A displayed inventory model using pentagonal fuzzy number
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1 Itrtol Jourl of Mthmts oft omut Vol.6 No IN rt : 9 8 IN Ol: 9 sly vtory mol us tol fuzzy umr K. hm M. rmlv rtmt of Mthmts Govrmt rts oll for om utoomous uuott Tmlu. hmsthvl@yhoo.o. rtmt of Mthmts Govrmt rts oll for om utoomous uuott Tmlu. vvsh@ml.om strt Ths r ls th Fuzzy Mult-tm sly vtory mol th ltrtv or suly ost or rtor. Th ost rmtrs th ostrt r rrst y th tol fuzzy umr. Th mol s solv y fuzzy omtr rormm mtho. Th otml orr qutty umr of sly qutty hv trm. umrl xml s v to llustrt th mol. Kyors: sly vtory oom orr qutty fuzzy omtr rormm tol fuzzy umr rst trvl roxmto. M ujt lssfto : 9B. Itrouto Mult tm lssl vtory mols ur vrous tys of ostrts suh s tl vstmt vll stor r umr of orrs vll st u tm r rst ll o oos rtt y hurhm off roff [ Hrly ht [ lvr trso. hl mol vtory rolm rlly thr tys of m r osr. Thy r ostt m tm t m sto t m. I th sto t m slly sly vtory lvl m hs fft o sls for my rtl routs. ht [6 stt tht for th rtl stors th vtory otrol rolm for styl oos s furthr omlt th th ft tht th vtory th sls r ot t to h othr. rs vtory my r rs sls of som tms. or to lvr trso [
2 K. hm M. rmlv th sl t th rtl lvl s roortol to th mout of sly vtory. Th most of th rtlrs sly som routs o shlf follo th rout vrty ho of th ustomrs tors r qulty hysl sz of th rout to flu th ustomr s ttto. Ur [ vlo mol to tfy thos routs hh shoul lu frm s rout l hh th m rt s olyoml futo of r vrts struto. orstjs oyl [ vlo shlf s lloto mol hh m rt s futo of shlf s llot to th rout. But ll ths vtory rolms r solv th th ssumto tht th o-fft or ost rmtrs r sf rs y. I rl lf thr r my vrs stutos u to urtty. Hr vtory osts r mrs tht s fuzzy tur. Erly ors us fuzzy ot so m r o y Zh [8 Bllm [ y trou fuzzy ols osts ostrts. tr th fuzzy lr rormm mol s formult roh for solv lr rormm mol th fuzzy umrs hs rst y Zmmrm [9. Gomtr rormm mtho s rltvly thqu to solv o-lr rormm rolm. uff trso Zr [ frst vlo o G mtho. Kothrr s th frst to us ths mtho o vtory rolms. tr o orrll Hll [ lyz mult-tm vtory mol th svrl ostrts us osyoml G mtho. tr th Gomtr rormm thqus r suss y ou-l-t Fry El-l [ Ml oy [8 [9 [ [. tly Ml oy [ rst sly vtory mol th trulr fuzzy umr. Th srty of or ffts th smll sl ustrs suh s Bry sturts foo rout oms tl shorooms. To solv ths rolm rtors r stll t urs ost. Ths r trous th ost s ltrtv or suly ost. lso th tol fuzzy umr s f. o th sly vtory mol y us tol fuzzy umr th ltrtv or suly ost hs osr. I ths r mult tm sly vtory mol ur shlf s ostrt fuzzy vromt s formult. lso or rtor hs us oth room stor r sly r. Th rmtrs volv ths r r ssum to mrs tur th rmtrs r rrst y tol fuzzy umrs th ffrt tys of lft rht mmrsh futos. Th mol s th ru to mult-ojtv so-m vtory rolm s solv y fuzzy omtr rormm mtho. Flly umrl xml s v to llustrt th mol. ssumtos Nottos mult-tm sly vtory mol th rtor ost s formult ur th follo ssumtos.
3 sly vtory mol us tol fuzzy umr ssumtos:. Th ut ost of th tm s t of.. Th sly ost os ot o th lth of yl tm T.. Th outst orr s vr mor th o.. tm s zro.. horts r ot llo. 6. m rt s o sly vtory for th tm Hr =.. r th sl sh rmtrs of th m futo. Full-shlf mrhs oly hs ot hr th sly r s lys t fully sto so th vtory s rlsh s soo s th room vtory rhs zro. Th sly vtory ll lys t ts mxmum. Th vtory lvl rss t ostt rt. 8. ltrtv or suly or rtor ost s llo. Nottos: t thr tms. Th follo r for th th tm T - umr of sly qutty so vrl... T - umr of orr qutty so vrl... - sttous vtory lvl of th tr systm lu oth th room stor th sly Ivtory t vtory - sll r r ut - urhs r r ut - hol ost r ut r ut tm - sly shlf ost r ut r ut tm - st u ost r yl - m rt - routo rt T - ltrtv or suly ost or rtor r ut r ut tm - fuzzy roft futo - yl tm - totl sly shlf s.
