Optimal Replenishment Policy for Hi-tech Industry with Component Cost and Selling Price Reduction

Transcription

1 Opmal Replenshmen Poly for H-eh Indusry wh Componen Cos and Sellng Pre Reduon P.C. Yang 1, H.M. Wee, J.Y. Shau, and Y.F. seng 1 1 Indusral Engneerng & Managemen Deparmen, S. John s Unversy, amsu, ape 5135 awan, ROC Indusral Engneerng Deparmen, Chung Yuan Chrsan Unversy, Chungl 33 awan, ROC Asra. Due o rapd ehnologal nnovaon and gloal ompeveness, he omponen os, he sellng pre and he demand rae of H-eh ndusres (suh as ompuers and ommunaon onsumer s produs usually delne sgnfanly wh me. From a praal vewpon, here s a need o develop a replenshng poly wh fne horzon when he omponen os, he sellng pre and he demand rae are redued smulaneously. A numeral example and sensvy analyss are arred ou o llusrae hs model. wo ases are dsussed n hs sudy: Case A onsders fxed replenshmen nerval, Case B onsders varyng replenshmen nerval. From Case A, he resuls show ha dereasng omponen os leads o smaller replenshmen nerval. However, dereasng sensve parameer of demand leads o larger replenshmen nerval. When oh he omponen os and he sensve parameer delne-raes derease smulaneously, he replenshmen nerval dereases. he soluons y Case A and B are su-opmal and opmal respevely. he ne-prof perenage dfferene eween Case A and B s.6% approxmaely, whle he ompuaonal proess of Case A s easer han ha of Case B. Keywords: replenshmen nerval, os/pre/demand reduon, H-eh ndusry. 1 Inroduon he rends of H-eh produs have he followng haraers: here are shorer produ lfe yle me, quker responsve me, nreasng need for gloalzaon and massve usomzaon. Moreover, he omponen os, he sellng pre and demand rae usually derease wh me due o ompeveness and ehnologal nnovaon. In some H-eh ndusres suh as ompuers and ommunaon onsumer s produs, some omponen os and sellng pre are delnng a aou 1% per week [1]. hs mples ha purhasng or sellng one-week earler or laer wll resul n aou 1% loss. Lee [] has made some ommen on he mporane of he aove suje. Many oher researhers lke Lev and Wess [3], Goyal [4] and Gason [5] have suded he orderng poly n he lass EOQ model for oh fne and nfne O. Gervas and M. Gavrlova (Eds.: ICCSA 7, LNCS 475, Par I, pp , 7. Sprnger-Verlag Berln Hedelerg 7

2 7 P.C. Yang e al. horzon. Buzao [6] assumed ompound nreasng pre and seup os wh nflaon n a fne horzon. Buzao [6] and Erel [7] onsdered a onnuous pre nrease due o nflaon. Dave and Pael [8] onsdered he nvenory of deerorang ems wh me proporonal demand. he onsderaon of exponenally dereasng demand was frs analyzed y Holler and Mak [9] who relaxed he fxed replenshmen yle assumpon n Dave and Pael, and oaned opmal replenshmen poles under oh onsan and varale replenshmen nervals. Erel [7] onsdered a ompound-nreasng pre EOQ model wh nflaon rae. Harga and Benkherouf [1] derved opmal and heurs nvenory replenshmen models for deerorang ems wh exponenal me-varyng demand. Wee [11] derved a jon prng and replenshmen poly for deerorang nvenory wh delnng marke. Yang and Wee [1] addressed a quk response produon sraegy wh onnuous demand and pre delnng n a fne horzon. Khouja and Park [13] derved an opmal lo sze model for a dereasng rae of omponen os n a fne horzon. Usng presen-value onep, euner [14] derved a modfed EOQ model o approxmae Khouja and Park s model [13]. None of hem onsders he smulaneous derease n he omponen os and sellng pre derease wh exponenal me-varyng demand. In hs sudy, a replenshmen poly wh fne plannng horzon s developed for a uyer when he omponen os, he demand rae and he sellng pre o he end-onsumer delne a a onnuous rae. Fxed and varyng replenshmen nervals are onsdered n hs sudy. A mahemaal model and s soluon proedure are developed n he nex wo seons. A numeral example s hen provded o demonsrae he dfferene eween he wo ases. Sensvy analyss s arred ou o derve he degree of sensvy for he yle me of eah parameer. he onludng remark s gven n he las seon. Mahemaal Modelng and Analyss he mahemaal model n hs paper s developed on he ass of he followng assumpons: (a Replenshmen rae s nsananeous. ( Componen os and produ sellng pre o he end onsumer delne a a onnuous rae per un me. ( Demand rae s onnuous and exponenally dereasng. (d Enre plannng horzon s fne. (e wo ases of oh denal and dfferen replenshmen nervals are onsdered. (f No shorage s allowed. (g Purhase lead-me s onsan. he deson varales are n numer of orders n he enre plannng horzon Q -1 lo sze durng h yle, = 1, n me pon when he nvenory level of h yle drops o zero

