Intrnational Journal of Oprations Rsarh Intrnational Journal of Oprations Rsarh Vol. 13, o., 035 046 (016) Optimal ordring poliy with non- inrasing dmand for tim dpndnt dtrioration undr fixd lif tim prodution and prmissibl anjit Kaur 1, Sarla Park 1, and R.P.ripathi * 1 Dpartmnt of athmatis, Banasthali Univrsity, Rajasthan India Dpartmnt of athmatis, Graphi Era Univrsity, Dhradun (UK), India Rivd Jun 016; Rvisd Jun 016; Aptd Jun 016 Abstrat ost of th itms in th univrs dtriorat ovr tim. any itms suh as pharmautials, high th produts and radymad food produts also hav thir xpiration dats. his papr dvlops an onomi ordr quantity modl for rtailr in whih dmand rat is linarly tim dpndnt and non inrasing funtion of tim, dtrioration rat is tim dpndnt having xpiration dats undr trad rdits..w thn show that th total avrag ost is snsitiv with rspt to th ky paramtrs. Furthrmor, w disuss svral sub- spial ass. Finally, numrial xampls and snsitivity analysis is providd to illustrat th rsults. athmatia 5. softwar is usd to find numrial rsults. Kywords Invntory, xpiration dats, dtrioration, optimality, trad rdit, tim-dpndnt dmand 1. IRODUCIO At prsnt, it is ommon that, th vndor oftn provids to his/ hr ustomr a trad rdit priod to rdu invntory and stimulat sals. hus trad rdit is bnfiial for both vndor and buyr point of viw. Goyal [1] is th first author who has stablishd th rtailr s optimal onomi ordr quantity undr prmissibl. Aggarwal and Jaggi [] xtndd modl [1] for dtriorating itms. Stohasti EOQ modl for dtriorating itms undr prmissibl dlay in paymnts was dvlopd by Shah [3]. Shinn t al. [4] xtndd modl [1] onsidring quantity disount for fright ost. Chu t al. [5] also xtndd modl [1] for dtriorating itms. Khanra t al. [6] stablishd an EOQ (Eonomi Ordr Quantity) modl for a dtriorating itm with tim-dpndnt dmand undr trad rdits. ng [7] modifid Goyal s [1] modl for th fat that unit slling pri is nssarily highr than purhas ost. Lou and Wang [8] stablishd an EPQ (Eonomi Prodution Quantity modl for a manufaturr (or wholsalr) with dftiv itm whn its supplir offrs an up- stram trad rdits whil it turn provids its buyrs a down-stram trad rdit. Huang [9] stablishd an EOQ modl for a supply hain in whih supplir offrs th wholsalr th prmissibl dlay priod and th wholsalr in turn provids th trad rdit priod to its rtailrs. Soni and Shah [10] dvlopd an EOQ modl with an invntory- dpndnt dmand undr inrasing paymnt shm. ng and Chang [11] prsntd optimal manufaturr s rplnishmnt poliis undr two lvls of trad rdit finaning. ripathi [1] prsntd an invntory modl for sllr with xponntial dmand undr prmittd rdit priod by th vndor. any rlatd rsarh paprs an b found in Chung [13], Dvis and Gaithr [14], Chung and Liao [15], Huang and Hsu [16], Ouyang t al. [17], Skouri t al. [18], Yang t al. [19] and thir itations. any produts lik mdiins, grn vgtabls, volatil liquids, milk, brad and othrs dtriorat ontinuously but also hav thir xpiration dats. Howvr, fw rsarhrs hav onsidrd th xpiration dat of dtriorating itms. Krng and an [0] stablishd th optimal rplnishmnt dision in an onomi prodution quantity modl with dftiv itm undr prmissibl dlay in paymnt. Wu t al. [1] proposd an onomi ordr quantity