International Journal of Supply and Operations Management

Transcription

1 Inernaional Journal of Supply and Operaions Managemen IJSOM May 05, Volume, Issue, pp ISSN-Prin: 8-59 ISSN-Online: 8-55 wwwijsomcom An EPQ Model wih Increasing Demand and Demand Dependen Producion Rae under rade Credi Financing Juanjuan Qin a* a Business School, ianjin Universiy of Finance & Economics, ianjin, China Absrac his paper invesigaes an EPQ model wih he increasing demand and demand dependen producion rae involving he rade credi financing policy, which is seldom repored in he lieraures he model considers he manufacurer was offered by he supplier a delayed paymen ime I is assumed ha he demand is a linear increasing funcion of he ime and he producion rae is proporional o he demand ha is, he producion rae is also a linear funcion of ime his sudy aemps o offer a bes policy for he replenishmen cycle and he order quaniy for he manufacurer o maximum is profi per cycle Firs, he invenory model is developed under he above siuaion Second, some useful heoreical resuls have been derived o characerize he opimal soluions for he invenory sysem he Algorihm is proposed o obain he opimal soluions of he manufacurer Finally, he numerical examples are carried ou o illusrae he heorems, and he sensiiviy analysis of he opimal soluions wih respec o he parameers of he invenory sysem is performed Some imporan managemen insighs are obained based on he analysis Keywords: EPQ; rade credi financing; increasing demand; demand dependen producion rae * Corresponding auhor address: jufeqin@6com 5

2 An EPQ Model wih Increasing Demand and Demand Dependen Producion Rae under rade Credi Financing Inroducion rade credi is a kind of credi which is offered by he buyer, allowing he buyer o buy he goods from he seller wihou he immediae paymen Since he globalizaion of he marke and increased compeiion, he enerprises always adop rade credi financing policy o boos sales, increase he marke share and reduce he on-hand invenory levels As a resul, he rade credi financing plays an imporan role as he source of business financing afer banks or oher financial insiue For example, Wal-Mar has adoped he rade credi as a larger source of capial han bank borrowings (Zhang and Zhou, 0) In he radiional invenory EPQ (Economic Producion Quaniy, EPQ) or EOQ (Economic Order Quaniy, EOQ) model, i is assumed ha he manufacurer mus pay for he producs when receiving hem However, in pracice, he suppliers ofen provide he rade credi for he paymen of he amoun owed herefore, he assumpions of he EPQ or EOQ will be relaxed considering he rade credi he opimal decisions for he invenory sysem will be influenced On he oher hand, he opimal replenishmen policies are also affeced by he marke demand and he producion rae In he radiional EPQ or EOQ model, i is assumed ha he demand is a consan and he producion rae is infinie or a consan Bu in realiy, for he marke demand, i will be influenced by many facors such as he ime for he shor life cycle producs And when he marke demand is beer, he manufacurer will provide a higher producion rae; when he marke demand is shrinking, he supplier will reduce he producion rae herefore, he producion rae should be demand dependen producion rae herefore, in-deph research is required on he invenory replenishmen decisions wih he rade credi considering he increasing demand and he demand dependen producion rae o exend he radiional EPQ model Lieraure Review he invenory replenishmen policies under rade credi financing have been researched inensively he EOQ model wih he rade credi financing is firs proposed by Goyal (985), which is exended by Chu e al (998) considering he deerioraed producs he order sraegies are sudied by Jamal e al (997) and Chang and Dye (00) allowing he shorage under delayed paymen and deerioraing condiions he economic producion quaniy model is researched by Chung and Huang (00) considering he manufacurer offering he reailer he delayed paymen policy Huang (00) firs exended Goyals model o analyse he wo levels of he rade credi policy Qiu e al (00), Huang (007), and Chung (00) exend he models of Huang (00) sao and Sheen (008) use EPQ o model he decisions under sysem mainenance and rade credi eng e al (0) develop a supply chain invenory model wih rade credi financing linked o order quaniy, and hen sudy he opimal policies for boh he vendor and he buyer under a non-cooperaive environmen firs, and hen under a cooperaive inegraed siuaion Liao e al (0) aemp o deermine economic order quaniy for deerioraing iems wih wosorage faciliies (one is an owned warehouse and he oher is a rened warehouse) where rade credi is linked o order quaniy I is clear ha he issue of rade credi is very popular in his field of research he papers above discussed he EOQ or EPQ invenory models under rade credi financing all based on he assumpions ha he demand rae is consan over ime Bu in pracice, he marke demand is always changing fas and effeced by many facors such as he price, he invenory level, he sage of he produc life ime, ec Some researchers realize his phenomenon and exend he sudies above o esablish he invenory models by assuming ha he demand is variable Chung and Liao (0) discussed he invenory replenishmen problems wih rade credi In J Supply Oper Manage (IJSOM), Vol, No 5