3 Opmal Replenshmen Poly for H-eh Indusry 73 yle me or lengh of he replenshmen nerval when eah replenshmen nerval s denal he oher relaed parameers are as follows: d( weekly demand rae, where d( = a exp(, a s he sale parameer and s he sensve parameer of demand C un omponen os, where C = C (1, C s he un omponen os when r =, r s he weekly delne-rae of omponen os S un sellng pre, wheres = S(1 r, S s he un sellng pre when =, r s s he weekly delne-rae of sellng pre o he end-onsumer H weekly lengh of he plannng horzon C 1 orderng os per order C holdng os per dollar per week NP ne prof n he plannng horzon s Invenory Q Q 1 Q -1 Q n n-1 n Fg. 1. Graphal represenaon of nvenory sysem wh demand derease Whou loss of generaly, -1 (= 1, n are he re-orderng mes over he enre perod H. Inal me ( = and fnal me ( n =H nvenores are oh zero. Invenory n h yle drops o zero a pon (= 1, n. he purpose of hs prolem s o oan opmal values of suh ha he oal ne prof over he fne horzon s a maxmum value. he ases for fxed and varyng replenshmen nervals are dsussed as follows:

4 74 P.C. Yang e al. Case A. For fxed replenshmen nerval For fxed replenshmen, =H/n, (1 =, = 1, n ( Sne sok s depleed y demand, he dfferenal equaon of nvenory level s di ( = ae d, 1 ( (3 Usng he oundary ondon I ( = when =, he nvenory level s a I( = ( e e (4 he lo sze durng h yle s I( when = (-1, ha s a Q 1 = e ( e 1 (5 Durng h yle, he un omponen os sc (1 ( 1, and he holdng os, HC s r HC = C ( 1 acc = C (1 r ( e ( 1 I( d 1(1 r ( 1 e (6 he holdng os n he whole plannng horzon, HC s he summaon of n yles of (6. ha s acc( e HC = 1[(1 r [(1 r e ] n e n 1] (7 he omponen os durng h yle, PC s he produ of Q -1 and un omponen os C (1 r a = (-1. ha s PC ( 1 ac e ( e 1(1 r = (8 he omponen os n he whole plannng horzon, PC s he summaon of n yles of (8. ha s

