AN EOQ MODEL FOR A DETERIORATING ITEM WITH NON-LINEAR DEMAND UNDER INFLATION AND A TRADE CREDIT POLICY

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1 Yugoslv Journl of Oprtons Rsrch 5 (005), Numbr, 09-0 AN EOQ MODEL FOR A DETERIORATING ITEM WITH NON-LINEAR DEMAND UNDER INFLATION AND A TRADE CREDIT POLICY S. K. MANNA, Dprtmnt of Mthmtcs, Sovrn Mmorl Collg, Jgtbllvpur, Howrh, Ind skmnn-5@hotml.com K. S. CHAUDHURI Dprtmnt of Mthmtcs, Jdvpur Unvrsty, Kolkt, Ind k-s-chudhur@yhoo.com Rcvd: Octobr 003 / Accptd: Aprl 004 Abstrct: Ths ppr dvlops n nfnt tm-horzon dtrmnstc conomc ordr quntty (EOQ) nvntory modl wth dtrorton bsd on dscountd csh flows (DCF) pproch whr dmnd rt s ssumd to b non-lnr ovr tm. Th ffcts of nflton nd tm-vlu of mony r lso tkn nto ccount undr trd-crdt polcy of typ " α /T nt T". Th rsults r llustrtd wth numrcl xmpl. Snstvty nlyss of th optml soluton wth rspct to th prmtrs of th systm s crrd out. Kywords: Infnt-tm horzon, dtrorton, non-lnr dmnd, nflton, trd-crdt polcy.. INTRODUCTION Th ffcts of nflton nd tm-vlu of mony wr gnord n th clsscl nvntory modls. It ws blvd tht nflton would not nflunc th cost nd prc componnts. Th conomc stuton of most of th countrs hs chngd consdrbly durng th lst 5 yrs du to lrg-scl nflton nd shrp dcln n th purchsng powr of mony. Buzcott (975) nd Msr (975) wr th frst to dvlop EOQ modls tkng nflton nto ccount. Both of thm consdrd constnt nflton rt for ll th ssoctd costs nd mnmzd th vrg nnul cost to drv n xprsson for th conomc lot sz. Thr work ws xtndd by rsrchrs lk Chndr nd Bhnr (985), Aggrwl (98), Msr (979), Brmn nd Thoms (977), Srkr nd Pn (994), tc. to covr consdrtons of tm vlu of mony, nflton rt, fnt

2 0 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm rplnshmnt rt, tc. All ths rsrchrs workd on constnt dmnd rt. Dtt nd Pl (990) nd Bos t l. (995) ntroducd lnrly trndd dmnd long wth nflton, tm-vlu of mony nd shortgs n nvntory. In dvlopng mthmtcl nvntory modl, t s ssumd tht pymnts wll b md to th supplr for th goods mmdtly ftr rcvng th consgnmnt. In dy-to-dy dlngs, t s obsrvd tht th supplrs offr dffrnt trd crdt polcs to th buyrs. On such trd crdt polcy s " α /T nt T" whch mns tht α% dscount on sl prc s grntd f pymnts r md wthn T dys nd th full sl prc s du wthn T (> T ) dys from th dt of nvoc f th dscount s not tkn. Bn-Horm nd Lvy (98), Chung (989), Aggrwl nd Jgg (994), tc., dscussd trd-crdt polcy of typ " α /T nt T" n thr modls. Aggrwl t l. (997) dscussd n nvntory modl tkng nto consdrton nflton, tm- vlu of mony nd trd crdt polcy " α /T nt T" wth constnt dmnd rt. Ll nd Stln (994) dvlopd n optml dscountng prcng polcy. Ldny nd Strnlb (994) trd to mk n ntrcton btwn conomc ordr quntts nd mrktng polcs. In ths ppr, n ttmpt hs bn md to xtnd th modl of Aggrwl t l. (997) to n nvntory of tms dtrortng t constnt rt (0 < <). Also th dmnd rt s tkn to b non-lnr nstd of constnt dmnd rt. Snstvty of th optml soluton s xmnd to s how fr th output of th modl s ffctd by chngs n th vlus of ts nput prmtrs.. ASSUMPTIONS AND NOTATIONS A dtrmnstc ordr-lvl modl wth n nfnt rt of rplnshmnt s dvlopd wth th followng ssumptons nd nottons. t () Dt () = bρ ( > b> 0,0< ρ < ) s th dmnd rt t ny tm t. It s n ncrsng concv functon of t, but th rt of ncrs s dmnshng. dd() t Mthmtclly, D () t = > 0 dt d D() t nd D () t = 0. dt Thus mrgnl dmnd s dcrsng functon of tm. () T s th rplnshmnt cycl lngth. () Q s th ordr quntty n th ( )th cycl. (v) Rplnshmnt s nstntnous. (v) Ld tm s zro. (v) Invntory crryng chrg s I pr unt pr unt tm. (v) Shortgs r not llowd. (v) C(0) nd A(0) r rspctvly th unt cost of th tm nd th ordrng cost pr ordr t tm zro. (x) h s th nflton rt pr unt tm. (x) r s th opportunty cost pr unt tm.

