assg two values U k ad L k as upper ad lower acceptable levels of achevemet for the k th objectve respectvely ad d k = U k L k = the degradato allowace for the kth objectve (k = 1, 2, ). Now the problem (9) defed crsp evromet s sutable for the applcato of T. The steps of the fuzzy programmg techque are as follows. Step1: Solve the mult objectveprogrammg problem as a sgle objectve problem usg oly oe objectve at a tme ad gorg the rest objectves subject to the costrats of storage space. Let be the optmal soluto for the th sgle objectve problem. Step2: rom the results of stepi, determe the correspodg values for every objectve at each optmal soluto derve. Usg all the above optmal values of the objectves step1, costruct a payoff matrx ( x ) as follows: 1 2 () () C() ( ) ( ) C( ) ( ) ( ) C( ) ( ) ( ) C( ) Here, The Dagoal Elemets represet the optmal values of the correspodg objectves. rom the payoff matrx we fd lower bouds L = M ( ( 1 ), ( 2 ), ( )), L = M ( ( 1 ), ( 2 ), ( )), L C = M (C ( 1 ), C ( 2 ), C ( )). Ad the upper bouds, U = Max ( ( 1 ), ( 2 ), ( )), U = Max ( ( 1 ), ( 2 ), ( )), U C = Max (C ( 1 ), C ( 2 ), C ( )). The the objectve summatos are estmated as L () U, L () U ad L C C () U C Step : rom step2, we may fd for each objectve the value L k ad U k correspodg to the set of solutos. Copyrght 21, Statperso ublcatos, Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6 Issue 1 21

4 Omprakash Jadhav, V.H. Bajaj or the multobjectve problem (9), the membershp fuctos (), () ad C () whch may be lear or olear, are defed below. or smplcty, we have cosdered lear membershp fuctos oly. () = 1, f () > U = ( ) L, f L () U L U (1) =, f () < L ( ) = 1, f () < L = U ( ), f L U L () U (14) =, f () > U C () = 1, f C () < L C = U C C ( ) U C L C, f L C C () U C (15) = f C () > U C. Step 4: Use the above membershp fuctos to formulated a crsp olear programmg model followg Zmmerma s approach as Maxmze α (Q,Q ) α ( Q,Q ) α (16) C ( Q,Q ) α f = 1 Q > Q, α 1, Q,Q > ; ) Weghted uzzy rogrammg Techque: Crsp weght: Sometmes decso makers may cosder the relatve weghts for objectve goals to reflect ther relatve mportace. Here, postve crsp weghts w ( =,",") for the model are used (weghts ca be ormalzed by takg ( www=1). To acheve more mportace of the objectve goals, we choose sutable weghts the fuzzy olear programmg techque. If w,wad w are sttutve weghts for the proft goal, total average cost ad wastage cost goals respectvely, the the crsp model (9) ca be wrtte as, Maxmze α w ( Q, Q ) L ( U L ) U ( Q, Q ) w U L α T U C ( Q, Q ) W C U C L C α α f Q = 1 1 Q > Q, α 1, Q, Q > ; (17) ) Goal rogrammg Techque: I the smplest verso of the goal programmg, the decso maker sets goal for each objectve that he/she wshed to atta. The optmum soluto * s the defed as the oe that mmzes the devatos from the set goals. Thus, the goal programmg formulato of the multobjectve optmzato problem leads to 1 /p k Mmze (d p j= 1 j d j ) g ( ) W ( ) d d = b objectve ad j j j j d, d = j j j j d d (18) Here b j s the goal set by the decso maker for the j th d ad d are respectvely, the uderj j ad over achevemets of the j th goal. The value of p s based upo the utlty fucto chose by the decso maker. I the preset case, the goals b j are take depedg upo the ature of the objectve W j obtaed from dvdual maxmzato/mmzato of W, sce overachevemet of the goals, d s defed for the maxmzato of the objectve. Thus, uderachevemet d s defed for the mmzato of the objectve. j Thus the followg the mxed equalty equalty costraed problem stated equato (16), (9) ca be restated as the followg equvalet equalty costraed problem: Mmze subject to j (d ) (d ) (d ) = 1 p p p 1 2 f Q 1 ( Q,Q ) d = b 1 1 (19) 2 2 ( Q, Q ) d = b 1 /p Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6 Issue 1 age 46

