International Journal on Recent and Innovation Trends in Computing and Communication ISSN: Volume: 5 Issue:

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1 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: Dvlopmn of n EPQ Modl for Drioring Produ wih Sok nd Dmnd Dpndn Produion r undr Vril Crrying Cos nd Pril Bklogging * Dhir Singh & ** S.R.Singh * Dp. of Mhmis, Shhd Hr Singh Gov. Dgr Collg,Dhnpur (Chnduli), Indi ** Dp. of Mhmis, C.C.S Univrsiy, Mru, Indi Emil: * dhirsingh7@yhoo.om, ** shivrjpundir@gmil.om Asr:- In h prsn ril, n onomi produion quniy (EPQ) modl is dvlopd for drioring produ wih im dpndn dmnd nd h im dpndn invnory rrying os. Hr,i is ssumd h h produion r ny insn dpnds on oh h sok nd h dmnd of h produ. o mk h modl mor rlisi, h shorgs r llowd nd prilly kloggd. h kordring r is kn s drsing funion of wiing im for h nx rfill.h min ojiv of h prsn sudy is o find h opiml ol os pr uni im of h produion sysm. o vlid h opiml rsus, numril xmpl is providd. o nlyz h ff of vriions in h opiml rsus wih rsp o hng in on prmr im, snsiiviy nlysis is rrid ou nd h rsuls r prsnd grphilly. Kywords EPQ, driorion, vril rrying os, sok nd dmnd dpndn produion, shorg, nd pril klogging. ***** INRODUCION A mjor onrn of invnory mngmn is o know whn nd how muh o ordr or mnufur so h h ol os of h sysm pr uni im is minimizd.h onomi produion quniy (EPQ) modl plys n imporn rol on mnging h invnory ffivly. Eonomi produion quniy (EPQ) modl is n xnsion of onomi ordr quniy (EOQ) modl. EOQ modl ssums h h quniy ordrd will rriv immdily in h sok. On h ohr hnd, EPQ modl ks mor rlisi pproh ssuming h ordrs r vill or rivd in n inrmnl mnnr onsidring mnufuring ompny will produ is own quniy or prs r going o shippd o h ompny whil ing produd. Ovr h ls fw dds, numrous rsrhs hv n don o xnd h EPQ modl mking i losr o rl lif siuions. Mndl nd Phujdr [] dvlopd n onomi produion quniy modl for drioring ims wih uniform produion r nd linrly sok dpndn dmnd. Widydn nd W [] dvlopd n onomi produion quniy (EPQ) modl for drioring ims in whih produion, rwork, driorion nd dmnd r r ssumd onsn. In rliy, h dmnd r of ny produ my vry wih im or wih pri or wih h insnnous lvl of invnory displyd in suprmrk. On of h unrlisi ssumpion is h ims sokd prsrv hir physil nd hmil hrrisis during hir sy in sok for long priod of im. hrfor,driorion of h sord produs is ky hrrisi nd is imp on modling of invnory sysms nno ngld. Driorion is dfind s dmg, dy or spoilg of h produs h r sord for fuur us, lwys loos hir originl vlu wih pssg of im. Invnory modls for drioring ims hving mulivri dmnd funions wr lso sudid y svrl rsrhrs. You [] prsnd n invnory poliy for produs wih slling pri nd im dpndn dmnd. so & Shn [] onsidrd dynmi priing, promoion nd rplnishmn poliis for drioring im undr prmissil dly in pymns. Modls for drioring ims hving sok lvl nd slling pri dpndn dmnd r wr sudid y ng nd Chng [] nd Khnr, l. [6]. Pl, l. [7] onsidrd singl drioring im wih h dmnd r dpndn on displyd sok lvl, slling pri of produ nd frquny of dvrismn. Singh l.[8] prsnd Mhmil produion invnory modl for drioring ims wih im dpndn dmnd r undr h ff of h inflion nd shorgs. Sivshnkri nd Pnyppn [9] ingrd os rduion dlivry poliy ino produion invnory modl wih dfiv ims in whih hr diffrn rs of produion r onsidrd. Singh l. [] dvlopd n IJRICC My 7, hp:// 78

