Discree Dynamics in Naure and Sociey, Aricle ID 839418, 18 pages hp://dx.doi.org/1.1155/214/839418 Research Aricle he Opimal Replenishmen Policy under rade Credi Financing wih Ramp ype Demand and Demand Dependen Producion Rae Juanjuan Qin 1 and Weihua Liu 2 1 Business School, ianjin Universiy of Finance and Economics, ianjin 3222, China 2 College of anagemen and Economics, ianjin Universiy, ianjin 3222, China Correspondence should be addressed o Juanjuan Qin; jufeqin@163.com Received 28 February 214; Acceped 12 ay 214; Published 19 June 214 Academic Edior: Xiang Li Copyrigh 214 J. Qin and W. Liu. his is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. his paper invesigaes he opimal replenishmen policy for he reailer wih he ramp ype demand and demand dependen producion rae involving he rade credi financing, which is no repored in he lieraures. Firs, he wo invenory models are developed under he above siuaion. Second, he algorihms are given o opimize he replenishmen cycle ime and he order quaniy for he reailer. Finally, he numerical examples are carried ou o illusrae he opimal soluions and he sensiiviy analysis is performed. he resuls show ha if he value of producion rae is small, he reailer will lower he frequency of puing he orders o cu down he order cos; if he producion rae is high, he demand dependen producion rae has no effec on he opimal decisions. When he rade credi is less han he growh sage ime, he reailer will shoren he replenishmen cycle; when i is larger han he breakpoin of he demand, wihin he mauriy sage of he producs, he rade credi has no effec on he opimal order cycle and he opimal order quaniy. 1. Inroducion Along wih he globalizaion of he marke and increasing compeiion, he enerprises always ake rade credi financing policy o promoe sales, increase he marke share, and reduce he curren invenory levels. As a resul, he rade credi financing play an imporan role in business as a source of funds afer Banks or oher financial insiuions. In he radiional invenory economic order quaniy (EOQ) model, i is assumed ha he reailer mus pay for he producs when receiving hem. In pracice he suppliers ofen provide he delayedpaymenimeforhepaymenofheamounowed. Usually, here is no ineres charged for he reailer if he ousanding amoun is paid in he allowable delay. However, if he paymen is unpaid in full by he end of he permissible delay period, ineres is charged on he ousanding amoun. he opimal invenory policy is also influenced by he marke demand and he producion rae wih he rade credi. For he marke demand, i is always changing fas and influenced by many facors such as he price [1, 2], he invenory level [3, 4], and he sage of he produc life ime [5, 6]. For he rapid developmen of he echnology and he quickly changing consumer ases, he producs are likely o face he problem of shor life cycle, which in reurn forces he suppliers o frequenly promoe new producs for a compeiion purpose. A demand model abou produc inroducion is widely used wih he ramp ype funcion of ime, which assumes ha he demand is a linear growh wih respec o ime (growh sage) and, up o a cerain ime, reaches a climax and evenually remains o be he peak sales (mauriy sage). In addiion o he new producs inroducion, holiday-relaed producs, such as hose for Chrismas season, are experimenally shown o fi his ype of demand rae. On he oher hand, when he marke demand is beer, he supplier will provide a higher producion rae; if he marke demand is shrinking, he supplier will reduce he producion rae. herefore, he producion rae is demand dependen producion rae. he opimal invenory policy is no only relaed o he marke demand bu also influenced by he producion rae.