5 Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6, Issue 1, 21 pp 45 C (Q,Q ) d = b Q > Q, p 1, Q, Q, d, d, d 1 2 v) Weghted Goal rogrammg Techque: If achevemet of certa goals s more mportat compared tha the others, the above problem (17) ca be restated as, Mmze p p p 1 /p w (d ) w (d ) w (d ) 1 2 f Q 1 = 1 (Q,Q ) d 1 = b 1 (2) (Q,Q ) d = b 2 2 C ( Q,Q ) d = b Q > Q,p 1,Q,Q,d,d,d 1 2 w w w = 1 v) uzzy Nolear rogrammg Techque: NL algorthm has bee llustrated ad used here to solve fuzzy multobjectve vetory model (1). I fuzzy set theory, the fuzzy objectves, costrats, costs, rate of deterorato ad rate of backloggg are defed by ther membershp fuctos whch may be lear or olear. Accordg to Zmmerma (1976), the lear membershp fuctos are (Q,Q ) =, fo r (Q,Q ) < B B B (21) B (Q,Q ) = 1, fo r (Q,Q ) > B = 1 ( ), fo r B (Q,Q ) B B B (Q,Q ) = 1, fo r (Q,Q ) < C C (22) = 1 ( ), fo r C (Q,Q ) C (Q,Q ) C C =, fo r (Q,Q ) > C C C (Q,Q ) = 1, fo r C (Q,Q ) < D D (2) = 1 ( ), for D C (Q,Q ) D C (Q,Q ) D D D =, fo r C (Q,Q ) > D D ( f Q ) = 1, fo r f Q < 1 1 = 1 = 1 f Q (24) 1 = 1 = 1, fo r f Q 1 = 1 =, for f Q > = 1 1 C (u ) = 1, fo r u < p p 1 1 = u p 1 1 ( ), f o r p u p p 1 (25) =, 1 1 p 1 for u > p 1 p 1 c u ) = 1, f o r u > l ( c c l u = 1 ( ), fo r c l c c u c l l l (26) =, f o r u < c l c l l = 1,2,. (u) = 1, for u > a a a u = 1, f o r a a a u a (27) =, f o r u < θ θ (u ) = 1, fo r u > u = 1 ( ), fo r u (28) =, for u < Here, objectve goals, total storage area, purchasg prce, setup cost, holdg cost, shortage cost, rate of deterorato ad rate of backloggg are respectvely B, C, D,, p 1, C, C 1, C 2, θ, havg ther respectve toleraces B, CO, DO,, p1, c, c1, c2, θ ad x whch are postve real umbers. Usg the above membershp fuctos, the fuzzy model (1) s trasformed to a equvalet crsp model, Maxmze α B ( Q, Q ) ( 1 ) α Where B ( Q,Q ) C ( 1 ) α C O (29) C (Q,Q ) D (1 ) α D O Q > Q, α 1, Q, Q >, ( ) = (m (α ) (α ) (α ) φ ( )) Q, Q p Q p a Q 1 1 = C 1 (α ) G ( Q ) C 2(α ) G ( Q ) C (α )) / T ( ) = ( (α ) (α ) φ ( ) (α ) G ( ) Q, Q p a Q C 1 = 1 1 Q 1 1 C 2 (α ) G ( Q ) C (α ))/ T l Copyrght 21, Statperso ublcatos, Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6 Issue 1 21

6 Omprakash Jadhav, V.H. Bajaj C ( Q, Q ) = 1 1 ( (α ) (α ) φ ( Q )) / T = 1 1 p a Let B be the mmum ad CO, DO, be the maxmum acceptable volato for the asprato levels U ad L, L C ad L respectvely. v) uzzy roduct Goal rogrammg (G) Techque: The above fuzzy problem (1) ca be formulated as Maxmze V (α 1, α 2, α, α ) = α 1 α 2 α α U ( Q,Q ) (1 ) =α 1 ( Q,Q ) L ( 1 ) = α 2 () C ( Q,Q ) L C ( 1 ) = α C Q 1 ( 1 = 1 ) = α α, α 1, Q Q ; Q, Q, = 1,2,. Where V(α 1, α 2, α, α) s a smple product achevemet fucto. 5. Numercal Examples or all models, let us assume, = 2, α 1= 9, α 2 = 1, Q 1 = 1, Q 2 =15, f 1 =.5 sq.ft. f 2 =.8 sq.ft. The above olear programmg problems (9), (15), (16) ad (17) are solved by computer algorthm based o gradet search techque (Geeralzed Reduced Gradet method) for the followg umercal data. 5.1 Crsp Model: To llustrate the model (9), We assume p = $ 1, p = $ 1 2, m = 1. 2, m = 1.2 5, c = $ 1, c = $.9, c = $ 5., c = $ 6., c = $ 1.5, c = $ 1. 6, a =.1, a =.1 1, =.8, =. 7 5, We frst solved the multobjectve programmg problem as a sgle objectve problem subject to the space costrats usg oly oe objectve at a tme ad gorg the rest objectves. Let be the optmal soluto whe oly th objectve s cosdered as objectve fucto. The result s show matrx [], gve by ( ) ( ) W C ( ) colum represet the Here, the values the th optmum value of the th objectve ad the values of the order objectves at. ) uzzy rogrammg Techque. rom the above matrx, the values of the bouds are L = $2., U = $41., L = $757.12, U = $862.22, L = $11.61, U = $57.6. C C Wth the above parametrc values, the optmal values of model (15) are α =.6, * = $ 2.5 5, * = $ , C * = $ ) Weghted uzzy rogrammg Techque. or the dfferet values of w, w", w ", the optmum values of the model (16) are gve Table1 Table 1 α w w w /$ /$ C /$ ) Goal rogrammg Techque: We solve the model (17) wth the same parametrc values of model (1) ad b 1=$5, b 2= $79., b = $15., the optmal values are gve Table2.. Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6 Issue 1 age 48

