International Journal of Modeling and Optimization, Vol. 2, No. 2, April 2012

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1 Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0 Fuzzy Invenory Models of Persle Mul-Iems for Inegred nd Non-negred Busnesses w Possly/Necessy Mesure of Trpezodl Fuzzy Gol Sv Pk nd Seem Srkr Mondl Asrc A mul ojecve nvenory models of deerorng ems ve een developed w Weull re of decy llowng sorges, n wc demnd s ken s funcon of me nd producon s proporonl o demnd re. Here ojecves re o mxmze e prof from dfferen ems w spce consrn on nfne plnnng orzon for nonnegred nd negred usness. Ojecves re lso mde fuzzy n nure for non- negred usness. Te compromsed soluons of e opmzon prolem re oned y e pplcon of Zmmermnn s ecnue nd Fuzzy Addve Gol Progrmmng ecnue. Crsp nd fuzzy wegs re used o ncorpore e relve mpornce of e ojecve nd consrn gols. Te models re llusred numerclly nd e resuls of ose models ec w crsp nd fuzzy wegs re compred. Te resuls for e model ssumng em o e Sngle House Inegred Busness SHIB re oned y usng Generlzed Reduced Grden meod. Te coss lke cos per un ems, oldng coss, se up coss, sorge coss, sellng prces re ken n fuzzy envronmen s rngulr fuzzy numers nd rpezodl fuzzy numers lso. Wen coss re mprecse, opmsc nd pessmsc euvlen of fuzzy ojecve funcon s oned y usng credly mesure of fuzzy even y kng fuzzy expecon. Te prolems ve een solved y formulng em s sngle ojecve w fuzzy coss. Te resuls of fuzzy SHIB model s llusred w numercl exmple nd ose re compred w e es possle soluon of e non- negred usness. Index Terms Mul ojecve, crsp/fuzzy wegs, mul em, expeced vlue w possly/necessy. I. INTRODUCTION In mos of e nvenory model, e producon re nd demnd re re unform rougou e perod. Bu s usully oserved n e mrke sles of e fsonle goods, elecronc gdges, sesonle producs, food- grns ec. cnge w me. So, e producon re nd demnd re s vred. Demnd vres ccordng o e me, uly of ems, fesvls, weers ec. For ese resons, dynmc models of producon nvenory sysems ve een consdered nd solved y ssumng demnd s connuous funcon of me wc my ncrese or decrese w me nd producon depends on mny fcors lke mn power, nroducng new ecnology, vlly of new merl, power supply, me, demnd ec. A numer of Mnuscrp receved Ferury 8, 0; revsed Mrc 6, 0. Te uors re w e Nonl Insue of Tecnology, Durgpur, W. B. Ind e-ml:sv.pk@gml.com, seemskrmondl@yoo.co.n. reserc ppers ve lredy een pulsed on e opc n wc producon of mny rel ems s proporonl o demnd []-[] nd oers]. Mrkeng resercers recognze e producon of mny rel ems s proporonl o e demnd long w fcors lke mn power, new ecnology, power supply, me ec. []. Mny mes, e oldng cos of persle ems ncreses w me. [5] developed n nvenory model for deerorng ems w prce-dependen demnd nd me-vryng oldng cos. Recenly muc work s een done regrdng nvenory models for deerorng ems. Te lfe me of persle ems lke perfumes, medcnes, lood ec. re fxed nd ey cnno e used fer e de of expry. [6] ve revewed nvenory models for deerorng ems. [7]-[9] ve een developed some nvenory models for persle ems lke perfumes, medcnes, lood ec. wc ve e fxed lfeme nd cnno e used fer e expred de. [0] developed n nvenory model w Weull re of decy vng sellng prce dependen demnd. However, ey consdered e cse of nsnneous replensmen. In mny prccl suons lke food processng ndusres, poocemcl ndusres, e producon s no nsnneous. [] dscussed persle nvenory model w fne re of replensmen vng Weull lfeme nd prce dependen demnd. [] developed n nvenory model for ems w Weull melorng. [] gve model on Weull dsrued deeroron. [], [5] nd oers presened nvenory models w Weull dsruon deeroron, me-vryng demnd nd sorges. In mny prolems more n one ojecve cn e consdered. Bu ere s mul em mul ojecve nvenory model for negred nd non negred usness developed y [6]. [7] Developed mul ojecve fuzzy nvenory model of deerorng ems w vlle sorge re. To gve e relve mpornce o e ojecve ey ve ssgned crdnl wegs crsp/fuzzy. [9] nd oers presened model w necessy/ possly consrn y usng expeced vlue of fuzzy vrle. Mu em clsscl nvenory model under resource consrns suc s cpl nvesmen, vlle sorge re, numer of orders nd vlle seup me ec. re presened n well known ooks [9], [0] nd oers]. Tkng spce lmon s consrns severl workers [], [] ve consdered mul em nvenory models n crsp nd fuzzy envronmens. [], [] ve developed wo nvenory models, n e frs model, e producon re s ssumed o e funcon of e on nd nvenory level nd n e second model, e producon re s ssumed o e funcon of demnd re. 9