4 K. hm M. rmlv Mthmtl Mol rs Evromt Th vtory mol s formult to mxmz th vr t roft hh lus th ross rvus ut urhs ost stu ost hol ost th sly ost ur th lmt sly s ostrt. vr roft = Gross rvus r ut urhs r r ut stu ost r ut tm hol ost r ut tm rtor ost r ut tm sly shlf s ost r ut tm H th roft futo s F hr th vr vtory s. vr roft futo s ru to F Th rolm s th stt s Mx F sujt to: ; Th str omtr rormm rolm s F M sujt to:. ; Ths rml rolm s ostr soml rolm th - r of ffulty. Th orrso ul rolm s 6 6 Mx sujt to: = -
5 sly vtory mol us tol fuzzy umr = + hr 6 > = By us omtr rormm thorm [ th lytl xrssos for th so vrls r ot. 6 tol Fuzzy umr ts Nrst Itrvl roxmto fto.. tol fuzzy umr s fuzzy sust o th rl l hos mmrsh futo x s f s follos: x x x x x x x x x x x hr.6 r rl umrs. Ths ty of fuzzy umr ot s = ; stsfs th follo otos: FN.. s otuous m from to th los trvl [.. s ovx futo.. =.. x = x s strtly rs o.. x = x=. 6. x = x s strtly rs o.
6 6 K. hm M. rmlv. x =. mrs:. If <.6 th oms trulr fuzzy umr.. If = th oms trzol fuzzy umr. Fur : Grhl rrstto of tol Fuzzy umr for =.. Nrst Itrvl roxmto: Hr roxmt fuzzy umr y rs mol. uos B r to fuzzy umrs th α -uts r [ α α [B α B α rstvly. Th th st t B s B - B - B Gv s tol fuzzy umr. hv to f los trvl hh s th rst to th rst to mtr. o t s h trvl s lso fuzzy umr th ostt α - ut for ll α [. H α =[. No hv to mmz - - th rst to.. I orr to mmz t s sufft to mmz th futo. Th frst rtl rvtvs r
7 sly vtory mol us tol fuzzy umr olv t. hv.. H H. Tht s s lol mmum. Thrfor th trvl s th rst trvl roxmto of fuzzy umr th rst to th mtr. t = tol fuzzy umr. Th α lvl trvl of s f s α = α α. h s lr fuzzy umr FN th lft rht α uts r x f x f x f x f By th rst trvl roxmto mtho th lor ur lmts of th trvl r rstvly. h s rol fuzzy umr th lmts r v y. mlrly h s hyrol fuzzy umr th lor ur lmts r lo lo
8 K. hm M. rmlv 8 t t Th roos vtory mol fuzzy vromt: If th ost rmtrs totl sly shlf s rmtrs r fuzzy umrs th th rolm s trsform to sujt to: ; hr rrsts th fuzzfto of th rmtrs. I our roos mol th ost rmtrs r osr s tol fuzzy umrs. Our roos mol s ru to F [ [ [ [ [ [ [ [ [ [ [ [ [ [ Mx = [F F sujt to: ; [
9 sly vtory mol us tol fuzzy umr 9 hr F F 6 ss of roos vtory mol th tol fuzzy umr s : ll th ost rmtrs r fuzzf th totl sly shlf-s rmtr s trmst. 8 sujt to:. ; Us th Nrst Itrvl roxmto th ov mol s fuzzf s follos: Mx F [F F sujt to:. ; Th mol s ovrt to mult-ojtv o-lr rormm rolm v lo. F Mx 9
10 K. hm M. rmlv F Mx sujt to:. ; Th mult-ojtv vtory rolm 9 s solv y th omtr rormm thqu y-off mtrx of orr s form. Th str omtr rormm rolm s F M sujt to: ; Ths rml rolm s ostr soml rolm th - r of ffulty. Th orrso ul rolm s 6 6 Mx sujt to: = = - + = hr 6 & >.