2 Discree Dynamics in Naure and Sociey As a resul, in-deph research is required on he invenory replenishmen decisions. herefore, his sudy exends he EOQmodelsinheseveralwaysasFirs,rade credi financing is inroduced o he radiional EOQ models. he reailer is offered by he supplier wih a delayed paymen ime. Second, he ramp ype demand is inroduced o he EOQ models wih he rade credi financing. he demand is nearly a consan in he mauriy sage. During he growh ime, he demand of he producs is increasing wih ime. Furhermore, he producion rae dependen on demand is inroduced o he EOQ models wih rade credi financing considering he ramp ype demand for he firs ime. 2. Lieraure Review 2.1. rade Credi. he rade credi is sudied by many scholars from he aspecs of finance, accouning, and operaions managemen. Finance and accouning research mainly focused on he sudy of rade credi enerprise s cash flow, discussing henaureofheradecrediandisimpac.inheareaof operaion managemen research, mainly from he angle of cash flow and logisics coordinaion, based on he radiional economic order quaniy model framework, weigh he cos of capial cos elemens such as fixed ordering cos, sorage cos, and profi (or oher elemens), o analyze he supply chain invenory conrol and coordinaion problems. Our research is mainly fel in he operaion managemen flow. he invenory replenishmen policies under rade credi financing have been sudied inensively. os papers discussed he EOQ or EPQ invenory models under rade credi financing all based on he assumpions ha he demand rae is a consan and he infinie producion rae. he EOQ model wih he rade credi financing is pu forward for he firs ime by Goyal [7], which is exended by Chu e al. [8] considering he deerioraed producs. he models wih he exponenial deerioraion rae of he iems are discussed by Aggarwal and Jaggi [9]. he order sraegies are sudied by Jamal e al. [1] and Chang and Dye [11]allowingheshorage under delayed paymen and deerioraing condiions. he uni selling price is no he same as he uni produc coss in he model of eng [12]. he economic producion quaniy model is researched by Chung and Huang [13] considering he manufacurer offering he reailer he delayed paymen policy. Uhayakumar and Parvahi [14] analyze he supply chain of a single vendor and a single buyer for a single produc, aking ino consideraion he effec of deerioraion and credi period incenives wih a consan demand and he infinie replenishmen rae. In addiion, some auhors considered he wo levels of he delayed paymen. Huang [15] firs exended Goyal s model o analyze he wo levels of he rade credi policy, he manufacurer s credi period available o reailers as, andhereailersprovidehecusomersa credi period N, >N. And hen, Hao e al. [16], Huang [17], and eng and Chang [18] exend he models of Huang [15]. Recenly, Du e al. [19] invesigae he coordinaion of wo-echelon supply chains using wholesale price discoun and credi opion. Wu e al. [2] research he opimal credi period and lo size for deerioraing iems wih expiraion daes under wo-level rade credi financing. 2.2. Ramp ype Demand Rae. A number of papers in published lieraure have exensively sudied invenory problem by assuming his ramp ype demand rae. Under he assumpion, Hwang and Shinn [21] proposed he opimal replenishmen policy for perishable seasonal producs in a season wih ramp ype ime dependen demand. Panda e al. [22] discussed he invenory models wih ramp ype demand rae, parial backlogging, and Weibull deerioraion rae. Skouri e al. [23] presened an economic producion quaniy models for deerioraing iems wih ramp ype demand. anna and Chaudhuri [24] inroduced an EOQ invenory model for Weibull disribued deerioraing iems under ramp ype demand and shorages. andal [25] analyzed he supply chain model wih sochasic lead ime under imprecise parially backlogging and fuzzy ramp-ype demand for expiring iems.s. R. Singh and C.Singh [26] invesigaed he invenory model for ramp ype demand, ime dependen deerioraing iems wih salvage value, and shorages. ishra and Singh [27] proposed wo-warehouse invenory model wih ramp-ype demand and parially backlogged shorages. Agrawal and Banerjee [28] discussed a finie ime horizon EOQ model wih ramp ype demand rae under inflaion and ime discouning wih he infinie replenishmen rae. Roy and Chaudhuri [29] sudied he compuaional approach o an invenory model wih ramp ype demand and linear deerioraion. Recenly, Agrawal e al. [3] analyze a wowarehouse invenory model, wih he demand rae being a general ramp ype funcion of ime and he shorages are parially backlogged a a consan rae. Ahmed e al. [31] focus on he invenory model wih he ramp ype demand rae, he parial backlogging, and general deerioraion rae. Saha [32] discusses he opimal order quaniy of he reailer wih quadraic ramp ype demand under supplier rade credi financing. However, he problems of paymen delay linked o he rampypedemandraehavenoreceivedmuchaenion. he mos relaed lieraures o our sudy are as ishra and Singh [33] discussed he opimal order policy for single period producs under paymen delay wih ramp ype demand rae; Huang e al. [34] made some effors o build an invenory sysem for deerioraing producs, wih ramp ype demand rae, under wo-level rade credi policy considering he shorages and he parially backlogged unsaisfied demand. Bu hey do no consider he demand dependen producion rae. 2.3. Demand Dependen Producion Rae. For he rade credi, he researchers aemping o solve he problems mosly assume ha he producion rae is infinie or a consan value. 