modl for rtailr whr (i) th supplir provids an up-stram trad rdit and th rtailr also offrs a down-stram trad rdit, (ii) th rtailr s downstram trad rdit to th buyr not only inrass sals and rvnu but also opportunity ost and dfault risk and (iii) dtriorating itms having thir xpiration dats. Ghar and Shradr [] stablishd an invntory modl by onsidring an xponntially daying invntory. Dav and Patl [3] dvlopd an onomi ordr quantity (EOQ) modl for dtriorating itms with linarly non drasing dmand with no shortags. h modl [] is xtndd by Sahan [4] to allow for shortags. Hariga [5] stablishd invntory modls for dtriorating itms with tim- dpndnt dmand. Goyal and Giri [6] studid a survy on th rnt trnds in modlling of dtriorating itms. ng t al. [7] dvlopd invntory modl to allow for partial baklogging. Skouri t al. [8] prsntd invntory modls with ramp typ dmand rat and Wibull * Corrsponding author s mail: tripathi_rp031@rdiffmail.om
36 dtrioration rat. ahata [9] onsidrd an onomi prodution quantity (EPQ) modl for dtriorating itms undr trad rdits. Dy [30] dvlopd an invntory modl for th fft of thnology invstmnt on dtriorating itm. Rntly, Wang t al. [31] proposd an EOQ modls for a sllr by inorporating th fats (i) dtriorating produts not only dtriorat ontinuously but also hav thir maximum lif and (ii) prmissibl dlay priod inrass with dmand and dfault risk. h rmaining part of th papr is framd as follows. Stion prsnts assumptions and notations followd by mathmatial formulation for diffrnt situations. Optimal solution is dtrmind in stion 4. umrial xampls and snsitivity analysis is disussd in stion 5. At last onlusion and futur rsarh is providd in stion 6.. ASSUPIO AD OAIOS h following assumptions ar usd throughout th manusript. 1. h dmand Rat is tim dpndnt and non inrasing funtion of tim 1. h dtrioration rat is tim dpndnt and θ( t) =, 0 t m. 1+ m-t In as of tim approahs to xpiration dat m, dtrioration rat los to 1. Dtrioration boms zro for vry larg xpiring dat, i.. m and θ( t) 0. 3. If yl tim is longr than rtailr s trad rdit priod thn rtailr pays for th intrst hargs on itm in stok with intrst harg I during tim [, ]. If yl tim is shortr than rtailr s trad rdit priod thn intrst hargs is zro in whol yl. But if sllr s prmissibl dlay priod () is gratr than, th rtailr an aumulat rvnu and arn intrst in [, ] with rat I. 4. Rnwal rat is instantanous. 5. im horizon is infinit. 6. Slling pri is nssarily gratr than purhas ost. In addition th following notations ar adoptd throughout th manusript: h : Unit stok holding ost / yar in dollars xluding intrst hargs. A : Ordring ost / ordr. : Unit purhas ost. s : Slling pri pr unit tim, s > : Rtailr s prmissibl dlay priod offrd by th supplir. : Customr s prmissibl dlay priod offrd by th rtailr. D D(t) = a-bt : Dmand rat, whr a > 0 and 0 b 1, a is th initial dmand. I I t : Intrst arnd /dollar. : Intrst hargd / $ in stoks pr yar. : h tim in yars. I(t) : Invntory lvl at tim t. 1 θ ( t) = : h dtrioration rat at tim t, 0 θ( t) 1 1 + m -t m Q C() : h xpiration dat of itm. : Rnwal tim (in yars). : Ordr quantity. : otal ost/ yar. * : Optimal rplnishmnt tim (in yars). Q* : Optimal ordr quantity. C* : Optimal total ost/yar (in dollars).