11 sly vtory mol us tol fuzzy umr By us omtr rormm thorm [ th lytl xrssos for th so vrls r ot. usttut ot. F F th otml vlus of F F r I smlr y th otml vlus of for F [ sujt to th sm ostrt r ot. 6 usttut F F th otml vlus of F F r ot. Us th otml solutos yoff mtrx of sz s form From th yoff mtrx th lor ous r = M[ =M[ th ur ous r U = Mx U =Mx[. Th rolm 8 formult s Mx V = [ Tht s [ sujt to: ; hr
12 K. hm M. rmlv. Th str omtr rormm rolm s [ sujt to:. ; Ths rml rolm s ostr soml rolm th - r of ffulty. Th orrso ul rolm s V Mx sujt to: = = - + = hr 6 & >. By us omtr rormm thorm [ th lytl xrssos for th so vrls r ot. 8 9 s : Th ost rmtrs r trmst th sly shlf s rmtr s tol fuzzy umr. Th th rolm s F Mx sujt to:. ;
13 sly vtory mol us tol fuzzy umr Us th Nrst Itrvl roxmto th ov mol s fuzzf s F Mx sujt to:. ; Th str omtr rormm rolm s F M sujt to: ; Ths rml rolm s ostr soml rolm th - r of ffulty. Th orrso ul rolm s Mx sujt to: = = - + = hr 6 >. By us omtr rormm thorm [ th lytl xrssos for th so vrls r ot.
14 K. hm M. rmlv s : Th ost rmtrs th totl sly shlf- s rmtr r osr s tol fuzzy umrs. sujt to:. ; 6 Us th rst trvl roxmto th ov mol s fuzzf s F Mx F Mx 8 sujt to:. By us th sm rour s ss mult-ojtv vtory rolm s solv y off mtrx s form. lso th mmrsh futo for th ojtv futo hs ostrut. Th rolm 6 formult s Mx V = [. Tht s [ sujt to:.
15 sly vtory mol us tol fuzzy umr Th str soml omtr rormm form stt s [ 9 sujt to:. By us omtr rormm thorm [ th lytl xrssos for th so vrls r ot. Numrl Exml ssum tht rl shoroom slls to tms. Th sho hs totl vll stor s of m. Th rlvt t for th to tms s v lo:. = uts = =. =. = = = = uts =.m =.6 uts = =.6 =. = = =. = uts =. m. Us th lyt xrsso & for F rs fuzzy vromt th follo rsults r ot. Tl : ft ht Brhs of Fuzzy rmtrs. Br ft H H H H ht H H h H H H Hr H sts for rol r Hyrol tol fuzzy mmrsh futo rstvly.
16 6 K. hm M. rmlv Tl : Nrst trvl roxmto to tol fuzzy umrs for Itm &. Br ft ht tr Br ft ht tr Tl : Otml olutos. ss F rs s s s [ [ Osrvto.. I Tl - th otml vlus r v for th fuzzy mol s ll s th rs mol from th sm th follo r osrv. I s th otml vlu of th vr roft s mor th tht of rs mol. I s th otml vlu of th vr roft s lss omr to tht of ss rs mol. I s th otml vlu of th vr roft s mor omr to tht of ss th rs mol. v mo th ov thr ss s vs th st otml soluto. frs [ M. O. ou-l-t H.. Fry M.F. El-l rolst mult-tm vtory mol th vry orr ost ur to rstrtos: omtr rormm roh Itrtol Jourl of routo Eooms 8 -. [. E. Bllm.. Zh so-m fuzzy vromt Mmt s 9 6.
17 sly vtory mol us tol fuzzy umr [... hurhm... off E.. roff Itrouto to Ortos srh N Yor: ly 9. [ M. orstjs. oyl mol for otmz rtl s llotos Mmt [. J. uff E.. trso. Zr Gomtr rormm-thory lton Yor: ly 96. [6. Grzorzs Nrst trvl roxmto of fuzzy umr Fuzzy ts ystms. [ G. Hly T. M. ht lyss of vtory systms. Eloo lfs NJ: rt Hll 98. [8 N.K. Ml T.K. oy M. Mt Mult-ojtv fuzzy vtory mol th thr ostrts: omtr rormm roh Fuzzy ts ystms 8 6. [9 N.K. Ml T.K. oy M. Mt Ivtory mol of trort tms th ostrt: omtr rormm roh Euro Jourl of Ortol srh [ N.K. Ml T.K. oy M. Mt sly vtory mol th fuzzy umr Fuzzy Otm s M 6. [. M. Mt Mult-ojtv fuzzy vtory mol th r t m ur flxlty rllty osrto rs s ostrt: omtr rormm roh Mthmtl omutr Mol [ T.K. oy M. Mt fuzzy EO mol th m-t ut ost ur lmt stor ty Euro Jourl of Ortol srh [ E.. lvr. trso so systms for vtory mmt routo l to N Yor: lly 98. [.. o Otmzto thory ltos ly Estr mt lh 98. [ G.. Ur mthmtl mol roh to rout l sos Jourl of Mrt srh [6 T. M. ht Th thory of vtory mmt rto NJ 9.
18 8 K. hm M. rmlv [ B. M. orrl M.. Hll Th lyss of vtory otrol mol us olyoml omtr rormm Itrtol Jourl of routo srh [8.. Zh Fuzzy ts Iformto otrol [9 H. J. Zmmrm Fuzzy lr rormm th svrl ojtv futos Fuzzy ts ystms 98 6.
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On Matrices associated with L-Fuzzy Graphs
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