2 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: ingrd produion invnory poliy for prishl produs wih rd rdi priod nd invsmn in prsrvion hnology in whih produion r is kn s funion of dmnd r. In ll hs modls, h produion r onsidrd o indpndn of oh sok nd dmnd of h produ. Bu in rliy if h produ is produd wihou onsidring h on hnd sok nd dmnd of h produ, h siuion my ld xss invnoris or hvy shorgs. hrfor, i is h ommon pri in svrl of h indusris dling wih drioring ims, h produion r is djusd dpnding on h sok on hnd nd dmnd i.. if mor sok is hr hn h produion r is rdud nd if h lss sok is hr h produion r is inrsd. his yp of produion r is known s sok nd dmnd dpndn produion. In his ppr, n onomi produion quniy modl for drioring produs wih sok nd dmnd dpndn produion r is dvlopd nd nlyzd. Hr, i is lso ssumd h h dmnd of h produ nd holding os r im dpndn.o vlid h opiml rsus, numril xmpl is providd. Finlly, snsiiviy nlysis of opiml soluion wih rsp o h mjor prmrs r lso sudid o drw som disions. ASSUMPIONS h Mhmil modl of h produion invnory prolm is dvlopd undr h following ssumpions.. h modl is dvlopd for singl produ.. h plnning horizon is fini.. h produ onsidrd in his modl is drioring in nur nd hr is no rpir or rplmn of h driord unis.. Rplnishmn is insnnous nd ld im is zro.. h dmnd of h produ is n inrsing funion of im nd givn y D whr, 6. h produion r, P is mulivri funion of posiiv sok lvl nd dmnd nd givn y h following xprssion: Whr D() I(),I P D(),I I is h invnory lvl ny im, nd r iniil nd sok-dpndn onsumpion r prmrs nd. 7. Shorgs r llowd nd kloggd prilly. h klogging r is vril nd is dpndn on h lngh of h wiing im for h nx rplnishmn. L h klogging r for ngiv invnory is η, Whr is known s klogging prmr wih nd is h wiing im up o h nx rplnishmn. And h rmining frion B - is los. 8. Holding os h pr uni pr uni im is ssumd o n inrsing funion of im nd onsidrd s h h whr,h NOAIONS h h h Mhmil modl of h produion invnory prolm is dvlopd undr h following noions., dmnd prmrs,, produion prmrs h, h holding os prmrs klogging prmr driorion prmr, whr I invnory lvl ny im IJRICC My 7, hp:// 79

3 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: I invnory lvl im during h im inrvl, I invnory lvl im during h im inrvl, I invnory lvl im during h im inrvl, I invnory lvl im during h im inrvl, I mximum invnory lvl during h im priod, S p s l d O mx.c. mximum shorg lvl of invnory during sok ou priod produion os pr uni shorg os pr uni los sl os pr uni driorion os pr uni sup os pr produion run h im whih produion sops h im whih invnory lvl oms zro nd shorg srs h im whih produion rsrs o rovr oh h prvious shorgs nd o sisfy dmnd in h im priod, yl im ol os pr uni im for produion yl MAHEMAICAL FORMULAION & SOLUION OF HE MODEL In his modl,h invnory lvl is zro iniilly (i.. im ). h produion srs im o sisfy h ourring dmnd nd driorion nd h invnory gins o pils up oninuously unil i rhs o is mximum lvl I im. h produion sops im nd fr h h invnory lvl dpls du o omind ff of dmnd nd driorion unil i rhs o zro im. hr fr shorgs our nd umul unil i rhs o is mximum lvl S im. h produion rsrs gin im o rovr oh h prvious shorgs in h im priod,.susqunly, h yl nds im wih zro invnory.h, nd o sisfy dmnd in h im priod nir pross rps islf. h ov dfind produion modl for on yl is shown in h figur.. Invnory lvl mx I mx im Los Sls Fig.: h grphil rprsnion of produion invnory modl diffrnil quions rprsning h invnory sus r givn y h IJRICC My 7, hp:// 8

4 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: di d di d di d P D, I () D, I () η D, I () di I d P D, () Wih h oundry ondiions,i,i, I I nd () h soluions of h ov diffrnil quions r givn y I, I I,, (6) (7) (8) Wih h hlp of quions (6) nd (8), w g h mximum posiiv invnory lvl I, I (9) mx I () And h mximum lvl of sok ou quniy S I () h ol os pr uni im of h produion sysm ompromiss h following omponns:.c. [Sup Cos +Produion Cos + Holding Cos + Driorion Cos + Shorg Cos + Los Sls Cos] () ) Sup Cos: h sup os pr produion run is givn y OC O () ) Produion Cos: h produion os of h sysm is luld s IJRICC My 7, hp:// 8