37 3. AHEAICAL FORULAIOS h invntory lvl I(t) drass to mt tim dpndnt dmand and tim dpndnt dtrioration. h diffrntial quation of stats at I(t) during th rplnishmnt yl [0,] is givn by di( t) + θ( t) I( t) = ( a bt), 0 t. (1) dt With th ondition I( ) = 0. h solution of (1) is h Rtailr ordr quantity is 1+ m - I( t) = b(1+ m - t)( - t ) + {( m + 1) b - a}(1+ m - t)log 1 + m - t h total annual ost ontains th following lmnts: Q I(0) (1 m) b { 1+ m- = = + + (m+1)b-a}(1+ m)log 1+ m () (3) 1. Ordring ost is = A. Purhas ost pr yl is qual to {( m + 1) b - a}(1 + m) 1+ m - (1 + m) b+ log 1+ m h 3. Stok holding ost is= I( t) dt 0 3 h b (1 m ) b (1 + m ) (1 + m ) (1 + ) 1 (1 ) {( 1) } m + log m m b a + m = + + + + 6 6 1+ m 4 (5) h two ass may aris to alulat th annual apital opportunity ost i.. (i) < and (ii). Cas 1: < W disuss two possibl sub- ass basd on th valus of and +. If +, thn th rtailr taks th full rvnu at th tim + and pay off total purhas ost at rtailr s rdit priod. h two sub ass ar dpitd in th following Figurs 1 and. Sub-as 1. (i): + In this situation, th last paymnt of rtailr in tim + is longr than rtailr s rdit priod, thrfor th rtailr finand all itm sold aftr at an intrst harg I (4) + I Intrst hargd pr yar = t( a bt) dt 0 I a b = + + ( ) ( ) (6)
38 Cumulativ rvnu D I(t) B C O + im Fig 1: < and +. h rtailr slls dtriorating goods at th bginning but rivs mony at tim. During to, th rtailr aumulat rvnu in an aount that arns I /$/yar. Hn, th intrst arnd/ yar is si si = t( a bt) dt = ( ) ( ) 0 hrfor, th annual apital opportunity ost is I si = ( + ) ( + ) ( ) ( ) otal annual ost for th rtailr an b xprssd as follows: A {(m +1)b-a}(1+ m) h b(1 m) b(1 m ) C1( ) (1 m) b 1+ m- + = + + + log + + 1+ m 6 3 (1 + m) (1 + m) 1 + m (1 + m) + {( m+ 1) b a} log + 6 1+ m 4 I si + ( ) ( ) + + ( ) ( ) {( m+ 1) b a}( m + m+ 1) A h b(1 m) b(1 + m ) = (1 m) b + + (1 m) + + + + 6 3 b(1 + m) {( m+ 1) b a}( m + m+ 1) {( m+ 1) b a} (1 + m) {( m+ 1) b a} + + 6 4 4 I ( + ) si ( ) ( + + ) ( ) Sub-as1.: + In this situation rtailr rdit priod is longr than tim at whih th rtailr rivs th payout from th ustomr, rtailr rivs th whol rvnu and thr is no intrst harg, si and Intrst arnd / yar = t( a bt) dt t( a bt) dt + 0 + si a b 3 3 = ( ) ( ) + ( a+ b+ b) (11) (7) (8) (9) (10)
39 Cumulativ rvnu D I(t) O + im Fig : < and > +. hrfor, th total annual ost of rtailr A {(m +1)b-a}(1+ m) h b(1 m) C1.( ) (1 m) b 1+ m- = + + + log + 1+ m 3 3 b(1 + m ) (1 + m) (1 + m) 1 + m (1 + m) + + {( m+ 1) b a} lo g + 6 6 1+ m 4 si a b 3 3 ( ) + ( ) + { a + b ( + )} 3 {( m+ 1) b a}( m + m+ 1) A h b(1 m) b(1 + m ) = (1 m) b + + (1 m) + + + + 6 3 b(1 + m) {( m+ 1) b a}( m + m+ 1) {( m+ 1) b a} (1 + m) {( m+ 1) b a} + + 6 4 4 si a b 3 3 ( ) + ( ) + { a + b ( + )} 3 (1) (13) Cas : In this as, ustomr s rdit priod is longr than rtailr s rdit priod, th rtailr dosn t arnd intrst, but hargd intrst during to +. hrfor, th intrst hargd pr yar 1 b + = I ( )( a b) + { a ( )} 1 = I ( )( a b) ( a bt) dt + (14)
