Inventory Model For Weibull Decaying Items With Exponentially Decreasing Demand And Permissible Delay In Payments

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1 nnionl Jounl of Engining cinc nvnion N (Onlin): N (Pin): Volum 6 ssu 8 Augus 7 PP nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil Dly n Pymns D Rvish Kum Ydv Associ Pofsso Hindu ollg odd dvishydv@gmil.com osponding Auho: D Rvish Kum Ydv. noducion h EOQ modl is widly usd y pciions s dcision-mking ool fo h conol of invnoy. h diionl EOQ modl ssums h h il mus pid fo h ims s soon s h ims civd. Howv in pcic h suppli will off h il dly piod which is d cdi piod in pying fo h moun of puchsing cos. Bfo h nd of d cdi piod h il cn sll h goods nd ccumul vnu nd n ins. A high ins is chgd if h pymn is no sld y h nd of d cdi piod. n mos of h liu dling wih invnoy polms ih in dminisic o poilisic modl i is ofn ssumd h h dioion ss onc s h ims poducd o puchsd. Bu in mos of h invnoy sysms his is no lly u. hs n osvd h dioion ss f cin piod clld lif im of h im. his lif im diffs fom im o im. n ody s usinss nscion i is mo nd mo common o s h h puchss llowd fixd im piod fo hy sl h ccoun wih h suppli. his povids n dvng o h puchs du o h fc h hy do no hv o py h suppli immdily f civing h ims u insd cn df hi pymn unil h nd of h llowd piod. hus pying l indicly ducs h puchss cos of h ims. On h oh hnd h pmissil dly in pymns poducs nfi o h suppli such s i should c nw puchss who consid i o yp of pic ducion. Anlysis of invnoy of goods whos uiliy dos no min consn ov im hs involvd num of diffn concps of dioion. innnc of such invnoy is polm of mjo concn o mng of modn usinss ognizion. h uliy of socks minind y n ognizion dpnds vy hvily on h im h invnoy spnd in h whous. Dioion in h sockd invnoy hs n xnsivly sudid y mny schs. Gh nd chd (96) w mongs h fis uhos o hv ddssd h polm of dioing invnoy. Aggwl (978) psnd n od lvl invnoy modl wih n xponnilly dcying invnoy. h modl psnd y Dv nd Pl (98) hd im dpndn dmnd fo n invnoy wih consn dioion. L Hig (995) Bhuni nd ii (999) Lio. l. () ng. l (5) nd mo cnly Yng (6). All hv dlid h ffcs of consn dioion on invnoy. Bu consn dioion is h concp which cn no jusify und mny cicumsncs. n fc dioion dpnds upon lo of fcos. Goyl (985) discussd pmissil dly siuion in his pp. ndl nd Phujd (989) xndd Goyl o incopo shogs nd considd h ins nd fom sl vnus. L Aggwl nd Jggi (995) xndd Goyl fo dcying ims. Jml. l. (997) nichd his sudy y considing shogs in h cycl. Lio. l. () sudid pmissil dly und inflion. hung nd Lio (4) xndd h concp of cdi limi y linking i wih oding uniy. hng (4) hs fuh xplod his gion y incopoing h ffcs of inflion on h fom pp. n his pp n invnoy modl is dvlopd in which dmnd is xponnilly dcsing wih im. Dioion is kn non insnnous. Rlisic siuion of pmissil dly is lso kn in considion. h diffn css hv n discussd fo diffn siuions. Expssions oind fo ol opiml cos in diffn siuions. h diffn lgoihms givn o oin h opiml soluion. os minimizion chniu is pplid o solv h modl. h following noions nd ssumpions pplid in h modl.. Noions. : Oding cos of invnoy p od.. : Holding cos xcluding ins chg p uni p uni im.. : hog cos p uni p uni im. 4. : Uni puchs cos. 5 Pg