7 Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6, Issue 1, 21 pp 45 Table 2 p d d d /$ /$ C /$ uzzy Model: To llustrate the fuzzy model (28), we assume that all the crsp parametrc values rema the same. I addto we take = $ 2., B = $ 6., C = $ 7 5., = $ 1 5.,D = $ 1, C = $., = 2 s q.f t, = $ 2, = $, =., =. 2, θ C = $.1 2, C = $ 2, C = $ 8., θ = $ 1, =. 5 =. 6. C The we obta the obta the followg optmal values of the objectves ad related parameters by NL ad G method as, ) NL: * = $ 4 9.9, * = $ , C * = $ 1. 6, * = $ 1 1.1, * = $ 1.5 2, s * = $ 1.2 1,s * = $ ,C * = $.9 4,C * = $.8,C * = $ 1.2 5, C * = $ 1.4, C * = $ , C * = $ , * = 6.7 s q.f t.,θ * =. 8 5,θ * =. 9 9, * =.7 7 5, * =.7 5,α * =. 5 1,Q * = u t s,q * = u t s, Q * = u t s,q * =.4 7 u t s ) G: α =.5 4, α =.8 5, α =.5 4, α =. 5 4, =.7 7, =. 7 2, a =. 8, a =. 9, * = $ 5.8 2, * = $ , C * = $ , * = s q.ft., 2 * * * * Q = u ts, Q = 4. 8 u ts, Q = u ts, Q = 9. 4 u ts, s = $ , s = $ , p = $ 9.4, p = $ , C = $.7 9, C = $ 1.2 1, C = $ 1. 7, C = $ , C = $ 5 4.2, Sestvty Aalyss Now, we perform some Sestvty Aalyses upo the proft, total cost, wastage costs, goals ad warehouse space fuzzy model due to the chages the tolerace lmts of,, ad ad the results are gve Table, Table4 Table5 ad Table6 respectvely. C Table : Effect o,, C ad due to cremetal chages of /$ /$ /$ C / $ /s q.f t. α * Copyrght 21, Statperso ublcatos, Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6 Issue 1 21

8 Omprakash Jadhav, V.H. Bajaj 7. Cocluso rom the above Tables to 6, t s observed that whe creases, all the objectve s values decreases; whe Table 4: Effect o,, C ad due to cremetal chages of /$ /$ /$ C /$ /s q.ft. α * Table 5: Effect o,, C ad due to cremetal chages of C /$ W C /$ /$ C /$ /sq.ft. α * Table 6: Effect o,, C ad due to cremetal chages of _ /sq.ft. /$ /$ C/$ /sq.ft. α * _ creases, all the objectve s values creases ad whe creases, the behavor C of the objectves remas same as the case for but very less sestve tha others. Whe we crease the value of, t s observed that all objectves rema more or less uchaged. The above behavors are tue wth the assumptos. Refereces 1. Balkh,Z.T., (24), A optmal soluto of a geeral lot sze vetory model wth deterorated ad mperfect products, takg to accout flato ad tme value of moey. Iteratoal Joural of Systems Scece, 5(2), p Bookder, J.H., ad Che, V.Y..,(1992) Mult crtera tradeoffs a warehouse retaler system, Joural of Operatoal research socety 4(7), p Chag, C.T, Ouyag, L. Y. ad Teg, J. T.,(2), A EOQ model for deteroratg tems uder suppler credts lked to orderg quatty, Appled mathematcal modelg, 27, p Chag, C.T, (24), A EOQ model wth deteroratg tems uder flato whe suppler credts lked to orderg quatty, Iteratoal Joural of roducto Ecoomcs, 88, p Che, S. H., Wag(1996), Backorder fuzzy vetory model uder fucto prcpal, Iformato Sceces, 95, p Datta, T.K. ad al, A.K.,(1991), A ftehorzo model for vetory returs ad specal sales of deteroratg tems wth shortages.asa acofc Joural Operatoal Research 8, p Goyal, S.K. ad Gr.B.C. (21), Recet treds modelg of deteroratg vetory. Europea Joural of Operatoal Research, 14, p apachrstos, S. ad Skour,K., (2), A vetory model of deteroratg tems, quatty dscout, prcg ad tmedepedet partal backloggg, It. Joural of roducto Ecoomcs, 8, p Roy,T.K. ad Mat,M., (2) A multtem dsplayed EOQ model fuzzy evromet, 1. Omprakash Jadhav ad V.H Bajaj: Multtem, multobjectve fuzzy model of Deterorat tems uder two Costrats wth hyperbolc ad lear membershp ; Joural of Statstcal Sceces, Vol. 1, No. 2, Dec29, p Omprakash Jadhav ad V.H Bajaj: Multobjectve Ivetory model of deteroratg tems wth fuzzy vetory cost ad some fuzzy costrats. It. Jr. of Agr. ad Stat. Scece, Vol.6, No.2, 21, pp Iteratoal Joural of Statstka ad Mathematka, ISSN: EISSN: , Volume 6 Issue 1 age 5