40 Cumulativ rvnu D I( t) O + im Fig 3: or ( ) C ( ) C A {(m +1)b-a}(1+ m) h b(1 m) (1 m) b 1+ m- = + + + log + 1+ m 3 3 b(1 + m ) (1 + m) (1 + m) 1 + m (1 + m) + + {( m+ 1) b a} log + 6 6 1+ m 4 b + I ( { )( a b) + a b + {( m+ 1) b a}( m + m+ 1) A h b(1 m) b(1 + m ) = (1 m) b + + (1 m) + + + + 6 3 b(1 + m) {( m+ 1) b a}( m + m+ 1) {( m+ 1) b a} (1 + m) {( m+ 1) b a} + + 6 4 4 b + I {( ) ( a b) + a b + ( Approximatly) (15) (16) 4. DEERIAIO OF HE OPIAL SOLUIO For as 1: < Sub-as1.1: +, Sub-as1.: + and Cas : Diffrntiating Equations (10), (13) and (16) with rspt to, two tims, w gt dc 1 ( ) {( m+ 1) b a}( m + + m+ 1) A hb(1 + m ) (1+ m+ ) b(1 + m) = + d (1 + m) 6 6 h{( m+ 1) b a}( m + + m+ 1) h{( m+ 1) b a} I a( + )( ) (17) 4 4 I b( + ) ( ) 1 + + si ( ) ( ) 3 dc 1. ( ) {( m+ 1) b a}( m + + m+ 1) A hb(1 + m ) (1+ m+ ) b(1 + m) = + d (1 + m) 6 6 3 h{( m+ 1) b a}( m + + m+ 1) h{( m+ 1) b a} si a( ) si b( ) SI b 4 4 (18)
41 and dc ( ) {( m+ 1) b a}( m + + m+ 1) A hb(1 + m ) (1+ m+ ) b(1 + m) = + d (1 + m) 6 6 h{( m+ 1) b a}( m + + m+ 1) h{( m+ 1) b a} I b( ) + I ( a+ b+ b) 4 4 ( 1 ) ( 1 ) d C ( ) 1 1 h hb + m b + m = {( m 1) b a} ( m 1) A d + + + + + + 3 3 b( )( ) a b I + a( ) ( ) s ( ) ( ) + 3 (0) hb(1 + m ) (1 + m) (1 + m) I + 8 + { a b( + ) }. 6 ( 1 ) ( 1 ) d C ( ) 1 1 h hb + m b + m = {( m 1) b a} ( m 1) A d + + + + + + 3 3 b ( + + ) hb(1 + m ) (1 + m) (1 + m) + si ( ) a( ) 8 + + 3 6 d C ( ) 1 d ( 1 ) ( 1 ) h hb + m b + m = {( m 1) b a} ( m 1) A bi + + + + + + 3 3 (19) (1). () h nssary ondition for C ( ), C ( ) and ( ) dc C to b minimum is ( ) dc 1 = 0, ( ) 1. = 0, 1 1. d d ( ) dc d C ( ) d C ( ) 1 = 0, providd > 0, 1. d C ( ) > 0 and > 0, whih is lar from (0), (1) and () d d d d dc1 that all sond drivativs ar positiv. Putting, ( ) dc1. = 0, ( ) dc = 0, and ( ) = 0, w gt d d d {( m+ 1) b a}( m + + m+ 1) hb(1 + m ) (1+ m+ ) b(1 + m) + A+ + (1 + m) 6 6 h{( m+ 1) b a}( m + + m+ 1) h {( m+ 1) b a } I a( + )( ) + + (3) 4 4 I b( + ) ( ) si = 3 ( ) ( ) 0 {( m+ 1) b a}( m + + m+ 1) hb(1 + m ) (1+ m+ ) b(1 + m) + A+ (1 + m) 6 6 h{( m+ 1) b a}( m + + m+ 1) h {( m+ 1) b a} si a( ) + + + (4) 4 4 3 3 si b( ) + + si b = 0, 3 and {( m+ 1) b a}( m + + m+ 1) hb(1 + m ) (1+ m+ ) b(1 + m) + A+ + (1 + m) 6 6 h{( m+ 1) b a}( m + + m+ 1) h {( m+ 1) b a} + + (5) 4 4,