3 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm (x) A constnt frcton (0< <) of th on-hnd nvntory dtrorts pr unt of tm. (x) Th tm horzon of th nvntory systm s nfnt. (x) Th trm of crdt polcy s " α / M nt M", whch mns tht M prcnt dscount of sl prc s grntd f pymnts r md wthn M dys nd th full sl prc s du wthn M dys from th dt of nvoc f th dscount s not tkn. 3. MATHEMATICAL FORMULATION Lt A(t) nd C(t) b th ordrng cost nd unt cost of th tm t ny tm t. Thn A(t) = A(0) ht nd C(t) = C(0) ht, ssumng contnuous compoundng of nflton. Lt I () t b th nstntnous nvntory lvl t ny tm t n th ( )th cycl. Th dffrntl quton govrnng th nstntnous stts of I () t n th ntrvl [ T,( ) T ] s di () t t I () t = ( bρ ), dt T t ( ) T () whr I ( T) = Q nd I (( ) T) = 0, = 0,,,... () Th soluton of quton () wth boundry condton I ( T) = Q (S Appndx I) s b I t Q log ρ ( T t) T ( T t) t ( T t) () = ( ) ( ρ ρ ), T t ( ) T, = 0,,,... (3) T b T ( ) T T whr Q = ( ) ( ρ ρ ), = 0,,,... (4) log ρ Lt t 0, t, t,... b th rplnshmnt ponts nd t t = T so tht t = T, = 0,,,... () Cs I: Whn dscount s tkn Hr purchss md t tm t r pd ftr M dys nd purchs prc n rl trms for Q unts t tm t n th ()th cycl s = Q( α) C( t). Th prsnt worth of csh-flows for th ()-th cycl s whr ( ) T ( d ) rt rt ( ) = [ ( ) ( )( α) ( )( α) ( ) ] T PV T At QCt ICt I t dt = A A A, sy 3 rt A = A( t ) RT = A(0), (ssumng R = r hnd usng t = T),

4 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm α rt A = QC( )( ) t C(0)( α) bc(0)( α) = ( ) ρ ( log ρ) bc(0)( α) ( log ρ) T RT T RT ρ T ( ) T RT, Now, ( ) T rt rt 3 [ ( )( α) ( ) ] T A = IC t I t dt. ( ) T T I () t rt dt b = [ ( ) ( ) ρ ( r ) r ( log ρ)( r ) ( r ) T ( r ) T rt rt rt ( ) T rt T rt b( ρ ) ρ ( log ρ)(log ρ r) ( r ) ( r ) T b ( ) T ρ ( log ρ)( r ) Thrfor, T rt T ( r ) T rt ( )( ) b ( ) ρ ( log ρ)( r ) rt T rt ( ) T rt C(0)( α) bc(0)( α) PV T A ( log ρ) ( d) T RT T RT ( ) = [ (0) ( )] ρ bc(0)( α) T ( ) T RT rt ρ IC(0)( α) [ ( ) ( log ρ) r rt bic(0)( α) ( )] [ ( ρ )] ρ ( r ) ( log ρ)(log ρ r) bic(0)( α) ρ = ( log ρ)( r ) rt T ( ) [ ( )], (tkng P r h). Th prsnt worth of ll futur csh flows s ( d) ( d) = = 0 PV PV ] T rt C(0)( α) T bc(0)( α) = [ A(0) ( )] ( RT ) ( log )( T RT ρ ρ ) T ρbc(0)( α) rt IC(0)( α) [ ( ) T RT ( log ρ) ( ρ ) r T rt rt T bic(0)( α) ( ρ ) ( )] ( r ) ( ) ( log ρ)(log ρ r) ( ρ ) bρic ( log ρ)( r ) ( ρ ) rt T (0)( α) ( ), (usng Appndx II). (5) (6) (7)