2 Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0 Nowdys, lmos every mporn rel world prolem nvolves more n one ojecve. So, decson mkers ry o model em s mul crer decson mkng MCDM prolems denfyng e dfferen crer. Te mpornce of suc models s o produce e es lernve ssfyng e ojecves nd e consrn wc es fulflls e reuremen of decson mkers. Recenly vrous new meods [cf. []-[6] ve een oulned o fnd e compromse soluons of MCDM prolems. In mny relsc suons, s dffcul o ssgn precse spron levels o ojecves. Moreover, n some cses, s no even possle o rcule precse oundres of e consrns. In suc suons fuzzy gol model s more ppropre. In ese cses, normlly, lner nd non- lner spes for e memersp funcons of e fuzzy ojecve nd consrn gols re proposed. To reflec e decson mker s performnces regrdng e relve mpornce of ec ojecve gol, crsp / fuzzy wegs re used [cf. [7]. Te fuzzy prores my e lngusc vrles suc s very mporn, moderely mporn nd mporn. Memersp funcons cn e defned for ese fuzzy prores n order o develop comned mesure of e degree o wc e dfferen gols re ended. Recenly, [8] presened mul-ojecve nvenory model of deerorng ems w spce consrn n fuzzy envronmen. Tll now e cos of ems re ken s consn, u n rel lfe suons ese coss my e of mprecse ype. Te coss of e ems my decrese or ncrese ccordng o e demnd nd sock ec. T s wy e coss uns oldng seup sorge deeroron re ken n fuzzy envronmen s rngulr nd rpezodl fuzzy numers. Te possly/necessy nd credly mesures of ojecve re lso consdered. [9], [0] nroduced e necessy nd possly consrns wc re very relevn o e rel lfe decson mkng prolems nd presened e process of defuzzfcon for er consrns. Roy e.l [], [] nd [8] ve developed models w necessy nd possly consrns y usng expeced vlue of fuzzy vrles [cf.[]]. In s pper, mul ojecve nvenory model of deerorng ems ve een developed w Weull re of decy llowng sorges, n wc demnd s ken s funcon of me, nd producon s proporonl o demnd re. Here e ojecves re o mxmze e prof from dfferen deerorng ems w spce consrn on nfne plnnng orzon for non- negred nd negred usness. I s ssumed e deerorng res of dfferen ems follow e wo prmeer Weull dsruons nd oserved e deerorng cos long w dsruon prmeers ve remendous nfluence on e opml prof for o ype of usnesses. Te ojecves for prof mxmzon for ec em re seprely formuled. Tese ojecves re lso mde fuzzy n nure for non-negred usness. Te mprecseness of nvenory prmeers nd gols for non-negred usness s een expressed y lner memersp funcons. Te compromsed soluons of e mul ojecve non-lner opmzon prolem re oned y e pplcon of wo dfferen fuzzy opmzon meods Zmmermnn s ecnue nd Fuzzy Addve Gol Progrmmng ecnue FAGP sed on grden meod. Crsp nd fuzzy wegs re used o ncorpore e relve mpornce of e ojecve nd consrn gols. Te models re llusred numerclly nd e resuls of ose models ec w crsp nd fuzzy wegs re compred. Te resuls for e model ssumng em o e Sngle House Inegred Busness SHIB re oned y usng Generlzed Reduced Grden meod GRG. Tll now e cos of ems re ken s consn. Bu n rel lfe suon ese cos my e of mprecse ype. So uncerny s o e mposed, s wy n s pper e coss lke cos per un ems, oldng coss, se up coss, sorge coss, sellng prces re ken n fuzzy envronmen s rngulr fuzzy numers nd for more relsc suons, ese re ken s rpezodl fuzzy numers lso. Te fuzzy SHIB model w mprecse nvenory cos s formuled o opmze e possly necessy mesure of fuzzy gol of e ojecve funcon. Wen coss re mprecse, opmsc nd pessmsc euvlen of fuzzy ojecve funcon s oned y usng credly mesure of fuzzy even y kng fuzzy expecon. Also e prolems ve een solved y formulng em s sngle ojecve w crsp nd fuzzy coss. Te resuls of crsp nd fuzzy SHIB model s llusred w numercl exmple nd ose re compred w e es possle soluon of e non- negred usness. II. ASSUMPTIONS AND NOTATIONS Te followng ssumpons nd noons ve een used n developng e model: n = numer of ems. Sorges re llowed nd cklogged. Te led me s zero. Plnnng orzon s nfne. 5 W= vlle floor or self spce s. f., For =,,, n em. 6 W = sorge spce reured per un em s. f., 7 K = sock level me, 8 S = sorges level me, 9 Iems re deerored nd s re s θ.e.,, 0 < α <, > 0, β. Holdng cos uns me s lnerly me dependen.e., = + were nd re posve consns $. 0 T = e me perod for ec cycle yers, H = e sorge cos per un per me $, C = e orderng cos per cycle $, Tme dependen demnd me s defned s R = -, > 0 nd 0 < re consns. N = e purcse cos per un per me $. Also, sellng prce per un s m N, were m >, 5 I = e ol verge prof $, 6 e e nvenory level me n cycle 0, T, 7 Demnd dependen producon me s P = γ R, γ >. 8 K= spce covered y ll e ems s. f..e. K n K w, 9 I = prof n oly $.e. I n I. 0