5 Opmal Replenshmen Poly for H-eh Indusry 75 ac( e PC = 1[(1 r [(1 r e n e ] n 1] (9 he sales revenue durng h yle, SR s he negraon of he produ of he un sellng pre and he demand quany. ha s SR ( 1 [ ln(1 rs ] [ ln(1 rs ] as ( e e = Sd( d = (1 ( 1 ln(1 r s he sales revenue n he whole plannng horzon, SR s he summaon of n yles of (1. ha s n as[1 (1 rs e SR = ln(1 r s n ] (11 he ne prof, NP s NP = (1 SR PC HC nc 1 Sne s equal o H dvded y n, he ne prof (1 s a funon of sngle varale n. Assumng n s any real numer, he suffen ondons for he onavy of he ne prof (1 are and dnp = dn (13 d NP < (14 dn he soluon of (13 anno e a lose form, u an e ompued numerally. Aually n s a posve neger, he opmal value of n, denoed y n *, mus sasfy he followng ondon * * * NP ( n 1 NP( n NP( n + 1 (15 * he value of n s loaed n he neghorhood of n sasfyng (13. Case B. For varyng replenshmen nerval he nvenory level dfferenal equaon s di ( = ae, 1 for = 1, n (16 d Usng he oundary ondon I(= when =, he nvenory level s a I( = ( e e (17

6 76 P.C. Yang e al. he lo sze durng h yle s I( when = -1, ha s Q 1 a 1 ( = e e (18 Durng h 1 yle, he un omponen os s (1 C r, and he holdng os, HC s HC = 1 C C (1 r 1 I ( d 1 CC(1 r a( e e = d 1 acc = (1 r [( 1 e + e 1 he omponen os durng h yle, PC s he produ of Q -1 and un omponen os C (1 a = -1. ha s r a 1 1 PC = ( e e C(1 r ( he sales revenue durng h yle, SR s he negraon of he produ of un sellng pre and he demand quany. ha s ] (19 SR = as Sd( d = ( e ln(1 rs ] e ln(1 r 1[ [ ln(1 r ] s (1 1 s he ne prof, NP s NP = n = 1 SR = 1 1 = 1 H as[1 (1 rs e = ln(1 r n n acc PC s n = 1 H (1 r HC nc ] 1 n = 1 ( e 1 ac [ e ( e 1 e d nc 1 (1 r 1 ] ( he ne prof ( has n deson varales: 1, n-1 and n. 3 Soluon Proedure Our am s o derve he opmal values of he deson varales o maxmze he oal ne prof n he enre plannng horzon.

7 Opmal Replenshmen Poly for H-eh Indusry 77 Case A. For fxed replenshmen nerval he soluon proedure s as follows: (A1 Derve he value from (13. (A he neger near he value from (A1 ha sasfes (15 s he opmal value. Case B. For varyng replenshmen nerval he soluon proedure s as follows: (B1 Le he opmal value of Case A as he sarng value of n. he ne prof ( s denoed y NP( (n, n. (B Equae he frs paral dervaves of he ne prof ( o zero wh respe o. NP( ( n, n =, = 1, n-1 (3 (B3 Solve (n usng he n-1 smulaneous equaons from (B. (B4 he opmal value of n, denoed y n *, whh s n he neghorhood of he sarng value from (B1, mus sasfy he followng ondon: * * * * * * NP( ( n 1, n 1 NP( ( n, n NP( ( n + 1, n + 1 (4 4 Numeral Example he preedng heory an e llusraed y he followng numeral example. he parameers are gven as follows: Un omponen os C = C(1 r, where C = $ 1 and r =. 1per week; Un sellng pre S = S(1 r s, where S = $ 1 and rs =. 1per week; Enre Plannng horzon onsdered, H= 6 weeks; Demand rae, d(= a exp(-, H, a= 1, =.1; Orderng os per order, C 1 = $75; Holdng os per dollar per week, C =.8. ale 1. Ne prof wh varous replenshmen mes for Case A & B n NP for Case A NP for Case B % error 4 $7,46 $7, % 5 $9,568 $9, % 6 $3,899 $3, % 7 $31,77 $31, % 8 $3,35 $3, % 9 $3,743 $3, % 1 $33,1 $33,8 -.8% 11 $33,161 $33, % 1 $33,49 $33, % 13* $33,81* $33,31* -.6% 14 $33,68 $33, % 15 $33,1 $33, % 16 $33,145 $33, %