3 Qin financing considering a price sensiive demand Kreng and an (0) propose a producion model for a lo-size invenory sysem considering he defecive qualiy under he condiion of wolevel rade credi policy wih a consan producion rae and demand Sarkar (0) analyzed he EOQ model wih delay in paymens considering demand dependen on ime and a consan replenishmen rae eng e al (0) build he economic quaniy model wih rade credi financing for he non-decreasing demand wih ime and infinie replenishmen rae Min e al (0) develop an invenory model under condiions of permissible delay in paymens, assuming ha he iems are replenished a a known consan rae and he demand rae of he iems is dependen on he curren invenory level Soni (0) discusses he opimal replenishmen policies wih he price and sock sensiive demand and infinie producion rae under rade credi Wu e al (04) explore he opimal credi period and lo size considering he demand dependen on he delayed paymen ime and he infinie producion rae However, he infinie or a consan replenishmen rae of he invenory models is inconsisen wih he acual indusrial pracices When he marke demand is beer, he manufacurer will provide a higher producion rae; if he marke demand is shrinking, he manufacurer will reduce he producion rae In he radiional EOQ or EPQ models wihou he rade credi financing, Manna and Chaudhuri (006) presened a producion-invenory sysem for deerioraing iems wih demand rae being a linearly ramp ype funcion of ime and producion rae being proporional o he demand rae he wo models wihou shorages and wih shorages were discussed Boh models were sudied assuming ha he ime poin a which he demand is sabilized occurs before he producion sopping ime Skouri e al (0) exends his model by considering ha: a) for he model wih no shorages; he demand rae is sabilized afer he producion sopping ime, and b) for he model wih shorages; he demand rae is sabilized afer he producion sopping ime or afer he invenory level reaches zero or afer he producion resaring ime Recenly, Soni and Pael (0) developed a more general inegraed supplier- reailer invenory model wih a demand rae ha is sensiive o he reailing price and a demand dependen producion rae wih wo level of rade credi financing Bu hey do no consider he siuaion ha he demand increases wih ime from he perspecive of he produc life cycle herefore, based on he lieraures above, his paper found ha none of he above models explore he opimal replenishmen policies of he manufacurer under rade credi financing combining wih he increasing demand and demand dependen producion rae his sudy exends he EPQ models in he several ways following Firs, rade credi financing is inroduced o he radiional EPQ models he manufacurer is offered by he supplier wih a delayed paymen ime Second, he increasing demand is inroduced o he EPQ models wih he rade credi financing he demand is nearly a consan in he mauriy sage from he perspecive of he produc life cycle During he growh ime, mos of he produc demand increases wih ime Furhermore, he producion rae dependen on demand is inroduced o he EPQ models wih rade credi financing considering he increasing demand for he firs ime Given he analysis above, his paper developed an invenory model under rade credi financing wih increasing demand and demand dependen producion rae he sudy hen demonsraes some easy-o-use heorems o idenify he opimal replenishmen policies for he manufacurer Numerical examples are provided o illusrae he heorems proposed Finally, sensiiviy analysis of he opimal soluion wih respec o he parameers of he sysem is carried ou and some imporan managemen insighs are obained Mahemaical modeling he invenory sysem can be considered as follows A he beginning, he sock level is zero he producion begins jus afer 0, coninues up o and sops as soon as he sock level In J Supply Oper Manage (IJSOM), Vol, No 54