3 III. MODEL AND FORMULATIONS Here, nvenory model for - deerorng em s sown n Fg.. Inlly, e sock s ssumed o e zero. Demnd dependen producon srs =0 nd smulneously supply lso srs o ssfy e me dependen demnd R. A =, e sock level reces K uns. Te producon s en sopped. Te nvenory ccumuled durng e producon perod 0, fer meeng e demnd durng e perod nd e deeroron, e nvenory reces o e zero level me =. Now, e sorges re ccumuled o e level S me = nd demnd dependen producon srs w e me dependen demnd R. Te cklog s flled durng e me, T, ll e cklog ecomes zero. Te cycle en repes self fer me T. Fg.. Pcorl represenon of e nvenory sysem for e - em. Te dfferenl euons descrng e nvenory level of - em n e nervl 0 T, s gven y: R d d,0 R d d, R d d, T R d d, Te condons re, = 0 = 0, nd T ; = K =. Usng e condons, e soluon of s were K 6, nd 7 Now, e soluons of nd re: 8 T T T T 9 Deerorng uns = DE = DE + DE, were deerorng uns n 0, nd, re : d DE 0 0, d DE Holdng cos = HC =HC +HC, were oldng coss n 0, nd, re: HC d , d HC And Tol sorge cos n, T = SC = H SC, were nvenory level n,t re : 6 H H d SC Relons eween,, nd T y euly condons nd re: 5 Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0

4 Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0 And, T T T T T T 6 Now, sellng prce - producon cos = Q were, R nd, Q T 0 m N R 7 R d 8 R d T T Tol verge prof per un me, 9 I = sellng prce- producon cos oldng cos sorge cos deerorng cos se up cos/t ; 0 Here, our ojecve s o mxmze e ove prof for ree ems w e followng ypes of models. IV. TYPES OF MODELS Tere re four ypes of models. Model Crsp, Inegred; Model -Fuzzy, Inegred; Model -Crsp, Non-negred; nd Model -Fuzzy, Non-negred. A. Model Crsp, Inegred Assumng e ems re del collecvely s sngle negred usness process, e correspondng sngle ojecve model s o Mxmze {I, I, I } Sujec o, euons 5, 6, K W, nd 0,,, B. Model -Fuzzy, Inegred In prccl suon, every cos s mprecse, so we ke {N, C, H, ; =, nd } s fuzzy numers.e. s N, C, H, ;,, nd. Ten due o s ssumpon, e crsp funcons I, I, I, I wll ecome e I fuzzy funcons, I, I, I. Te opmzon of fuzzy ojecve s no well defned. So nsed I, I, I, I of one cn opmze s euvlen opmsc nd pessmsc reurns. Accordngly opmzon of s model - cn e descred s follows: To opmze e opmsc nd pessmsc euvlen of I, I, I, I y lemm- c.f. Appendx-A, e prolem reduces o, Mxmze E I E I Sujec o, euons 5, 6, K W, nd 0,,, n wc, E N, E C, E H, E ;,, nd re expeced vlues of N, C, H, ;,, nd respecvely. If N e rngulr fuzzy numer.e. N N, N, N en E N = ½ N N N nd f N e rpezodl fuzzy numer.e. N N, N, N, N, E N = ½ N N N N, were ρ 0 < ρ < s e mngerl ude fcor. ρ = nd ρ = 0 represen mos opmsc nd pessmsc ude respecvely nd ρ = 0.5 represens e credly mesure. Smlr wll e e cse for C, H, ;,, nd. C. Model -Crsp, Non-Inegred In crsp envronmen mul-ojecve producon nvenory prolem w spce consrn s o Mxmze {I, I, I } Sujec o, euons 5, 6, K W, nd 0,,, D. Model -Fuzzy, Non-Inegred Wen e ove verge prof of every em nd vlly of spce re ecome fuzzy, e sd crsp model 0 s rnsformed o fuzzy model s: Mxmze {I, I, I } Sujec o, euons 5, 6 nd K W, nd 0,,, V. MULTI-OBJECTIVE MATHEMATICAL PROGRAMMING A generl mulple ojecve non-lner progrmmng prolems re of e followng form: Mnmze Fx = [f x, f x, , f k x] Sujec o g x c, =,,..,T j x = j,j=,,..,j x є S, were S = [x / x є Rn]. Here, x = [x, x, , x n ] T s n n-dmensonl vecor of decson vrles, f x, f x, , f k x re k dsnc ojecve funcons. S s e se of fesle soluons. An opml soluon of sngle ojecve prolem s defned s one mnmzes e ojecve funcon f x, sujec o e consrn se x є S. To defne vecor mnml pon wc ll componens of e ojecve funcon vecor f x re smulneously mnmzed, s no n deue generlzon. Snce suc pons re seldom nle. Zmmermn [978] sowed fuzzy progrmmng ecnue cn e used effecvely o solve e mul-ojecve progrmmng prolem s follows: VI. FUZZY PROGRAMMING TECHNIQUE TO SOLVE CRISP MULTI-OBJECTIVE PROBLEM Te ove mul-ojecve progrmmng prolem s