5 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm 3 Th soluton of th quton provdd t stsfs th condton ( d dpv ) = 0 gvs th optmum vlu of T dt d PV ( d ) dt ( T) > 0 (8) Now ( d dpv ) = 0 ylds th quton dt RT T C(0)( α) T R [ A(0) ( )] C(0)( α) RT RT ( ) ( ) T T RT bc(0)( α)(log ρ R ) ρbc(0)( α)(log ρ R ) ρ [ ] T RT ( log ρ) ( log ρ) ( ρ ) T ρbc(0)( α) rt T PIC(0)( α) [ ( ) T RT ( log ρ) ( ρ ) ( r ) IC(0)( α) ( )] [ ( r )] r rt rt rt T ( ) ( r ) ( ) bic α ρ ρ r ρ ρ ρ P ρ ( log ρ)(log ρ r) ( ρ ) T rt T rt (0)( ) ( )(log ) ( )(log ) rt T rt T bρic(0)( α) ( ρ )( r ) ( )( P log ρ) ρ = 0. ( log ρ)( r ) ( ρ ) (9) Ths quton bng hghly nonlnr cn not b solvd nlytclly. It cn b solvd numrclly for gvn prmtr vlus. Its soluton gvs th optmum vlu T* of th rplnshmnt cycl tm T. Onc T* s obtnd, w cn gt optmum ordr quntts * ( d ) * Q ( = 0,,,...) from (4) nd PV, th optmum prsnt vlu of ll futur csh flows from (7). Cs II: Whn dscount s not tkn Hr purchss md t tm t r pd ftr M dys nd purchs prc n rl trms for Q unts t tm t n th ()th cycl s hm = QC( t ). Th prsnt worth of csh flows for th ()th cycl s

6 4 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm ( wd ) ( ) T rt = T PV ( T) [ At ( ) QCt ( ) ICt ( ) I( t ) dt ] C(0) bc(0) = [ A(0) ( )] ρ ( log ρ) bc(0) (0) [ ( ) ( log ρ) r T ( ) T RT rt ρ IC ( )] ( r ) rt T RT T RT bic(0) [ ( ρ )] ρ ( log ρ)(log ρ r) rt T rt bic(0) ρ = ( log ρ)( r ) rt T ( ) [ ( )], (tkng P r h). (0) = 0,,,... Th prsnt worth of ll futur csh flows s ( wd ) ( wd ) = = 0 PV PV C(0) T bc(0) = [ A(0) ( )] RT T RT ( ) ( log ρ) ( ρ ) T ρbc(0) rt IC(0) [ ( ) T RT ( log ρ) ( ρ ) r T rt rt T bic(0) ( ρ ) ( )] ( r ) ( ) ( log ρ)(log ρ r) ( ρ ) rt T bρic(0) ( ), (usng Appndx II). ( log ρ)( r ) ( ρ ) () Th soluton of th quton stsfs th condton ( wd dpv ) = 0 gvs th optmum vlu of T provdd t dt d PV ( wd ) dt ( T) > 0. () Now ( wd dpv ) = 0 ylds th quton dt