4 An EPQ Model wih Increasing Demand and Demand Dependen Producion Rae under rade Credi Financing becomes S hen he invenory level decreases due o demand, unil i becomes zero a Hence, he variaion of he invenory level I (), wih respec o ime can be described by he following differenial equaions di () K( ) f ( ), 0 () d And di () f(), () d Where K( ) rf ( ), f () b follows: Using he values, he equaion () and () become respecively as di () ( r )( a b), 0 () d Wih he condiions I (0) 0 and I( ) S di () ( a b), (4) d Wih he condiions I( ) S and I ( ) 0 Solving equaion () and (4), he values of I () and I () are as follows: I( ) ( r )( a b ), 0 (5) I( ) ( a b ) ( a b ), (6) In addiion, using he boundary condiion I ( ) I ( ) S, we obain ( r )( a b ) ( a b ) ( a b ) (7) By solving he equaion (7), we can obain: ar a r br( a b ) (8) br a+ b br a b ( ) b r( a b ) ( a+ b) [ br( a b )] / he oal profi per cycle consiss of he following elemens: revenue, ordering cos, holding cos, ineres payable and ineres earned he componens are evaluaed as follows: s s () Revenue per cycle: R D()d ( a b ) s( a b ) 0 () Ordering cos per cycle: A () Sock holding cos per cycle (excluding he ineres charges): In J Supply Oper Manage (IJSOM), Vol, No 55

5 Qin c h [ I 0 ( ) d+ I( ) d] (9) ch { r( a b ) ( a b )( ) a b } 6 6 (4) here are hree cases o occur in ineres earned and ineres charged for he iems kep in sock per cycle a a brm ( bm a) Case : M < <, ie M< <, as shown in Fig b S Invenory level and revenue M M < < Figure he oal ineres earned and ineres charged when Ineres earned per cycle is: si si si 6 M M e e e ( M ) D( ) d ( M )( a b) d ( am + bm ) 0 0 (0) Ineres charged per cycle for he manufacurer from he supplier is: ci p c [ I( ) d+ I( ) d] M () ci p c r r [ b a ( a b ) + a + b ( r )( am bm )] 6 6 Case : < M < ie a a brm ( bm a) M< <, as shown in Fig b In J Supply Oper Manage (IJSOM), Vol, No 56

6 An EPQ Model wih Increasing Demand and Demand Dependen Producion Rae under rade Credi Financing S Invenory level and revenue M Figure he oal ineres earned and ineres charged when < M < Ineres earned per cycle is: si e M sie ( M ) D( ) d ( am + bm ) () 0 6 Ineres charged per cycle for he manufacurer from he supplier is: c I c I 6 p c p c Q ( ) d [( a b )( M ) a( M ) b( M )] M () Case : M, as shown in Figure S Invenory level and revenue M Figure he oal ineres earned and ineres charged when M In J Supply Oper Manage (IJSOM), Vol, No 57

7 Qin Ineres earned per cycle due o he credi balance is: si si e [ ( ) D( ) d K( )( M ) d] 0 0 e (4) [ a b r( M )( a b)] 6 Ineres charged per cycle is zero From he above argumens, he oal profi per cycle for he manufacurer can be expressed as: ( ) Revenue Ordering cos sock holding cos ineres payable + ineres earned ( ); if m ( ) ( ); if M ( ); if 0< M m (5) Wih A ch + cpic r r ( ) s( a b ) [ b a + a + b ] (6) cpic( r ) sie ( am bm ) 6 A ch r r ( ) s( a b ) [ b a + a + b ] si ci 6 6 e p c ( am + bm ) [( a b )( M ) a( M ) b( M )] (7) A ch r r ( ) s( a b ) [ b a + a + b ] (8) sie [ a b r( M )( a b)] 6 m a a brm ( bm a) (9) b 4 he opimal replenishmen policy Mos srikingly, in order o opimize he soluions o Eq (6-8), i is necessary o sudy ( ) ( i,,), respecively i In J Supply Oper Manage (IJSOM), Vol, No 58