5 Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0 defned compleely n crsp envronmen. To solve s crsp prolem y fuzzy ecnue, we frs ve o ssgn wo vlues U k nd L k s upper nd lower ounds of e k- ojecve for ec k =,,. Here L k = spred level of cevemen, U k = ger cceple level of cevemen nd d k = U k - L k = degrdon llownce. Te seps of e fuzzy progrmmng ecnue re s follows: Sep-: Ec ojecve funcon I, I nd I of e mul ojecve progrmmng prolem s opmzed seprely sujec o e consrn of e prolem. Le ese opmum vlues e I *, I * *. nd I Sep-: A ec opml soluon of e ree sngle-ojecves progrmmng prolem solved n sep- fnd e vlue of e remnng ojecve funcons nd consruc py-off mrx of order s follows: Usng e ove memersp funcons, e fuzzy non-lner progrmmng model s formuled nd solved y followng e meods of Zmmermnn 978. VIII. CRISP WEIGHTS Somemes decson mkers re le o provde crsp relve wegs for ojecve gols o reflec er relve mpornce. Here, posve crsp wegs w =,,.., n for crsp model re used wc cn e normlzed y kng n w. To ceve more mpornce of e ojecve gol we cose sule nverse weg n e fuzzy non-lner progrmmng ecnue. Smlrly, n fuzzy nvenory model we my coose e smlles of e nverse weged memersp funcon correspondng o e mos mporn ojecve gol. From e py-off mrx, fnd lower ounds LI, LI, LI, nd upper ounds UI, UI, UI s follows: e lower ounds LI = Mn{ I, I, I }, LI = Mn{ I, I, I }, LI = Mn{ I, I, I }. And e upper ounds UI = Mx { I, I, I }, UI = Mx { I, I, I },UI = Mx { I, I, I }. Sep-: To solve s crsp prolem y Zmmermnn [978] meod, we ke e memersp funcons I, I, nd I respecvely of e ojecve funcons I, I, I n e lner form s follows: for I UI ; I LI =,,. for LI I UI ; I U I LI 0 for I LI; Sep-: Usng e ove memersp funcons, e crsp non-lner progrmmng model s formuled nd solved y Zmmermnn s ecnue nd Addve Gol Progrmmng ecnue. VII. FUZZY NON-LINEAR PROGRAMMING FNLP ALGORITHM TO SOLVE FUZZY MULTI OBJECTIVE PROBLEM Tkng e prof gol s B w olernce P =,, nd spce consrn gol s W w olernce P w e lner memersp funcons μ =,, nd μ w for ree ojecves nd one consrn re s follows: 0 for I BP; BI for B P I B w ; P for I B; fork W; K W forw K W P ; w W PW 0 fork W PW; were, K = K w + K w + K w, IX. FUZZY WEIGHTS Wen e decson mker cn only provde lngusc or mprecse weg e.g. prof of frs ojecve s very mporn, prof of second ojecve s very mporn ec, we my use fuzzy wegs ccordng o Nrsmn [980]. Here, memersp funcons of fuzzy wegs re nroduced o develop comned mesure of e degree o wc ojecve gols re ended. Le w {µ x} represen e weged conruon of e - gol o e overll ggreged ojecve, were w µ x s e memersp funcon correspondng o e fuzzy wegs ssoced w e - gol. Ten y usng mn operon, e memersp funcon µ D x of e decson D s: µ D x = w µ x ^ w µ x ^ ^ w n µ n x = mn { w µ x, w µ x, w n µ n x} Te mxmzed decson x* s oned y: µ D x* =mx {mn { w µ x}}, =,,,n. Noe e memersp funcons of fuzzy wegs re funcons of e memersp funcon of e gol. Te ronly for consrucng ese memersp funcons s e more mporn e gols re, e ger re e degrees of er memersp, nd so e ger re e memersp grde of er fuzzy wegs. X. CLASSIFICATION AND FORMULATION OF MODEL CRISP, NON-INTEGRATED Ts model s furer developed w crsp nd fuzzy wegs s follows: Crsp weged crsp model, Fuzzy weged crsp mode A. Crsp Weged Crsp Model Le w, w, nd w re e nuve crsp wegs for e frs em, second em nd rd em respecvely. Ten e model n cn e formuled y usng wo ecnues: Zmmermnn s ecnue nd Addve Gol Progrmmng

6 Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0 ecnue s follows: Zmmermnn s model: Mxmze λ Sujec o, w ; euons 5, 6, U I LI I LI K W nd 0,,,; w, 0 λ., Addve Gol Progrmmng model: Mxmze V λ, λ, λ = w λ + w λ + w λ I L I Sujec o,, euons 5, 6, U L I I 5 K W nd 0,,,, w, 0 λ., 6 Oservng e opml resuls of Zmmermnn s model nd Addve Model w crsp wegs, we ve developed furer only Zmmermnn s model w crsp nd fuzzy wegs n fuzzy envronmens nd w fuzzy wegs n crsp envronmens. B. Fuzzy Weged Crsp Model le w, w, nd w re e nuve fuzzy wegs for e frs em, second em nd rd em respecvely, en e model cn e formuled y usng Zmmermnn s ecnue s follows: Mxmze λ Sujec o, ; euons 5, 6, w U I LI W, nd 0, I LI,, K, 0 λ. 7 XI. CLASSIFICATION AND FORMULATION OF MODEL FUZZY, NON-INTEGRATED Ts model s furer developed w crsp nd fuzzy wegs s follows: Crsp Weged Fuzzy Model Fuzzy Weged Fuzzy Model A. Crsp Weged Fuzzy Model Le w, w, w nd w re e nuve crsp wegs for e frs em, second em, rd em nd floor spce respecvely, nd en e model cn e formuled y usng Zmmermnn s ecnue s follows: Mxmze λ B I K W Sujec o, w ; w ; P PW w + w + w + w =, euons 5, 6, K, 0 λ. 8 W, nd 0,,, B. Fuzzy Weged Fuzzy Model Le w, w, w nd w re e nuve fuzzy wegs for e frs em, second em, rd em nd floor spce respecvely, en e model cn e formuled y usng w crsp wegs.e. 5 nd 6 re presened n TABLES III nd IV respecvely. Zmmermnn s ecnue s follows: Mxmze λ Sujec o, BI K W ; ; w w P PW K w + K w + K w W, euons 5, 6, 0 λ, 0, =,,. 