5 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: IJRICC My 7, hp:// PC d P d P p PC p () ) Holding Cos: Invnory holding os pr yl will d I h d I h HC h h () d) Driorion Cos: h os ssoid wih h driord unis n luld s DC d D d P d DC d (6)

6 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: IJRICC My 7, hp:// ) Shorg Cos: h shorg os of h invnory produion sysm is givn y SC d I d I s SC 6 s (7) f) Los Sls Cos: h os riss du o h los sls pr produion run during h sokou priod n luld s LSC d D η l LSC l (8) Sin I I, nd I I hn w hv, f (9),, g () Hn, h ol os pr uni im of h produion sysm will

7 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: IJRICC My 7, hp:// l 6 s d h h p O.C. () From quion (), w osrv h h.c. of his produion invnoy sysm is funion of h dision vrils,,, nd. Wih h hlp of h quions (9) nd (), w n find h vlu of nd in rms of nd. Consqunly, h.c. of his produion invnory modl is funion of h dision vrils nd only.

8 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: h ol os of h produion sysm pr uni im will minimiz if nd.c..c. () () providd h.c..c..c. () Sin h quions () nd () r highly nonlinr, hrfor for givn s of prmri vlus, quions () nd () n solvd y Mhmi sofwr. NUMERICAL EXAMPLE h following inpu d of diffrn prmrs r usd o illusr h modl.,.,., d.,,.,.,., h, h p,s, nd in ppropri unis. Corrsponding o hs inpu vlus, h opiml vlu of,,., O, l, nd r,.766, nd.76. I mx, S. Susiu hs opiml vlu in quions (), (), nd () w g.979.c SENSIIVIY ANALYSIS E W will now prform h snsiiviy nlysis o xmin h ffs of hngs in h inpu prmrs,,, d, l, p, s,o,,,h,h,, nd on h opiml rsuls oind in h modl. Snsiiviy nlysis will prformd y hnging h of prmrs y -%,-%,% nd %, nd king on prmr im nd kping h rmining prmrs unhngd. h rsuls r shown in l.., nd IJRICC My 7, hp:// 8

9 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: C Fig.: Convxiy of h.c. funion. C.. Fig.: Convxiy of h.c. funion Grphil Rprsnion of h Snsiiviy Anlysis IJRICC My 7, hp:// 86

10 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: Prmr % hng in Chng in Chng in Chng in Chng in Chng in Prmr I mx S.C E E E E E E E E E E E E E E E+ d E E E E E+ l E E E E E+ O E E E E E E E E E E E E E E E+ IJRICC My 7, hp:// 87

11 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: h E E E E E+ h E E E E E E E E E E E E E E E+ p E E E E E+ s E E E E E+ IJRICC My 7, hp:// 88

12 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: Fig.: Vriion in,,imx nd S wih rsp o h prmr -% -% % % % Imx S Fig.: Vriion in,,imx nd S wih rsp o h prmr -% -% % % % Imx S Fig.6: Vriion in,,imx nd S wih rsp o h prmr -% -% % % % Imx S Fig.7: Vriion in,,imx nd S wih rsp o h prmr d -% -% % % % d Imx S IJRICC My 7, hp:// 89

13 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: Fig.8: Vriion in,,imx nd S wih rsp o h prmr l -% -% % % % l Imx S Fig.9: Vriion in,,imx nd S wih rsp o h prmr O -% -% % % % O Imx S Fig.: Vriion in,,imx nd S wih rsp o h prmr -% -% % % % Imx S Fig.: Vriion in,,imx nd S wih rsp o h prmr -% -% % % % Imx S IJRICC My 7, hp:// 9

14 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: Fig.: Vriion in,,imx nd S wih rsp o h prmr h -% -% % % % h Imx S Fig.: Vriion in,,imx nd S wih rsp o h prmr h -% -% % % % h Imx S Fig.: Vriion in,,imx nd S wih rsp o h prmr -% -% % % % Imx S Fig.: Vriion in,,imx, nd S wih rsp o h prmr -% -% % % % Imx S IJRICC My 7, hp:// 9