7 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm 5 C(0) R [ A(0) ( )] C(0) RT T T RT RT ( ) ( ) T RT T ( R) T bc(0)(log ρ R) ρ ρbc(0)(log ρ R) ρ ( log ρ) ( ρ ) ( log ρ) ( ρ ) T ρbc(0) PIC(0) [ ( T RT ( log ρ) ( ρ ) r T RT T RT ) rt T rt rt T IC(0) ( )] [ ( r )] ( r ) ( ) ( r ) ( ) bic(0) ( ρ )(log ρ r) ρ ( ρ )(log ρ P) ρ ( log ρ)(log ρ r) ( ρ ) bρic(0) ( ρ )( r ) ( )( P log ρ) ρ ( log ρ)( r ) ( ρ ) rt T rt T rt rt T rt T = 0. (3) As commntd bfor, ths quton lso nds to b solvd numrclly. 4. NUMERICAL EXAMPLES () Exmpl Lt = 50, b = 5, C(0) = 0, r = 0.04, I = 0.0, A(0) = 000, M = 30, h = 0.0, α = 0., = 0.0, ρ = 0.5 n pproprt unts. Solvng th hghly non-lnr quton (9) for Cs I by Bscton Mthod, w gt th optmum vlu of T s T* = ( d Substtutng T* n qutons (4) nd (7), w gt th optmum vlu of Q nd PV ) * * ( d ) * s Q 0 = 97.75, Q = , ( =,,3,...) nd PV = (b) Exmpl Lt = 50, b = 5, C(0) = 0, r = 0.04, I = 0.0, A(0) = 000, M = 35, h = 0.0, = 0.0, ρ = 0.5 n pproprt unts. Solvng th quton (3) for Cs II by th sm Mthod s n Cs I, w gt th optmum vlu of T s T* = Substtutng T* n ( wd qutons (4) nd (), w gt th optmum vlu of Q nd ) * PV s Q 0 = , Q * = , ( =,,3,...) nd PV ( wd ) * = SENSITIVITY ANALYSIS Bsd on th numrcl xmpls consdrd bov, snstvty nlyss of T*, ( d) * ( wd) * PV, PV s prformd by chngng (ncrsng or dcrsng) th prmtrs by 5% nd 50% nd tkng on prmtr t tm, kpng th rmnng prmtrs t thr orgnl vlus. In Tbl, t s sn tht th prcntg chng n ( d ) * PV s lmost qul for both postv nd ngtv chngs of ll th prmtrs

8 6 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm xcpt r nd h. It s somwht mor snstv for ngtv chng thn n qul postv chng of prmtr r; but t s mor snstv for postv chng thn n qul ngtv chng n h. Du to postv nd ngtv prcntg chngs n prmtrs, ( d ) * C(0), I, A(0), h, nd ρ, PV ncrss nd dcrss rspctvly. But ths trnd s rvrsd for th prmtrs b, r, M nd α. From Tbl, t s clr tht r s hghly snstv mong ll th prmtrs., C(0), M, h r modrtly snstv. I, A(0), α, hv lttl snstvty whl b nd ρ r nsnstv. Bhvours of th prmtrs n Tbl (α = 0) r nrly th sm s n Tbl. Tbl : Dscount cs Chngng (%) T* ( d ) * PV (%) chng (%) chng n prmtrs chng n T* ( d ) * PV b C(0) r I A(0) M h

9 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm 7 Tbl : Dscount cs contnu... α ρ Tbl : Wthout dscount cs Chngng (%) T* ( wd ) * PV (%) chng (%) chng n prmtrs chng n T* ( wd ) * PV b C(0) r I A(0) M