2 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil 5. : ns pid p up invsd in socks p y >. 6. : ns which cn nd p up p y. 7. () : nvnoy lvl im. 8. : Pmissil dly piod fo sling ccouns in im <<. 9. : im which shogs ss.. : Lngh of plnishmn cycl.. : Lif piod of im h nd of which dioion ss.. Q : ol moun of invnoy poducd o puchsd h ginning of ch poducion cycl.. (<Q) : niil moun of invnoy f fulfilling ck ods. 4. ( ) : h ol vg cos of h invnoy sysm p uni im. 5. () ( ) : h ol vg cos of h invnoy sysm p uni im fo nd. 6. () ( ) : h ol vg cos of h invnoy sysm fo nd. 7. ( ) : h ol vg cos of h invnoy sysm p uni im fo >.. Assumpions. h invnoy sysm consiss of singl im only.. h is no pi o plcmn of h diod uni.. h plnishmn occus insnnously n infini. 4. Whn poducd o puchsd ims iv in sock hy fsh nd nw. hy gin o dio f fixd im invl. h dioion funcion () is kn in h following fom () = - H ( - ) ( < << ) > And H ( -) is hvisid funcion dfind H 5. Dmnd is known nd incss xponnilly im. D() = H is h iniil dmnd nd is consn govning h incsing of dmnd. 6. Duing h fixd cdi piod h uni cos of gnd sls vnu is dposid in n ins ing ccoun. h diffnc wn sls pic nd uni cos is ind y h sysm o m h dy o dy xpnss of h sysm. A h nd of h cdi piod ccoun o sld. hn ins is gin nd duing h piod ( ). f ins chgs pid on h sock hld yond h pmissil piod. 7. hogs llowd nd hy fully ckloggd. V. hmicl odl And Fomulion Of h ysm H Q is h ol moun of invnoy poducd o puchsd h ginning of ch poducion cycl. Also ( <Q) is h iniil invnoy f fulfilling ck ods. Duing h piod [ ] h invnoy lvl dcss du o h mk dmnd only. Af his duing h piod [ ] h invnoy lvl fuh dcss du o h comind ffc of mk dmnd nd dioion. A im h invnoy lvl flls o zo nd shogs ss. Dmnd is ckloggd in h invl [ ]. A im moun of invnoy is lf fo h nx plnishmn cycl. h diffnil uions showing viions of invnoy lvl duing h piod [ ] s follows d () d d d...() d d...() 5 Pg

3 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil 54 Pg Boundy condiions oluion of uion (6.) using oundy condiion is givn y-...(4) Also uion (6.4) i ducs o...(5) oluion of uion () using oundy condiion on cn g-..(6) Using oundy condiion fom uion (6.6) h vlu of is givn y-...(7) Now susiuing h vlu of fom uion (7) in uion (6) on cn g-.(8) oluion of uion () is givn y...(9) ol moun of holding unis H duing h piod is H d d

4 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil 55 Pg usiuing h vlu of fom () w hv H () ol moun of diod unis D duing h piod is d D () Amoun of shog unis duing h piod is givn y d () Now h wo possiiliis gding h piod of pmissil dly in pymns. AE AE AE : h cs () is fuh dividd ino wo su css i.. cs () nd cs () AE () AE () AE () : inc h h lngh of piod wih posiiv invnoy sock is lg hn h cdi piod h uy cn us sl vnu o n h ins wih n nnul duing h piod [ ]. h uni cos of h gnd sls vnu is dposid in n ins ing ccoun. h diffnc wn sls pic nd uni cos is ind y h sysm o m h dy o dy xpnss of h sysm. A h nd of h cdi piod

5 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil 56 Pg h ccoun is sld. Af sing h ccoun im gin h uni cos of gnd sls vnu is dposid in n ins ing ccoun o n ins wih n nnul duing h piod [ ]. Byond h fixd cdi piod poduc sill in sock is ssumd o finncd wih n nnul. Now h ol ins nd E duing h piod is givn y- () d d E () ol ins pyl P ()_ is givn y () d () P...(4) hfo ol vg cos in his cs is E P ) ( () () D H ()

6 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil 57 Pg...(5) o minimiz h ol vg cos p uni im () ( ) h opiml vlus of nd (sy * nd *) cn oind y solving h following wo uions simulnously- ) ( ()...(6) nd ) ( () (7) Povidd hy sisfy h sufficin condiions- ) ( () ) ( () nd ) ( ) ( ) ( () () ()

7 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil 58 Pg Euions (6.6) nd (6.7) uivln o (8) nd

8 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil...(9) o g h opiml vlu of nd which minimizs ol cos () ( ) on nd o dvlop h following lgoihm o find h opiml ( ) ALGORH (): EP - Pfom () --- (V) () wih = () usiu () in uion (6.8) o oin () () Using () dmins () fom uion (9) (V) Rp () nd () unil no chng occus in h vlu of nd. EP - o comp nd () f is fsil hn go o sp (). () f is no fsil s = nd vlu h cosponding vlus of fom uion (9) nd hn go o h sp (). EP - lcul h cosponding ol cos. () ( * * ) AE (): nd his cs is simil o cs (). Bu s > h ins nd duing E () [ ] is givn y. E d...() ns pyl P () fo h piod [ ] is givn y- P d Now h ol vg cos () ( ) in his cs is givn y- () H D P E 59 Pg

9 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil 6 Pg...() o minimiz h ol vg cos p uni im ( ) ( ) h opiml vlus of nd (sy * nd *) cn oind y solving h following wo uions simulnously- ) ( ()...() nd ) ( () (4) Povidd hy sisfy h sufficin condiions- ) ( () ) ( () nd ) ( ) ( ) ( () () ()

10 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil 6 Pg Euion (6.) nd (6.4) uivln o-...(5) nd...(6) Now o dvlop h lgoihm o find h opiml vlus of nd.