15 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: Fig.6: Vriion in,,imx, nd S wih rsp o h prmr p -% -% % % % s Imx S Fig.7: Vriion in,,imx, nd S wih rsp o h prmr s -% -% % % % s Imx S 8E+ 6E+ E+ E+ Fig.8: Vriion in.c. wih rsp o h prmr -% -% % % %.C. 8E+ 6E+ E+ E+ Fig.9: Vriion in.c. wih rsp o h prmr -% -% % % %.C. IJRICC My 7, hp:// 9

16 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: E+ Fig.: Vriion in.c. wih rsp o h prmr 6E+ E+ E+.C. -% -% % % % 8E+ Fig.: Vriion in.c. wih rsp o h prmr d 6E+ E+ E+ d.c. -% -% % % % 8E+ Fig.: Vriion in.c. wih rsp o h prmr l 6E+ E+ E+ l.c. -% -% % % % 8E+ Fig.: Vriion in.c. wih rsp o h prmr O 6E+ E+ E+ O.C. -% -% % % % IJRICC My 7, hp:// 9

17 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: E+ E+ E+ E+ E+ E+ Fig.: Vriion in.c. wih rsp o h prmr -% -% % % %.C. 8E+ 6E+ E+ E+ Fig.: Vriion in.c. wih rsp o h prmr -% -% % % %.C. 8E+ 6E+ E+ E+ Fig.6: Vriion in.c. wih rsp o h prmr h -% -% % % % h.c. E+ 8E+ 6E+ E+ E+ Fig.7: Vriion in.c. wih rsp o h prmr h -% -% % % % h.c. IJRICC My 7, hp:// 9

18 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: E+ Fig.8: Vriion in.c. wih rsp o h prmr 6E+ E+ E+.C. -% -% % % % E+ 8E+ 6E+ E+ E+ Fig.9: Vriion in.c. wih rsp o h prmr -% -% % % %.C. 8E+ 6E+ E+ E+ Fig.: Vriion in.c. wih rsp o h prmr p -% -% % % % p.c. 8E+ 6E+ E+ E+ Fig.: Vriion in.c. wih rsp o h prmr s -% -% % % % s.c. IJRICC My 7, hp:// 9

19 mx Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: OBSERVAIONS: On h sis of h rsuls shown in l. nd figurs no. o, h following osrvions n md:. h shorg priod, mximum posiiv sok lvl I, nd mximum shorg lvl S inrs whil h non produion priod drss wih h inrs in h vlu of h produion prmr. hrfor, h non produion priod, shorg priod, nd mximum shorg lvl posiiv sok lvl Imx is highly snsiiv o hng in h produion prmr.. h non produion priod, nd mximum posiiv sok lvl S r slighly snsiiv nd h mximum Imx rmin qul wih h inrs in h vlu of h klogging prmr. hrfor, h non produion priod, nd mximum posiiv sok lvl I mx r insnsiiv o hng in h klogging prmr. h shorg priod inrss whil mximum shorg lvl S drss wih h inrs in h vlu of h klogging prmr. hrfor,h shorg priod, nd mximum shorg lvl S r slighly snsiiv o hng in h klogging prmr.. h non produion priod, nd mximum shorg lvl posiiv sok lvl S inrs whil h shorg priod, nd h mximum Imx drs wih h inrs in h vlu of h dmnd prmr. hrfor, h non produion priod, nd shorg priod r slighly snsiiv,h mximum shorg lvl snsiiv, nd h mximum posiiv sok lvl Imx is highly snsiiv o hng in h dmnd prmr. S is modrly. h non produion priod, shorg priod, mximum posiiv sok lvl I mx, nd mximum shorg lvl S r slighly hngd wih h inrs in h vlus of h prmrs,,,nd. hrfor, h non produion priod, shorg priod, mximum posiiv sok lvl mx slighly snsiiv o hng in h prmrs,,,nd.. h non produion priod, shorg priod, mximum posiiv sok lvl mx S rmin qul wih h inrs in h vlus of h prmrs produion priod, shorg priod, mximum posiiv sok lvl mx d, l, O,h,h, p, nd s I, nd mximum shorg lvl S r d, l, O,h,h, p, nd s I, nd mximum shorg lvl. hrfor, h non I, nd mximum shorg lvl S r insnsiiv o hng in h prmrs..c. of h sysm drss wih h inrs in h vlu of h driorion 6. h vlu of ol os pr uni im prmr. hrfor,.c. is highly snsiiv o hngs in h prmr. 7. h vlu of ol os pr uni im.c. of h sysm inrss modrly wih h inrs in h vlus of h prmrs,. h nd. hrfor,.c. is modrly snsiiv o hngs in h prmrs h, nd. 8. h vlu of ol os pr uni im.c. of h sysm rmins qul wih h inrs in h vlus in h prmrs,,, d,l, O,,h,, p, nd s. hrfor,.c. is insnsiiv o hngs in h prmrs,,, d,l, O,,h,, p, nd s. CONCLUSION h onomi produion invnory modl dvlopd hr inorpors h following rlisi furs:. I is pplil o n invnory whih driors ovr im.. I is onrnd wih linrly im vrying dmnd of h produ.. h produ in h sok is mnufurd sok nd dmnd of h produ dpndn r.. I is lso onrnd wih h shorgs of h produ in h sok.. h produ in h sok is sord im vrying holding os. In his ril, produion invnory modl for drioring produ wih im dpndn dmnd nd holding os hs n dvlopd. h produion r is onsidrd hr s funion of sok nd dmnd of h produ. Hr, shorgs r llowd IJRICC My 7, hp:// 96