4 I b ( ) I ( a + b + b ) = 0. 5. UERICAL EXAPLES AD SESIIVIY AALYSIS Exampl: 1 Lt us onsidr th paramtr valus a = 5 units/yar, b = 0.5, s = $15 pr unit, = $10 pr unit, A= $50/ordr, h = $/unit/yar, = 100/365 yar, = 50/365yar, m = 1 yar, I = $0.17 / $ / yar, I = $0.10 / $ yar. Substituting ths valus in th quation (0), w gt, th optimum solutions for = * = 0.319 yar and orrsponding optimum total annual ost C = C* = $11.3159. Cas 1: < Sub-as1.1: + abl 1.h snsitivity analysis will b hlpful in dision making to analyz th fft of hang of ths variations. Using th sam abov data (Exampl 1) th snsitivity analysis of diffrnt paramtrs has bn don. W study th fft of th variations in a singl paramtr kping othr systm paramtrs sam on th optimal solutions. Changing paramtrs hang * C* a 4.9 4.8 4.7 4.6 0.49478 0.63397 0.757551 0.877638 17.577 1.453 4.7180 7.351 s 17 19 1 3 0.34 0.3346 0.3701 0.31938 11.856 11.579 11.9 11.004 9.5 9 8 7 0.46911 0.58779 0.794300 0.988557 15.6140 18.563.4641 4.7753 A 53 56 59 6 0.51588 0.653646 0.76840 0.868848 18.4356 3.5398 7.768 31.3571 h.1..3.4 0.88785 0.48577 0.01936 0.14658 9.59314 7.67585 5.46005.6838 15/365 150/365 175/365 00/365 0.35699 0.37035 0.39057 0.33187 10.78 9.75954 9.7845 9.7795 m 0.9 0.8 0.7 0.6 0.464599 0.565615 0.645577 0.710943 17.1607 1.8798 6.15 30.1416
43 Exampl: Lt a = 5 units/yar, b = 0.5, s = $15 pr unit, = $10 pr unit, A= $50 pr ordr, h = $/unit/yar, = 375/365 yars, = 50/365 yars, m = 1 yar, I = $0.17 / $ yar, I = $0.10 / $ yar. Substituting ths valus in (1), w gt, th optimum solutions for * = 0.619185 yar and orrsponding optimum total annual ost C* = $ 5.55998. Sub-as1.: + abl. h snsitivity analysis will b hlpful in dision making to analyz th fft of hang of thss variations. Using th sam abov data (Exampl ) th snsitivity analysis of diffrnt paramtrs has bn don. W study th fft of th variations in a singl paramtr kping othr systm paramtrs sam on th optimal solutions. Changing paramtrs hang * C* s 17 19 1 3 0.645715 0.67167 0.695946 0.719844 5.16105 4.79179 4.44699 4.175 A 51 5 53 54 0.67408 0.71797 0.768087 0.811806 8.03404 10.053 1.153 13.958 h.1..3.4 0.591773 0.563970 0.535674 0.506757 4.00894.35804 0.59033 --------- 400/365 45/365 450/365 475/365 0.645695 0.6759 0.699809 0.7790 5.30010 5.05806 4.8374 4.6303 m 0.95 0.90 0.85 0.80 0.67030 0.715880 0.756805 0.793748 8.36814 10.897 13.31 15.4163 ot: Dottd data shows th non fasibl valu. Exampl: 3 Lt us onsidr th paramtr valus a = 5units pr yar, b = 0.5, s = $15 pr unit, = $10 pr unit, A = $50 pr ordr, h = $/unit/yar, = 50/365 yars, =100/365 yars, m = 1 yar, I = $0.17 / $ yar, I = $0.10 / $ Substituting ths valus in () th optimum solution for * = 0.90311 yar and th orrsponding optimum total annual ost C* = $14.865. Cas : > yar.