11 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil ALGORH (): EP - Pfom () --- (V) () wih () = () usiu () ino uion (5) o vlu () () Using () o dmin () fom uion (6) (V) Rp () nd () unil no chng occus in h vlu of nd. EP - o comp nd () f hn is fsil hn go o sp (). () f > hn is no fsil. = nd vlu h cosponding vlus of fom uion (6) nd hn go o h sp (). EP - ompu h cosponding. () ( * * ) AE (): < n his cs sinc < h uy pys no ins nd ns h ins duing h piod [ ] h ins nd in his cs is dnod y E () nd is E d h ol vg cos p uni im ( ) in his cs is H D (.7) E...(8) o minimiz h ol vg cos p uni im () ( ) h opiml vlus of nd (sy * nd *) cn oind y solving h following wo uions simulnously- nd ( )...(9) 6 Pg

12 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil 6 Pg ) ( () Povidd hy sisfy h sufficin condiions- ) ( ) ( nd ) ( ) ( ) ( Euions (6.9) nd (6.) uivln o () nd...() Now o dvlop h following lgoihm o find h opiml vlus of nd.

13 nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil ALGORH EP - Pfom () --- (V) () wih ( ) = () usiuing () = ino uion () o vlu () () Using () o dmin () fom uion () (V) Rp () nd () unil no chng occus in h vlu of nd. EP - omp nd () f < is fsil hn go o sp (). () f is no fsil. = nd vlu h cosponding vlus of fom uion () nd hn go o h sp (). EP - As sd li h ojciv of his polm is o dmin h opiml vlus of nd so h ( ) is minimum. As h discussion cid ou so f on cn g- * * * * * * * * in V. onclusions n his pp n ppopi picing nd lo sizing modl fo il whn h suppli povids pmissil dly in pymns is dvlopd nd discussd. h modl incopos som lisic fus lik dioion shogs nd suppli cdis which cn ssocid wih num of diffn yps of invnois. h modl cn usd fo goods lik domsic ims; lconic uipmns c. nd cn find vious pplicions in h il usinss. h polm hs n fomuld nlyiclly nd cos minimizion ppoch hs n usd o find h opiml soluion. W dsi h fis nd scond od condiions fo finding h opiml cos nd hn dvlopd n lgoihm o solv h polm. h modl is poposd fo non insnnous dioing ims wih xponnil dmnd. hogs llowd nd hy pilly ckloggd. Duing h shog piod only fcion of h dmnd is lf. h lgic pocdu nd cos minimizion pocdu is pplid o find h diffn opiml vlus. om picul css s consn dmnd insnnous dioion cn oind fom h dvlopd modl. Rfncs []. Aggwl.P. (978): A no on n od lvl invnoy modl fo sysm wih consn of dioion. Opsch []. Aggwl.P. Jggi.K.(995):Oding policis of dioing ims und pmissil dly in pymns. Jounl of h Opionl Rsch ociy []. A.. Jml B.R. k..wng (997):An oding policy fo dioing ims wih llowl shog nd pmissil dly in pymn.j. Op. Rs. oc. 48 () [4]. Bhuni A. K. nd ii. (999):An invnoy modl fo dcying ims wih slling pic funcy of dvismn nd linly im-dpndn dmnd wih shogs.apqr nscions [5]. hng.. (4): An EOQ modl wih dioing ims und inflion whn suppli cdis linkd o od uniy. n. Jou. Pod. Eco [6]. hung K.J. Lio. J.J (4): Lo-sizing dcisions und d cdi dpnding on h oding uniy. omp. nd Op. Rs. () [7]. hung K.J. Lio. J.J (6):h opiml oding policy in DF nlysis fo dioing ims whn d cdi dpnds on h od uniy. n. J. Pod. Eco. ( in pss). [8]. Dv U. Pl L.K. (98): (; i) policy invnoy modl fo dioing ims wih im popoionl dmnd. Jou. Op. Rs. oc [9]. Gh P.. chd G.P. (96): A modl fo n xponnilly dcying invnoy.jou. nd. Eng []. Goyl.K. (985): Economic od uniy und condiions of pmissil dly in pymns. Jou. Op. Rs. oc []. Hig. A. (995): Lo sizing modls fo dioing ims wih im-dpndn dmnd. n. J. ys. ci []. Lio H.. si.h. u.. (): An invnoy modl wih dioing ims und inflion whn dly in pymn is pmissil. n. Jou. Pod. Eco []. ndl B. N. nd Phujd. (989): A no on n invnoy modl wih sock-dpndn consumpion. Opsch [4]. ng J.... hng nd.k. Goyl. (5):Opiml picing nd oding policy und pmissil dly in pymns. n. J. Pod. Eco [5]. Yng Hui-Ling (6): wo-whous pil cklogging invnoy modls fo dioing ims und inflion.nnionl Jounl of Poducion Economics -9 nnionl Jounl of Engining cinc nvnion (JE) is UG ppovd Jounl wih l. No. 8 Jounl no. 4. *D Rvish Kum Ydv" nvnoy odl Fo Wiull Dcying ms Wih Exponnilly Dcsing Dmnd And Pmissil Dly n Pymns." nnionl Jounl of Engining cinc nvnion (JE) 6.8 (7): Pg