8 78 P.C. Yang e al. Applyng he soluon proedure, he ompuaonal resuls are gven as follows: For Case A and B, he opmal replenshmen mes are oh 13 n he plannng horzon of 6 weeks. he ne profs are $33,81 and $33,31 respevely. he perenage error of ne prof (denoed y % error eween Case A and Case B s defned as % error= [(NP for Case A-NP for Case B/(NP for Case B]1% From ale 1 and Fgure, he value of % error s -.6%. he mnus sgn means Case A s ne prof s less han Case B s ne prof (.e, $33,81<$33,31. Bu he ompuaonal proess of Case A s easer han ha of Case B. he relevan values for 13 replenshmen yles are shown n ale. Wh fxed replenshmen nerval n Case A, he replenshmen nerval nreases wh me. For Fg.. Ne Prof wh varous n for Case A & B h ale. Relaed resuls for 13 replenshmen yles for Case A & B Case A Case B yles Inerval Lo Inerval Lo sze ( - -1 sze Q -1 ( - -1 Q

9 Opmal Replenshmen Poly for H-eh Indusry 79 Case B, he lo sze dereases slghly wh me and nearly manans he same level for all yles due o nreasng nerval and dereasng demand. 5 Sensvy Analyss For ease of ompuaon under aepale error, Case A s used o arry ou he sensvy analyss when one or wo parameers n a se of parameer values Φ {C, C 1, a, C, r, H,, S and r s } hanges %, 4% and 6%. he resuls are shown n ale 3-5 and Fgure 3. ale 3. Sensvy analyss of yle me when one parameer hanges parameer -6% (I -4% -% % +% +4% +6% (II Degree of sensvy a, C hgh C hgh r, C medum H (If n s low real (n=5.7 (n=8.3 (n=1.8 (n=13. (n=15.5 (n=17.6 (n=19.7 H (If n s low neger (n=6 (n=8 (n=11 (n=13 (n=16 (n=18 (n= low r s, S none Noe: Degree of sensvy = ( I II 1% max( I, II =he asolue value of he yle-me peren dfferene eween hangng +6% and -6% for eah parameer In ale 3, we defned he degree of sensvy as he asolue value of he yle-me peren dfferene eween hangng +6% and -6% for eah parameer. From ale 3 and Fgure 3, s oserved ha C, C 1 and a are hghly sensve; r and C are medum, H and are low, and r s and S are none-sensve. When C 1 nreases, he yle me nreases, hus redung he un orderng os. When a, C and C nrease, he yle me dereases o ounera he nreasng holdng os. When r nreases, he yle me nreases o allow more low-os sok. If n s neger, he yle me fluuaes up and down, u n he rend of nreasng wh he lengh of me

10 73 P.C. Yang e al. horzon due o dsree value of n. If n s assumed o e real, he yle me mono-nreases wh he lengh of me horzon (see rows 5-6 n ale 3. Fg. 3. Cyle me wh % hange of eah parameer ale 4. Sensvy analyss of yle me when oh r and a hange r a {.1} {1} , ale 5. Sensvy analyss of yle me when oh r and hange r {.1} {.1}

11 Opmal Replenshmen Poly for H-eh Indusry 731 When wo parameers are hanged smulaneously, he parameer wh hgher degree of sensvy wll e domnan o he parameer wh lower degree of sensvy. For example, when oh r and a hange, he sensvy analyss of yle me s shown n ale 4. When r nreases, he yle me dereases. When r nreases and a dereases smulaneously, he yle me nreases (see he old numer n ale 4 eause he parameer a s more sensve han he parameer r. From ale 5, he yle me ends o nrease when nreases. Whle when oh and r nrease smulaneously, he yle me ends o derease (see he old numer n ale 5 eause he parameer r s more sensve han he parameer. he exra ne prof, denoed y Δ NP, s defned as Δ NP = NP when r or s onsdered NP when r or s gnored he ompuaonal resuls of Δ NP are gven n ale 6-7. he parameer r s more sensve o exra ne prof han he parameer. When he value of r or nreases, he exra ne prof eomes more sgnfan. r ale 6. Δ NP when r s onsdered When r s gnored (r = When r s onsdered NP NP Δ NP {.1} Noe: Δ NP =NP when r s onsdered-np when r s gnored ale 7. Δ NP when s onsdered Δ NP When s gnored (= When s onsdered NP NP {.1} Noe: Δ NP =NP when s onsdered-np when s gnored