44 abl 3. h snsitivity analysis will b vry hlpful in dision making to analyz th fft of hang of thss variations. Using th sam abov data (Exampl 3) th snsitivity analysis of diffrnt paramtrs has bn don. W study th fft of th variations in a singl paramtr kping othr systm paramtrs sam on th optimal solutions. Changing paramtr hang * C* a 4.9 0.370967 18.4641 4.8 0.439593 1.3604 4.7 0.50131 3.8154 4.6 0.558600 5.9550 9.5 9.0 8.5 8.0 0.418399 0.5858 0.616741 0.705 19.5789.7987 5.1919 7.0191 A 5 54 56 58 0.410014 0.501651 0.578761 0.646588 0.504 4.8651 8.5313 31.7641 h.1..3.4 0.5849 0.757 0.181043 0.17443 1.9934 10.9103 8.50185 5.47454 55/365 60/365 65/365 70/365 0.9040 0.90169 0.90098 0.9007 14.751 14.6391 14.561 14.4131 m 0.9 0.8 0.7 0.6 0.415136 0.505855 0.578344 0.638416 1.3080 6.4788 31.098 35.457 All th abov obsrvations an b summd up as follows: From abl 1, following infrns an b mad: (i). Inras of initial dmand a, unit purhas ost, ordring ost A and xpiration dat m will rsult inras in total annual ost C. hat is, hang in a, A and m will lad positiv hang in C. (ii). Inras of unit slling pri s unit stok holding ost h and rtailr s prmissibl dlay priod will lad dras in total annual ost C. hat is hang in s, h and will lad ngativ hang in C. From abl, th following infrns an b mad: (i). Inras of ordring A and xpiring tim m will lad inras in total annual ost C. hat is hang, in A and m will aus positiv hang in C. (ii). Inras of unit slling pri s, unit stok holding ost h and rtailr s rdit priod will lad dras in total annual ost C. hat is hang in s, h and will aus ngativ hang in C. From abl 3, th following infrns an b mad: (i). Inras of initial dmand a, unit purhas ost, ordring A and xpiring tim m will lad inras in total annual ost C. hat is hang, in a,, A and m will aus positiv hang in C. (ii). Inras of unit stok holding ost h and rtailr s rdit priod will lad dras in total annual ost C. hat is hang in h and will aus ngativ hang in C.