9 XII. ILLUSTRATION OF THE MODELS To llusre e ove crsp nd fuzzy models for negred nd non-negred usnesses, we ssume e followng npu d sown n Tle I. Opml resuls of non-negred usnesses Usng LINGO Sofwre re gven elow: For e ove d, e followng py-off mrx cf. TABLE II s consruced nd en e opmum resuls for e dfferen represenons of e crsp nvenory model Here, opmum resuls of e crsp model y wo dfferen meods re presened. In ec meod, four dfferen cses ve een mde ou, dependng upon e relve mpornce gven mong e ree ojecves. In cse- eul wegge o ll ojecves; n cse, more mpornce o s ojecve n e oer wo ojecves; n cse-, more cre o mxmzon of nd ojecve n oers; nd smlrly n cse-, rd ojecve receved more wegge n oers. As expeced, cse- gves mxmum reurn wen mxmum enon s pd o e s ojecve; smlrly cse nd cse gve eer resuls f e decson mker gves mxmum mpornce o e mxmzon of nd nd rd ojecves respecvely. Now, we fnd e opmum resuls of e crsp nvenory model w fuzzy wegs.e. 6, wc re sown n TABLE V. Fuzzy d: I = $9,$, I = $66,$78, I = $60,$660, W =s.f.75,s.f.5w npu d. Here, wo dfferen cses ve een mde ou, dependng upon e relve mpornce gven mong e ree ojecves. In cse- eul fuzzy wegge o ll e ojecves; n cse, more mpornce o s ojecve n e oer wo ojecves. As expeced, o cse nd of fuzzy weged crsp model gve eer reurn n e crsp weged crsp model Zmmermnn s Model n oly nd wen specl enon s pd o prculr ojecve respecvely. Now, we fnd e opmum resuls of e fuzzy ojecves.e. 7 w e crsp wegs, wc re sown elow: Te py-off mrx s mde y eung of e res wo ojecves w of e ojecve consdered n TABLE II. Te opmum resuls of e crsp weged crsp models re presened n TABLE III nd IV respecvely. TABLE III: n cse- eul wegge o ll ojecve; cse- gves mxmum reurn wen mxmum enon s pd o e s ojecve; smlrly cse- nd cse- gve eer resuls f DM gves mxmum mpornce o e nd nd rd ojecves

7 respecvely. TABLE V: s expeced, gves mxmum reurns wen mxmum enon s pd o e s, nd nd rd ojecve n cse, nd respecvely n TABLE III. Models presened n TABLES VI nd VII gve mxmum reurn wen mxmum enon s pd o e s ojecve n cse-, n cse- of Tle III nd V respecvely. Fuzzy wegs: fuzzy model gves more prof n crsp model n o e cses. In erms of ol prof, TABLE VIII gves eer resul n TABLE VII. Ts model cn e exended o N nclude dscoun, rndom plnnng orzon, slvge of deerored ules; ec. Deermnon of exc wegs for mul-em mul-ojecve fuzzy model nd er soluon my e e opc of e furer reserc. Ovously, crsp model for negred usnesses gves es resul n ll ype of models for e non-negred usnesses. Now, for negred usnesses, ll coss re ken s rngulr fuzzy numers TFNs s sown n TABLE IX.,8,, 5,65,85, 6,9,,,.5,, 6,0,, 55,75,95, C 7,0,,.,.7,., H 9,,7, 65,85,05, N 8,,, C H N.5,,.5 C H Te opmum resuls for e negred represenon of e fuzzy nvenory model.e. re presened n TABLE IX. N H Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0 For e more relsc suons, we ere consder ll e coss o e rpezodl fuzzy numers TrFNs s sown elow:,7,9,, 5,60,70,85, 6,7,0,,,.,.7,, 6,9,,, 55,70,80,95, 7,9,,, C.,.5,.9,., H 9,,,7, N 65,80,90,05, C 8,0,,,.5,.8,.,.5 N C H And e correspondng opmum resuls for e negred represenon of e fuzzy nvenory model gven n re presened n TABLE X. Here, opml resuls of fuzzy model for negred usnesses re presened w possly, necessy nd credly mesures w rngulr nd rpezodl fuzzy numers sown n TABLE IX nd X. Decson Mkers cn ke decson ccordng o e vlle suons. XIII. CONCLUSION Tll now, n e feld of nvenory, some mul-ojecve models of deerorng ems w wo or more ojecves re vlle n crsp nd fuzzy envronmen. Here, nvenory models vng Weull re of decy w ree ojecves llowng sorges ve een presened n crsp nd fuzzy envronmens for negred nd non-negred usnesses. Te models of non-negred usnesses ve een solved y FNLP ecnues nd e model of negred usnesses s een solved y Generlzed Reduce Grden Meod y nlyzng possly/necessy mesure. Te resuls ve een presened w dfferen ypes of wegs dmssle o e ojecves for non-negred usnesses. Ec weg, wc mples e relve mpornce of e ojecve gols, cn e deermned roug e prccl experences. Toug e prolem s een formuled n e feld of nvenory, e presen meodology n formulon nd soluon cn e doped for fuzzy non-lner decson mkng prolem n ny dscplne. Moreover, n s pper, model s een formuled w me-dependen demnd, demnd-dependen producon nd me-vryng oldng cos llowng sorges for negred nd non-negred usnesses. All coss re ken s rngulr fuzzy