20 Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: nd h r of h kloggd dmnd drss xponnilly s h wiing im of h nx rplnishmn inrss. h mhmil formulion o ompu h diffrn ssoid os nd ol os pr uni im of h produion sysm hs n prsnd. A numril xmpl hs n givn o illusr h sudy. Finlly, h snsiiviy of h opiml rsuls o hngs in h vlus of diffrn prmrs hs n disussd nd prsndd grphilly. For fuur sop, his modl n xndd for diffrn prmissil dly in pymns, diffrn dmnd nd im vlu of mony. I lso n xndd in h fuzzy nviromns. REFERENCES: [] Mndl, B.N., & Phujdr, S.(989). An invnory modl for drioring ims nd sok-dpndn onsumpion r. Journl of h Oprionl Rsrh Soiy, (),8 88. [] Widydn,G. A.,& W, H.M. (). An onomi produion quniy modl for drioring ims wih mulipl produion sups nd rwork. Inrnionl Journl of Produion Eonomis,8, [] You, P.S. (). Invnory poliy for produs wih pri nd im-dpndn dmnds. Journl of h Oprionl Rsrh Soiy, 6(7), [] so,y.c., & Shn, G.W. (8). Dynmi priing, promoion nd rplnishmn poliis for drioring im undr prmissil dly in pymns. Compurs & Oprions Rsrh,, 6 8. [] ng, J.., & Chng, C.. (). Eonomi produion quniy modls for drioring ims wih pri nd sok dpndn dmnd. Compurs & Oprions Rsrh,, [6] Khnr, S., Snkr, S., & Chudhuri, K.S. (). An EOQ modl for prishl im wih sok nd pri dpndn dmnd r. Inrnionl Journl of Mhmis in Oprions Rsrh, (),. [7] Pl, A.K., Bhuni, A.K. & Mukhrj, R.N. (6). Opiml lo siz modl for drioring ims wih dmnd r dpndn on displyd sok lvl nd pril kordring. Europn Journl of Oprionl Rsrh, 7(), [8] Singh, S.R., Agrwl, A.,& Rni, S. (). Mhmil produion invnory modl for drioring ims wih im dpndn dmnd r undr h ff of inflion nd shorgs. Inrnionl Journl of Compur & Mhmil Sins,, 8-8. [9] Sivshnkri, C.K., & Pnyppn, S.(). Produion invnory modl for hr lvls of produion wih ingrs os rduion dlivry poliy. Europn Journl of Sinifi Rsrh, 6 (), 7-86 [] Singh, S.R., yl, S.,& Shrm, R. (6). An ingrd produion invnory modl for prishl produs wih rd rdi priod nd invsmn in prsrvion hnology. Inrnionl Journl of Mhmis in Oprionl Rsrh,8(), 7-6. IJRICC My 7, hp:// 97