12 73 P.C. Yang e al. 6 Conludng Remarks In hs sudy, models for fxed (.e., Case A and varyng (.e., Case B replenshmen nervals are developed for dereasng omponen os, sellng pre and demand rae. he soluon of Case A s su-opmal, and Case B s opmal. he ne prof perenage error s.6% approxmaely. Bu he operaon of fxed replenshmen nerval s smpler, and he ompuaonal proess of Case A s easer han ha of Case B. Based on hs sudy, we reommend o nvenory praoners o adop he mehod of Case A. When he omponen os delne-rae nreases, he replenshmen nerval ends o derease resulng n more frequen and jus-n-me delveres. When he demand rae delne-rae nreases, he replenshmen nerval ends o nrease, resulng n lesser un orderng os. I s known ha when he sellng pre delne-rae nreases, he replenshmen nerval s denal due o he same sales revenue. I s oserved ha he orderng os has a hgh degree of sensvy o he yle me. he omponen os delne-rae has a medum sensvy degree, and he demand-sensve parameer s none-sensve o he yle me. When oh he omponen os and he demand-sensve parameer delne-raes nrease smulaneously, he replenshmen nerval ends o derease. I s eause he degree of sensvy for he omponen os delne-rae s hgher han ha of he demand-sensve parameer. he resuls gve helpful manageral nsghs n produon plannng. Referenes 1. Sern, L.C.: Presen and fuure of supply han n nformaon and eleron ndusry. In: Supply Chan Managemen Conferene for Eleron Indusry, Naonal sng Hua Unversy Hsnhu, awan, pp. 6 7 (3. Lee, C.H.: Invene Group worldwde operaon. In: Supply Chan Managemen Conferene- wh Noeook Compuers as Example. Chung Yuan Chrsan Unversy, Chungl, awan, pp ( 3. Lev, B., Wess, H.J.: Invenory models wh os hanges. Operaons Researh 38, ( Goyal, S.K.: A noe on nvenory models wh os hanges. Operaons Researh 4, ( Gason, A.: On he fne horzon EOQ model wh os hanges. Operaons Researh 43, ( Buzao, J.A.: Eonom order quanes wh nflaon. Operaonal Researh Quarerly 6, ( Erel, E.: he effe of onnuous pre hange n he EOQ. Omega, ( Dave, U., Pael, L.K. (,S poly nvenory model for deerorang ems wh me proporonal demand. Journal of he Operaonal Researh Soey 4, ( Holler, R.H., Mak, K.L.: Invenory replenshmen poles for deerorang ems n a delnng marke. Inernaonal Journal of Produon Researh 1, ( Harga, M.A., Benkherouf, L.: Opmal and heurs nvenory replenshmen models for deerorang ems wh exponenal me-varyng demand. European Journal of Operaonal Researh 79, (1994

13 Opmal Replenshmen Poly for H-eh Indusry Wee, H.M.: Jon prng and replenshmen poly for deerorang nvenory wh delnng marke. Inernaonal Journal of Produon Eonoms 4, ( Yang, P.C., Wee, H.M.: A quk response produon sraegy o marke demand. Produon Plannng & Conrol 1, (1 13. Khouja, M., Park, S.: Opmal lo szng under onnuous pre derease. Omega 31, (3 14. euner, R.: A noe on Khouja and Park, opmal lo szng under onnuous pre derease, Omega 31(3. Omega 33, (5