8 An EPQ Model wih Increasing Demand and Demand Dependen Producion Rae under rade Credi Financing Case M < < o find he opimal replenishmen policy, ake he firs-order parial derivaive of (N,) wih respec o We can obain (N,) Where f () (0) r r f() sb A ( ch + cpic)[ b a a b ( rb ra)] () 6 ( cpic( r ) sie)( am bm ) I is easy o obain f () sb ( ch + cpic)[ a b ( rb ra ) ( ) ( rb ra)] () aking he second-order parial derivaive of () wih respec o, we can obain f ( )+ f ( )= ( f ( )- f ( )) If is he roo of 0 (his may or may no exis) and f ( ) 0, hen f ( )<0 And his corresponds o a maximum value of We can obain lim f() 0 lim f() A ( cpic( r ) sie)( am bm ) If lim f() 0, hen exiss for [0, ) If is feasible, ie m, he opimal replenishmen cycle corresponds o max{ ( ), ( m)} On he oher hand, if does no exis or is infeasible, hen he opimal replenishmen cycle corresponds o max ( m) he opimal order quaniy is Q a / b herefore, based on he analysis above, i is easy o obain he heorem o decide he opimal replenishmen cycle for Case heorem () If m, he opimal replenishmen cycle corresponds o max{ ( ), ( m)} In J Supply Oper Manage (IJSOM), Vol, No 59

9 Qin () If does no exis or is infeasible, hen he opimal replenishmen cycle corresponds o max ( ) m Case < M < he firs derivaive of ( ) wih respec o is ( ) f( ) () Where r r f( ) sb A ch[ b a a b ( rb ra)] (4) 6 ( sie + cpic )( am bm ) cpic[ bm + a b ] I is easy o obain f() sb ch[ a b ( rb ra ) ( ) ( rb ra)] (5) c I bm a b p c[ + ] aking he second-order parial derivaive of () wih respec o, we can obain f ( )+ f ( )= ( f ( )- f ( )) If is he roo of 0 (his may or may no exis) and f ( ) 0, hen f ( )<0 And his corresponds o a maximum value of herefore, if is feasible, ie m M, he opimal replenishmen cycle corresponds o max{ ( ), ( m), (M )} On he oher hand, if does no exis or is infeasible, hen he opimal replenishmen cycle corresponds o max{ ( m), (M)} he opimal order quaniy is Q a / b herefore, based on he analysis above, i is easy o obain he heorem o decide he opimal replenishmen cycle for Case heorem () If m M exis, he opimal replenishmen cycle corresponds o max{ ( ), ( ), (M )} m In J Supply Oper Manage (IJSOM), Vol, No 540

10 An EPQ Model wih Increasing Demand and Demand Dependen Producion Rae under rade Credi Financing () If does no exis or is infeasible, hen he opimal replenishmen cycle corresponds o max{ ( ), (M)} m Case M he firs derivaive of ( ) wih respec o is ( ) f( ) (6) Where r r f( ) sb A ch[ b a a b ( rb ra)] sie( a bm b ) (7) I is easy o obain f () sb ch[ a b ( rb ra) ( ) ( rb ra)] (8) si a bm b e( ) aking he second-order parial derivaive of () wih respec o, we can obain f ( )+ f ( )= ( f ( )- f ( )) If is he roo of 0 (his may or may no exis) and f ( ) 0, hen f ( )<0 And his corresponds o a maximum value of I is easy o obain lim 0 f( ) A 0, lim f( ) 0, herefore, exiss for [0, ) If is feasible, ie 0 M for Eq (9), he opimal replenishmen cycle corresponds o max{ ( ), (0), (M )} On he oher hand, if is infeasible, hen he opimal replenishmen cycle corresponds o max{ (0), (M)} he opimal order quaniy is Q / a b Based on he analysis, i is easy o obain he heorem o decide he opimal replenishmen cycle for Case heorem () exiss for [0, ) In J Supply Oper Manage (IJSOM), Vol, No 54