10 8 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm Tbl : Wthout dscount cs contnu... h ρ CONCLUDING REMARKS Th modl dvlopd hr dls wth th optmum rplnshmnt polcy of dtrortng tm n th prsnc of nflton nd trd crdt polcy. It s dffrnt from th xstng modls n tht th dmnd rt s tkn to b tm-dpndnt n contrst to constnt dmnd rt n th othr modls. Tm dpndnt dmnd s pplcbl to th goods whos dmnd chngs stdly ovr tm. Dmnd of consumr goods chngs stdly long wth stdy chng n th populton dnsty. A tm-dpndnt dmnd rt s crtnly mor rlstc thn constnt dmnd rt. W cn mk comprtv study btwn th rsults of th dscount cs nd wthout dscount cs. In th numrcl ( d ) * xmpls, t s found tht th optmum prsnt worth of ll futur csh flows PV n cs I s 0.385% lss thn tht of cs II. Hnc th dscount cs s consdrd to b bttr conomclly. Thr s no pprcbl chng n th optmum vlu of th prsnt worth of ll futur csh flows for chngs md n th vlu of thr n th dscount cs or n th no-dscount cs. Acknowldgmnts: Th uthors xprss thr sncrst thnks to Jdvpur Unvrsty, Kolkt for ts nfrstructurl support to crry out ths work. REFERENCES [] Aggrwl, K.K., Aggrwl, S.P., nd Jgg, C.K., "Impct of nflton nd crdt polcs on conomc ordrng", Bull. Pur Appl. Sc., 6() (997) [] Aggrwl, S.C., "Purchs-nvntory dcson modls for nfltonry condtons", Intrfcs, (98) 8-3. [3] Aggrwl, S.P., nd Jgg, C.K., "Crdt fnncng n conomc ordrng polcs of dtrortng tms", Intrntonl Journl of Producton Economcs, 34 (994) [4] Bn-Horm, M., nd Lvy, H., "Inflton nd th trd crdt prod", Mngmnt Scnc, 8(6) (98) [5] Brmn, H., nd Thoms, J., "Invntory dcsons undr nfltonry condtons", Dcson Scncs, 8() (977) 5-55.

11 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm 9 [6] Bos, S., Goswm, A., nd Chudhur, K.S., "An EOQ modl for dtrortng tms wth lnr tm-dpndnt dmnd rt nd shortgs undr nflton nd tm dscountng", Journl of Oprtonl Rsrch Socty, 46 (995) [7] Buzcott, J.A., "Economc ordr quntts wth nflton", Oprtonl Rsrch Qurtrly, 6(3) (975) [8] Chndr, M., nd Bhnr, M.L., "Th ffcts of nflton nd th tm-vlu of mony on som nvntory systms", Intrntonl Journl of Producton Rsrch, 34 (985) [9] Chung, K.H., "Invntory control nd trd-crdt rvstd", Journl of Oprtonl Rsrch Socty, 4(5) (989) [0] Dtt, T.K., nd Pl, A.K., "A not on n nvntory modl wth nvntory-lvl dpndnt dmnd rt", Journl of Oprtonl Rsrch Socty, 4(0) (990) [] Ldny, S., nd Strnlb, A., "Th ntrcton of conomc ordrng quntts nd mrktng polcs", AIIE Trnsctons, 6 (994) [] Ll, R., nd Stln, R., An pproch for dvlopng n optml dscount prcng polcy, Mngmnt Scnc, 30 (994) [3] Msr, R.B., "A study of nfltonry ffcts on nvntory systms", Logstcs Spctrum, 9(3) (975) [4] Msr, R.B., "A not on optml nvntory mngmnt undr nflton", Nvl Rsrch Logstcs Qurtrly, 6 (979) [5] Srkr, R.B., nd Pn, H., "Effcts of nflton nd th tm vlu of mony on ordr quntty nd llowbl shortg", Intrntonl Journl of Producton Economcs, 34 (994) 65-7.

12 0 S.K. Mnn, K.S. Chudhur / An EOQ Modl for Dtrortng Itm APPENDIX I From quton (), w hv b t t I() t = ρ c, c log ρ bng constnt. From th boundry condton (), w hv b T T c = ( Q ρ ). log ρ Thrfor, b I () t = ( ) ( ρ ρ ) Q log ρ ( T t) T ( T t) t ( T t) whr b Q = ρ ρ log ρ T T ( ) T T ( ) ( ). APPENDIX II Assumng r > h, w hv = = RT,, = 0 RT = 0,, ( ρ ) ( ρ ) T RT T PT ρ = ρ = = 0 T RT = 0 ρ ρ,. ( ρ ) ( ρ ) ( ) T RT ( ) T PT ρ = ρ = = 0 T RT = 0