45 6. COCLUSIO AD FUURE RESEARCH his study was motivatd by th obsrvation of daily prati in th fild of matrial and pharmautial managmnt. In this papr, w hav dvlopd gnral approah to dtrmin optimal ordring poliy for drasing dmand with tim dpndnt dtrioration undr fixd lif tim prodution and trad rdits. By adopting th tim dpndnt dtrioration, th dtrioration boms zro for larg xpiring dat and boms on for tim approahs to xpiration dat. In this papr, w hav built up an optimal ordr quantity modl to obtain optimal total annual ost onsidring (i) ustomr s trad rdit priod offrd by th rtailr is lss than rtailr s prmissibl dlay priod offrd by th supplir (i.. < ) and (ii) ustomr s prmissibl dlay priod offrd by th rtailr is gratr than or qual to rtailr s trad rdit priod offrd by th supplir (i.. ). athmatial modls hav drivd to find optimal solution. orovr, w hav shown that th variation is quit snsitiv with rspt to diffrnt to ky paramtrs h possibl xtnsion of th prsnt modl ould b allowabl shortags and inflation. h modl an also b gnralizd for adding th fright hargs and othrs. REFERECES 1. Goyal,S.K. (1985). Eonomi ordr quantity undr prmissibl. Journal of Oprational Rsarh Soity, 36, 335-338.. Aggarwal,S.P. and Jaggi,C.K.(1993). Ordring poliis of dtriorating itms undr prmissibl. Journal of Oprational Rsarh Soity, 46, 658-66. 3. Shah,.H. (1993). Probabilisti tim- shduling modl for an xponntially daying invntory whn dlay in paymnt is prmissibl. Intrnational Journal of Prodution Eonomis, 3(1), 77-8. 4. Shinn,S.W. Hwang,H.P. and Sung,S.(1996). Joint pri and lot-siz dtrmination undr onditions of prmissibl dlay in paymnts and quantity disount for fright ost. Europan Journal of Oprational Rsarh, 91, 58-54. 5. Chu, P. Chung, K.J. and Lan, S.P. (1998). Eonomi ordr quantity of dtriorating itms undr prmissibl dlay in paymnts. Computrs and Oprations Rsarh, 5, 817-84 6. Khanra, S., Ghosh, S.K. and Chaudhuri, K.S. (011). An EOQ modl for dtriorating itms with tim-dpndnt quadrati dmand undr prmissibl. Applid athmatis and Computation, 18, 1-9. 7. ng, J.. (00). On th onomi ordr quantity undr onditions of prmissibl. Journal of Oprational Rsarh Soity, 53, 915-918. 8. Lou, K.R. and Wang, L. (013). Optimal lot-sizing poliy for a manufaturr with dftiv itm in a supply hain with up- stram and down-stram trad rdits. Computrs and Industrial Enginring, 66, 115-1130. 9. Huang, Y.F. (003). Optimal rtailr s ordring poliis in th EOQ modl undr trad rdit finaning. Journal of Oprational Rsarh Soity, 54, 1011-1015. 10. Soni,. and Shah,.H. (008). Optimal ordring poliy for stok-dpndnt dmand undr progrssiv paymnt shm. Europan Journal of Oprational Rsarh, 184, 91-100. 11. ng, J.. and Chang, C.. (009). Optimal manufaturrs rplnishmnt poliis in th EPQ modl undr two lvls of trad rdit poliy. Europan Journal of Oprational Rsarh, 195(), 358-363. 1. ripathi, R.P. (014). Optimal paymnt tim for a rtailr with xponntial dmand undr prmittd rdit priod by th wholsalr. Applid athmatis and Information Sin Lttr, (3),91-101. 13. Chung, K.J., Chang, S.L. and Yang, W.D. (001). h optimal yl tim for xponntially dtriorating produts undr trad rdit finaning. h Enginring Eonomist, 46, 3-4. 14. Davis, R.A. and Gaithr,. (1985). Optimal ordring poliis undr onditions of xtndd paymnt privilags. anagmnt Sin,31, 499-509. 15. Chung, K.J. Liao,J.J. (004). Lot sizing disions undr trad rdits dpnding on th ordring quantity. Computrs and Oprations Rsarh,31, 909-98. 16. Huang,Y.F. and Hsu,K.H. (008). An EOQ modl undr rtailr partial trad rdit poliy in supply hain. Intrnational Journal of Prodution Eonomis,11,655-664. 17. Ouyang,l.Y., Chang,C.. and Shum,P. (01). h EPQ with dftiv itm and partially prmissibl linkd to ordr quantity drivd algbraially. Cntral Europan Journal of oprations Rsarh, 0(1),141-160. 18. Skouri, K. Konstantaras, I. Papahristos, S., and ng, J..(011). Supply hain modls for dtriorating produts with ramp typ dmand rat undr prmissibl. Exprt systms with Appliations, 38, 14861-14869. 19. Yang,C.. Pan,Q.H. ouyang,l.y. and ng,j.. (01). Rtailr s optimal ordr and rdit poliy for dtriorating itm
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