11 Qin () If 0 M, he opimal replenishmen cycle corresponds o max{ ( ), (0), (M )} () If M, hen he opimal replenishmen cycle corresponds o max{ (0), (M)} Finally, combining he hree cases above, we can obain he Algorihm for deermining he opimal replenishmen policy in he following algorihm Algorihm SepInpu he values for all parameers Sep Find he global maximum of ( ), says, as follows: Sep compue, if m and f ( ) 0, find he max{ ( ), ( m)} And accordingly se Compue Q and ( ) Else, go o sep Sep find he max ( m) and accordingly se Compue Q and ( ) Sep Find he global maximum of ( ), says, as follows: Sep compue, if M m and f ( ) 0, find he max{ ( ), ( m), (M )} and accordingly se Compue Q andrc( ) Oherwise, go o sep Sep find he max{ ( m), (M)} and accordingly se Compue Q and ( ) Sep4 Find he global minimum of ( ), says, as follows: Sep4 compue, if 0 M, and f ( ) 0, find he max{ ( ), (0), (M )} and accordingly se Compue Q and ( ) Oherwise, go o sep 4 Sep 4 find he max{ (0), (M)} and accordingly se Compue Q and ( ) * * Sep5 Find he max{ ( ), ( ), ( )} and accordingly selec he opimal value for and Sop 5 Numerical analysis he purposes of he numerical analysis are as follows: give he numerical examples o obain he opimal soluions of he profi funcions for he manufacurer Carry ou he sensiiviy analysis o highligh he influence of he parameers associaed wih he model on he opimal decisions he sensiiviy analysis is shown in able In J Supply Oper Manage (IJSOM), Vol, No 54

12 An EPQ Model wih Increasing Demand and Demand Dependen Producion Rae under rade Credi Financing Example he inpu parameers are: A $0 per order, a 00 unis, r, s $ 0 per uni, c $5 uni per year, c p $0 per uni, I 009, I 04, M 05 e c year, and b=5 Using he Algorihm, we can calculae ( ) 976 wih 0988, ( ) 98 wih 0988, ( ) 54 wih 088 herefore, he opimal order cycle for he * * manufacurer is 0988 year and he opimal order quaniy is Q 0 unis he maximum profi is * $98 Example he inpu parameers are he same as example, excep M 0year Using he Algorihm, we can calculae ( ) 949 wih 000, ( ) 970 wih 0995, ( ) 905 * wih 0 herefore, he opimal order cycle for he manufacurer is 000 year and he opimal order quaniy is * Q * 05unis he maximum profi is $949 For he sensiiviy analysis, he basic parameers are he same as example he following managemen insighs are consisen wih he expecaions ) As he ordering cos increases, he replenishmen cycle ime and he order quaniy per cycle significanly increase, while he profi decreases hese analyical resuls sugges ha he manufacurer needs o order more quaniy o reduce he number of orders if he ordering cos is more cosly ) If he iniial demand increases, he replenishmen cycle ime per cycle decreases, while he order quaniy and he profi increase I implies ha when he iniial marke demand is greaer, he reailer can make more profis ) he larger he value of s,he more he values of he opimal cycle ime, he order quaniy and he profi increase ha is, when he uni selling price increases, he manufacurer can make more profi 4) When he holding cos and he purchasing cos increases, i is obained he opimal cycle lengh, he order quaniy and he opimal oal profi per cycle decrease herefore, i is reasonable ha when he relevan cos increases, he manufacurer will reduce he cycle ime and order quaniy o lower he on-hand invenory level or he purchasing cos o cu down he cos I means he reailer can adop some measuremens o reduce he holding cos or he purchasing cos o make more profi 5) he larger he value of M, he more he value of he opimal cycle ime and he value of he profi increase ha is, when he manufacurer s rade credi period offered by he supplier increases, he manufacurer can use he rade credi period o accumulae more ineres 6) As he rae of producion dependen on demand increases, he opimal replenishmen ime and he order quaniy firs decrease and hen increase, while he profi decreases ha is, when he dependence beween he producion and he demand is weak, he manufacurer will order fewer producs Bu if he dependence beween he producion and he demand is srong, he manufacurer will improve he order quaniy o mee he demand 7) When he demand rae increases, he opimal replenishmen cycle, he opimal order quaniy and he opimal profi increase his implies ha when he marke demand becomes larger fas, he manufacurer will improve he replenishmen quaniy o mee he marke s demand and is profi is improved h per In J Supply Oper Manage (IJSOM), Vol, No 54