numers nd rpezodl fuzzy numers n e cse of negred usness. Te presen nlyss cn e esly exended o oer ypes of nvenory models w consn demnd, nfne replensmen, fxed - me orzon ec. Our proposed model s wde rnge of pplcon n e feld of usnesses nd mngemen n wc e economc condons cn lso e developed. Ts model cn e furer exended o nclude dscoun, rndom plnnng orzon, slvge of deerored ules; ec. Deermnon of exc wegs for mul-em mul-ojecve fuzzy model nd er soluon my e e opc of e furer reserc. TABLE I: INPUT DATA FOR THREE ITEMS Iems m N $ H $ C $ α β γ w W S. F TABLE II: PAY-OFF MATRIX I $ I $ I $ TABLE III: CRISP WEIGHTED CRISP MODEL ZIMMERMANN S MODEL Cse w w w I $ I $ I $ T I$ SC S. F. / / /

8 Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0 TABLE IV: CRISP WEIGHTED CRISP MODEL ADDITIVE MODEL\ Cse w w w I $ I $ I $ T I$ SC S. F. / / / TABLE V: CRISP WEIGHTED FUZZY MODEL ZIMMERMANN S MODEL Cse w w w w I $ I $ I $ T I$ SC S. F. / / / / TABLE VI: FUZZY WEIGHTED CRISP MODEL ZIMMERMANN S MODEL Cse w w w I $ I $ I $ T I$ SC S. F. [.,.] [.,.] [...] [.8,] [.,.8] [0,.] TABLE VII: FUZZY WEIGHTED FUZZY MODEL ZIMMERMANN S MODEL w w w W I $ I $ I $ T I$ SC S. F. [0,] [.5,] [.,.999] [.55,.98] [.8,] [.,.8] [.,.5] [.5,] TABLE VIII: INTEGRATED MODEL GRG METHOD I $ I $ I $ T I$ SC S. F TABLE IX: INTEGRATED FUZZY MODEL FOR TFNS NECESSITY/POSSIBILITY ATTITUDE ρ I $ T SC S. F. ρ I $ T SCS. F. ρ I $ T SC S. F TABLE X: INTEGRATED FUZZY MODAL FOR TRFNS NECESSITY/POSSIBILITY ATTITUDE ρ I $ T SC S. F. ρ I $ T SC S. F. ρ I $ T SC S. F APPENDIX A A.: Any fuzzy suse of were represens se of rel numers w memersp funcon µ ã x : [0,] s clled fuzzy numer. Le nd e wo fuzzy unes w memersp funcons x nd y respecvely. Ten ccordng o Duos nd Prde 987, Lu nd Iwmur 998, M nd M 006 Pos sup mn x. y, x, y, xy Nes nf mx x. y, x, y, xy were e revon `Pos represens possly nd `Nes represens necessy nd `* s ny of e relons >, <, =,,. Te dul relonsp of possly nd necessy reures Pos Nes Also necessy mesures ssfy e condon, MnNes, Nes 0 Te relonsps eween possly nd necessy mesures ssfy e followng condons cf. Duos nd Prde 988}: Pos Nes Nes > 0 Nes 0 If, Є nd c f,, en Pos nd Pos. were f : Be nry operon, en memersp funcon c of c s defned s: sup mn x y x y z.,, ndz f x yz c, Recenly sed on possly mesure nd necessy mesure, e rd se funcon Cr, clled credly mesure, nlyzed y Lu nd Lu 00 s s follows: Cr A =/[Pos A +Nes A ] for ny A n, were s e power se of. I s esy o ceck Cr ssfes e followng condons: 6

9 Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0 Cr 0ndCr ; CrA Cr B wenevera, B nda B Tus Cr s lso fuzzy mesure defned on,. Besdes, Cr s self dul,.e. c. Cr ACr fornya n A In s pper, sed on e credly mesure e followng form s defned s Cr A Pos A Nes A cf. Lu nd Lu 00 for ny A n nd 0 < ρ <. I lso ssfes e ove condons. A.. Trngulr Fuzzy Numer: Trngulr fuzzy numer TFN A see Fg. A - s e fuzzy numer w e memersp funcon A x, connuous mppng: Ax 0,, 0 for x x for x x A x for x 0 for x Lemm : Te expeced vlue of rngulr fuzzy numer A,, s E A = / Proof. Le, numer. A, e rngulr fuzzy Cr 0fr r f r Ar r f r fr Bsed on e credly mesure, Lu nd Lu 00, 00 presened e expeced vlue operor of fuzzy vrle s: Le A e normlzed fuzzy vrle. Te expeced vlue of e fuzzy vrle A s defned y 0 0 E A Cr A r dr Cr Ar dr Wen e rg nd sde of e ove euon s of form -, e expeced vlue cnno e defned. Also, e expeced vlue operon s een proved o e lner for ounded fuzzy vrle,.e., for ny wo ounded fuzzy vrles XndY,we ve EX Y EX EY for o rel numers nd. Ten e expeced vlue of fuzzy vrle A s defned s: E A Cr A r dr CrArdr 0 CrAr CrAr E A / A.. Trpezodl Fuzzy Numer: Trpedodl fuzzy numer TrFNs A see Fg. A - s e fuzzy numer w e memersp funcon A x, Fg. A. Memersp funcon of Trngulr Fuzzy Numer TFN A,, fr r PosAr f r 0fr fr r NesAr f r 0fr fr r f r CrAr r f r 0fr Fg. A. Memersp funcon of rpezodl fuzzy connuous Numer TrFN A,,, mppng: x 0, A 0 for x x for x A x for x x for x 0 for x fr r PosAr f r 0fr fr r NesAr f r 0fr 7