13 Qin able he sensiiviy analysis of parameers parameers * Q * * A a s c p c h I e I c M r b In J Supply Oper Manage (IJSOM), Vol, No 544

14 An EPQ Model wih Increasing Demand and Demand Dependen Producion Rae under rade Credi Financing 6 Conclusion Mos of he exising invenory models under rade credi financing are assumed a consan demand and he infinie or a consan replenishmen rae However, in pracice, he demand rae is an increasing funcion of ime in he growh sage from he perspecive of he producion life cycle On he oher hand, he producion rae is relaed o he marke demand in some siuaions When he marke is beer, he producion rae will be improved herefore, in his paper, an EPQ model is buil under rade credi financing wih increasing demand and he demand dependen producion rae Subsequenly, he opimal replenishmen cycle and he opimal order quaniy are discussed, and he Algorihm is proposed o obain he opimal soluions of he model Finally, he numerical analysis is demonsraed o illusrae he heorems and he sensiiviy analysis is carried ou o give some managemen insighs he research presened in his paper can be exended in several ways For example, he ramp demand or he quadraic demand can be furher discussed considering he demand dependen producion rae in he EPQ invenory model wih he rade credi financing he deerioraion rae can be inroduced for he iems Addiionally, he models can be generalized o consider he shorage or he parial backlogging Furhermore, he influence of he poor qualiy producs on he EPQ model can be discussed o obain some managemen insighs Acknowledgemen he paper is funded by he Naional Naural Science Foundaion of China (7708, 705), Young Scholar Program of ianjin Universiy of Finance& Economics, and ianjin Universiy Innovaion eam raining Program (D-505) References Chang H J, Dye C Y (00) An invenory model for deerioraing iems wih parial backlogging and permissible delay in paymens Inernaional Journal of Sysems Science, (), 45-5 Chu P, Chang K H, Lan S P(998) Economic order quaniy of deerioraing iems under permissible delay in paymens Compuer and Operaions Research, 5(0), Chung K J, Huang Y F(00) he opimal cycle ime for EPQ invenory model under permissible delay in paymens Inernaional Journal of Producion Economics, 84(), 07-8 Chung K J, Liao J J (0)he simplified soluion algorihm for an inegraed supplier buyer invenory model wih wo-par rade credi in a supply chain sysem European Journal of Operaional Research, (), Chung Kun-Jen (00) he complee proof on he opimal ordering policy under cash discoun and rade credi Inernaional Journal of Sysems Science, 4 (4), Goyal S K(985) Economic order quaniy under condiions of permissible delay in paymens [J] Journal of he Operaional Research Sociey, 6(4), 5-8 Huang Y F (00) Opimal reailer s ordering policies in he EOQ model under rade credi financing [J] Journal of he Operaional research sociey, 54(9), 0-05 In J Supply Oper Manage (IJSOM), Vol, No 545

15 Qin HuangY F(007) Opimal reailer s replenishmen decisions in he EPQ model under wo levels of rade credi policy European Journal of Operaional Research, 76(), Jamal A M, Sarker B R,Wang S(997) An ordering policy for deerioraing iems wih allowable shorage and permissible delay in paymen Journal of he Operaional Research Sociey, 48(8), 86-8 Kreng V B, an S J (00) Opimal replenishmen decision in an EPQ model wih defecive iems under supply chain rade credi policy Exper Sysems wih Applicaions, 8(8), Liao Jui-Jung, Huang Kuo-Nan, Chung Kun-Jen (0) Lo-sizing decisions for deerioraing iems wih wo warehouses under an order-size-dependen rade credi Inernaional Journal of Producion Economics, 7(), 0-5 Manna S K, Chaudhuri K S (006) An EOQ model wih ramp ype demand rae, ime dependen deerioraion rae, uni producion cos and shorages European Journal of Operaional Research, 7(), Min Jie, Zhou Yong-Wu, Liu Gui-Qing(0) An EPQ model for deerioraing iems wih invenory-level-dependen demand and permissible delay in paymens Inernaional Journal of Sysems Science, 4(6),09-05 Qiu Hao, Liang Liang, Yu Yu-gang, Du Shao-fu(007) EOQ model under hree Levels of ordersize-dependen delay in paymens Journal of Sysems & Managemen, 6(6), Sarkar Biswaji (0)An EOQ model wih delay in paymens and ime varying deerioraion rae Mahemaical and Compuer Modelling, 545(-4), Skouri K, I Konsanaras, Manna S K (0)Chaudhuri Invenory models wih ramp ype demand rae, ime dependen deerioraion rae, uni producion cos and shorages Annals of Operaions Research, 9(), 7-95 Soni H N, Pael K A (0) Opimal sraegy for an inegraed invenory sysem involving variable producion and defecive iems under reailer parial rade credi policy [J] Decision Suppor Sysems, 54(),5-47 Soni N Hardik (0)Opimal replenishmen policies for non-insananeous deerioraing iems wih price and sock sensiive demand under permissible delay in paymen Inernaional Journal of Producion Economics, 46,59-68 eng J, Min J, Pan Q H (0) Economic order quaniy model wih rade credi financing for non-decreasing demand Omega, 40(), 8-5 eng J (00) On he economic order quaniy under condiions of permissible delay in paymens Journal of he Operaional Research Sociey, 5(8), In J Supply Oper Manage (IJSOM), Vol, No 546

16 An EPQ Model wih Increasing Demand and Demand Dependen Producion Rae under rade Credi Financing eng Jinn-sair, Chang Chun-ao, Chern Maw-Sheng (0)Vendor buyer invenory models wih rade credi financing under boh non-cooperaive and inegraed environmens Inernaional Journal of Sysems Science, 4(), sao Y C, Sheen G J (008) Dynamic pricing, promoion and replenishmen policies for a deerioraing iem under permissible delay in paymens Compuers & Operaions Research, 5(), Wu J, Ouyang L Y, Cárdenas-Barrón L E (04) Opimal credi period and lo size for deerioraing iems wih expiraion daes under wo-level rade credi financing [J] European Journal of Operaional Research, 7(), Zhong Y G, Zhou Y W(0) Improving he supply chains performance hrough rade credi under invenory-dependen demand and limied sorage capaciy Inernaional Journal of Producion Economics, 4(),64-70 Appendix A Noaions and assumpions Noaions A: ordering cos per order ch: he invenory holding cos per uni per uni ime (excluding ineres charges) cp: he purchasing cos per uni Ie: he ineres rae earned per uni per uni of ime Ic: he ineres rae charged per uni per uni of ime M: he credi period offered by he supplier I(): he invenory level a ime [0, ] : he ime a which he invenory level increases o maximum value (a decision variable) S: he maximum invenory level a each scheduling period (cycle) : he replenishmen cycle ime Q: he order quaniy s : he uni selling price for he manufacurer () : he oal relevan profi per cycle Assumpions () he lead ime is zero Shorage is no allowed () During he growh sage of a produc life cycle especially for he high- ech produc, he marke demand for he iem is assumed o be ime dependen and is defined as R D() a b a any ime wih 0, a 0, b 0 () he producion rae for he manufacurer is K( ) rd( ) where r( ) is a consan (4) he manufacurer would sele he accoun a M and pay for he ineres charges on iems in sock a he rae Ic over he inerval [M, ] as M Alernaively, he manufacurer seles he accoun a he M and is no required o pay any ineres charges for iems for he sock during he whole cycle as M Before he selemen of he accoun, he manufacurer can accumulae he sales revenue o earn he ineres up o he end of he period M a he rae Ie In J Supply Oper Manage (IJSOM), Vol, No 547