10 Te credly mesure for TrFNs cn e defned s: fr r f r CrAr f r r f r 0fr 0fr r f r CrAr f r r f r fr Bsed on e credly mesure, Lu nd Lu00, 00, s descred n rngulr fuzzy numer. Ten e expeced vlue of rpezodl fuzzy vrle s defned s: A / E. A.. Mul Ojecve Prolem Under Fuzzy Expeced Vlue Model: A generl mul - ojecve memcl progrmmng prolem w fuzzy prmeers n e ojecve funcon s of e followng form: Mx f x, f x,, f x,..., f,, n x Sujec o, x, 0 g, j=,,, k, were x nd re decson vecor nd fuzzy vecor respecvely. To conver e fuzzy ojecve nd consrns o er crsp euvlens, Lu & Lu 00 proposed new meod o conver e prolem no n euvlen mul-ojecve fuzzy expeced vlue model.e. e euvlen crsp model s: Mx E f x, E f x, Sujec o, g x Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0, Ef x,..., Ef n, 0, x E, j=,,, k., REFERENCES [] A. Roy nd G. P., Smn; And EOQ model of deerorng ems w me vryng demnd nd sorges, AMSE 7-9, 006. [] Co-Ton Su nd Cng-Wng. Ln; A producon nvenory model wc consders e dependence of producon re on demnd nd nvenory level, Producon Plnnng nd Conrol, Vol.,, 69-75, 00. [] M. M; Deermnsc nvenory models for vrle producon, Journl of e Operonl Reserc Socey, 8, [] A. K. Bun nd M. M; Deermnsc nvenory models for vrle producon, Journl of Operons Reserc, 8, -, 997. [5] A. Roy; An Invenory model for deerorng ems w prce dependen demnd nd me vryng oldng cos, Advnced Modelng nd Opmzon, Vol. 0, 008. [6] S. K. Goel nd B. C. Gr; Inved revew recen rends n modelng of deerorng nvenory, Europen Journl of Operonl Reserc, Vol., -6, [7] G. Pdmnn nd P. Vr; EOQ model for persle ems under sock dependen sellng re, Europen Journl of Operon Reserc, 8 9. [8] P. L. Ad;Opml prcng nd lo szng under condons of persly nd prl ck- orderng, Mgm. Sc,, 09 0, 996. [9] W. Luo; An negred nvenory sysem for persle goods w ck orderng, Compuer Ind. Engg,, 85 69, 998. [0] S. P. Aggrwl nd V. P. Goel; order level nvenory sysem w demnd pern for deerorng ems, Econ. Comp. Econ. Cyerne, sud. Res,, 57-69, 98. [] J. Ro Lksmnryn, K. Srnvs, nd Jon R. Mew; Persle Invenory Model w fne re of replensmen vng Weull lfeme nd Prce dependen demnd. IAPQR Trnscons, Vol. 0, No.,, 005. [] H. Hwng nd S.W. Snn; Reler s prcng nd lo-szng polcy for exponenlly deerorng producs under e condon of permssle dely n pymens, Compuer nd Operon Reserc,,59-57, 997. [] A. Al kedr nd L. Tdj; Opml conrol of producon nvenory sysem w Weull dsrued deeroron, Appled memcl scences, Vol., no. 5, 70 7, 007. [] S. K. Gos nd K. S. Cwdry; An order level nvenory model for deeroron em w Weull dsruon deeroron, me udrc demnd nd sorges, Advnced Modelng nd Opmzon, 6, 5, 00. [5] J. M. Cen nd S. C. Ln; Opml replensmen scedulng for nvenory ems w Weull dsruon deeroron nd me vryng demnd, nformon nd Opmzon scence,,.00. [6] N. K. Mpr nd M. My; Mul-ojecve nvenory models of mul-ems w uly nd sock-dependen demnd nd socsc deeroron, Advnced Modelng nd Opmzon, 7,, 005. [7] S. Kr nd M. My; Mul ojecve nvenory model of deerorng ems w spce consrn n fuzzy envronmen, Tmsu Oxford Journl of memcl Scences,, 7 60, 008. [8] A. Roy nd K. My, S. Kr, nd M. My; A producon nvenory model w remnufcurng for defecve nd usle ems n fuzzy envronmen, Compuers nd Indusrl Engneerng 56,87-96, 009. [9] E. Nddor; Invenory sysems, Jon Wley, New Yrk, 966. [0] D. Levs; Scenfc Invenory Conrol, Buerwors, Lndon, 970. [] T. K. Roy nd M. My; A Fuzzy EOQ model w demnd dependen un cos under lmed sorge cpcy, Europen Journl of Operon Reserc, 99, 5, 997. [] M. Mndl nd M. M; Invenory for dmgele ems w vrle replensmen nd sock-dependen demnd, As-Pcfc of Operonl Reserc, 7, -5, 000. [] A. K. Bun nd M. M; An nvenory model for decyng ems w sellng prce, freuency of dversemen nd lnerly me dependen demnd w sorge, IAPQR rnscons,, - 9, 997. [] G. A. Grel nd K. M. Rgsdell; e generlzed reduced grden meod, AMSE Journl of Engneerng for Indusry, 99, 8-00, 977. [5] R. N. Twr, S. Drmr, nd J. P. Ro; Fuzzy gol progrmmng-n ddve model, Fuzze Ses nd Sysem,, 7-, 987. [6] H. J. Zmmermnn; Fuzzy lner progrmmng w severl ojecve funcons; Fuzzy Ses nd Sysems,, 6-55, 978. [7] P. A. Run nd R. Nrsmn; Fuzzy Gol Progrmmng w Nesed Prores, Fuzzy Ses nd Sysems,, 5 9, 98. [8] S. Kr, A. K. Bun nd M. My; Invenory of mul deerorng ems sold from wo sops under sngle mngemen w consrns on spce nd nvesmen, Compuers nd Operonl Reserc, 8, 0-, 00. [9] I. A. Zde; Fuzzy ses s ss for eory of possly, Fuzzy Ses nd Sysems,, -8, 978. [0] D. Duos nd H. Prde; Possly Teory, New Yrk: Plenum, 988. [] A. Roy, S. Kr, nd M. M; Volume Flexle Invenory Conrol Sysem w Imperfec Quly nd Mcne Relly n Socsc nd Fuzzy Socsc Envronmens, Tmsu Oxford Journl of Mngemen Scences,, 7-6, 007. [] A. Roy; An Invenory model for deerorng ems w prce dependen demnd nd me vryng oldng cos, Advnced Modelng nd Opmzon, Vol. 0, 008. [] B. Lu nd Y. K. Lu; Expeced vlue of fuzzy vrle nd fuzzy expeced vlue model. IEEE Trnscon of Fuzzy Ses nd Sysems 0, 5-50, 00.

11 Inernonl Journl of Modelng nd Opmzon, Vol., No., Aprl 0 Sv Pk ws orn n Mrzpur, U. P., Ind on.0.97; Qulfcon: Hg Scool 98 nd Inermede from U. P. ord, Alld, U. P., Ind. B.Sc.Hons. n Memcs, 99 nd M.Sc.Ms speclzon n dfferenl geomery nd Specl funcon 99, from Bnrs Hndu Unversy, Vrns, U. P., Ind, compleed ree mons C Progrmmng rnng course, ulfed n Cred Course Ms., 00, nd currenly sumed P. D. ess Nonl Insue of Tecnology, Durgpur, W. B., Ind Se ended e sor erm course on Compuonl Inellgence nd s Applcons, eld Nonl Insue of Tecnology, Durgpur, seleced s reserc scolr under TEQIP sceme on Ocoer, 006. Afer seleced s n nsue reserc scolr on rd Aprl 008 n e Deprmen of Memcs, Nonl Insue of Tecnology, Durgpur W.B., Ind; prcped n e Nonl Worksop on E-Lernng for pedgogcl nnovons nd prcped n e nonl worksop on dvnces n compuonl opmzon nd pplcons durng 8 - novemer 00 nd n e nernonl worksop on fuzzy ses, roug ses, uncerny nlyss nd pplcons, fru 0 durng s -5 Novemer, 0 Deprmen of Memcs, Nonl Insue of Tecnology, Durgpur, W.B., Ind. Her pulsed rcles re: Sv Pk, Seem Srkr Mondl. 00; A Tree Pln Opml Producon Prolem Under Vrle Inflon nd Demnd w Necessy Consrn, Imperfec Quly nd Lernng Effecs, Journl of Compuer nd Memcl Scences, ISSN , Vol. 7, Sv Pk, Seem Srkr Mondl. 0; Posslsc lner progrmmng pproc o e mul-em ggrege producon plnnng, pulsed n Inernonl Journl of Pure nd Appled Scence nd Tecnology IJPAST w ISSN Sv Pk, Seem Srkr Mondl. 0; A fuzzy EOQ nvenory model for rndom Weull deeroron w Rmp Type demnd, prl ckloggng nd nflon under rde cred fnncng, Inernonl Journl Of Reserc In Commerce, IT &Mngemen w ISSN Curren nd prevous reserc neress re furer conruons n e feld of nvenory mngemen, supply cn ggrege producon plnnng w pplcon of fuzzy, roug nd uncerny eory. Mrs. Pk receved scolrsp n junor g scool, mer n Memcs n g scool nd grduon, leer of pprecon from e nernonl journl of reserc n commerce, &mngemen. Seem Srkr Mondl ws orn n Kolk, W. B., Ind on 0..96, Kolk. Secondry Exmnon, 979, Wes Bengl ord of Secondry Educon Insue-Holy Cld Grls Hg Scool, nd Hger Secondry Exmnon, 98, Wes Bengl Councl of Hger Secondry Educon Insue-P Bvn, from Kolk, W. B., Ind; B.Sc.Hons.-Ms, 985, Insue-Presdency College, M.Sc. Appled Ms, 987, M. Pl. Appled Ms, 989, nd P.D. Scence n Ms, 996, from Unversy of Clcu, W. B., Ind. Her Tecng Experence: ffeen yers + o u.g. nd p.g s ssoce professor, deprmen of memcs, nonl nsue of ecnology, durgpur, w.., Ind. Nne ppers pulsed n nonl nd nernonl journls; conduced nonl semnr nd wner scool n e deprmen of memcs, nonl nsue of ecnology, durgpur, w.., Ind. Se ended more n sxeen wner scools, sor erm courses, semnrs, conferences, nd ldes osel wrden Se ws ed of e deprmen ms., , Nonl Insue of Tecnology, Durgpur, W.B., Ind. Some pulcons re: Seem Skr Mondl e. l., 00; Assocon sudy eween led nd copper ccumulons dfferen Pysologcl sysems of cle y cnoncl co-rrelon nd cnoncl correspondence nlyses; Envronmen nd Ecology, 8,-5. Seem Srkr Mondl e. l., 0; Assocon sudy eween led nd copper ccumulon dfferen pysologcl sysems of go y pplcon of cnoncl co-rrelon nd cnoncl corresponce nlyses, Indn Journl of ppled reserc, vol, ssue 5, 7-9. Curren nd prevous reserc neress: Operons Reserc, Porfolo, Supply Cn Mngemen, Invenory, Aggrege Producon Plnnng, Sscl Anlyss nd Geodesy nd Geopyscs. Dr. Mrs. Mondl s Lfe Memer of Clcu Memcs Socey, W.B., Ind, receved BEST PAPER AWARD for presenon of pper n Inernonl Conference on Informon nd Mngemen Scence, eld n Urumc, Cn, durng Augus -9, 00, nonl scolrsp n secondry exmnon 979, leer of pprecon from e nernonl journl of reserc n commerce, &mngemen. 9