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 supplier will provide a higher producion rae; if he marke demand is shrinking, he supplier will reduce he producion rae. In he radiional EOQ or EPQ models wihou he
Discree Dynamics in Naure and Sociey 3 rade credi financing, Darzanou and Skouri [35] presened a producion invenory sysem for deerioraing iems wih demand rae being a linearly ramp ype funcion of ime andproducionraebeingproporionalohedemandrae. 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. anna and Chiang [36] exend his model by considering ha (a) for he model wih no shorageshedemandraeissabilizedaferheproducion sopping ime and (b) for he model wih shorages he demandraeissabilizedaferheproducionsoppingime or afer he ime when he invenory level reaches zero or afer he producion resaring ime. Recenly, Skouri e al. [37] developed a more general inegraed supplier reailer invenorymodelwihademandraewhichissensiiveohe reailing price and a demand dependen producion rae wih wo-level of rade credi financing, which did no consider he ramp ype demand rae. herefore, based on he lieraures above, we find ha none of he above models explore he opimal replenishmen policies of he reailer under rade credi financing wih he ramp ype demand and demand dependen producion rae. Given he analysis above, his paper developed invenory models under rade credi financing wih ramp ype demand and demand dependen producion rae. In his invenory sysem, hey migh aribue o inricae correlaions among he period o mauriy sage,heperiodofdelayedpaymen, he producion ime 1, and he planning ime inerval. We assume ha he break poin of he demand funcion is less han. Oherwise, he models in he siuaion of he ramp ype demand are he same as he models wih he linear increasing demand funcion eng e al. [6], which is no he subjecive of his research, and wihou doub 1 <holds. Sill, here exis eigh possible subcases discussed in his paper. he remainder of his paper is organized as he noaions and assumpions are inroduced in Secion 3. he invenory levels for 1 and 1 are presened in Secion 4. odel formulaions are developed where we have wo invenory models and including eigh subcases in Secion 5,forwhichhealgorihmsareprovided o obain he opimal decisions. he numerical examples and sensiiviy analysis hen are conduced in Secion 6 in associaion wih managerial insighs. Finally, Secion 7 closes he paper wih a conclusion. 3. Noaions and Assumpions o build he mahemaical models, he following noaions andassumpionsareadopedinhispaper. 3.1. Noaions A: Ordering cos per order 1 :heimeawhichheinvenorylevelisamaximum for he siuaion 1 ; 1 : he ime a which he invenory level is a maximum for he siuaion 1 S: he maximum invenory level a each scheduling period (cycle) c h : he invenory holding cos per uni per uni ime (excluding ineres charges) c p : he purchasing cos per uni s:heunisaleprice k:heproducionraewihk=rf() R: he marke demand wih R=f() :hebreakpoinofhedemandfuncion D :heraeofdemandwihheimeduringhe demand increasing sage I i () (i = 1, 2, 3): he invenory level a ime [,] I e : he ineres rae earned per uni per uni of ime I c : he ineres rae charged per uni per uni of ime :heofferedcrediperiodbyhesupplier : he replenishmen cycle ime RC ():heannualoalrelevancos. 3.2. Assumpions. he following assumpions are used hroughou in his paper. (1) here is a single supplier and single reailer and hey deal wih a single produc. (2) he lead ime is zero. he planning horizon is infinie. he shorage is no allowed. (3) he iniial and final invenory levels are boh zero. (4)hemarkedemandforheiemisassumedobe a ramp ype funcion of ime. A he firs sage, he demand is increasing wih he ime, such as he inroducion sage of he new producs; bu as he ime increases, he demand will keep a consan, such as he mauriy sage of he new produc. ha is, R= f() = D [ ( )H( )] a any ime,where D and are posiive consans and H( ) is he Heaviside s funcion defined as follows: H( )={ 1, if, if <. (5) For he producion rae, when he marke demand is high, he producion will be improved; bu when he marke demand is low, he producion rae will be down. herefore, he producion rae is relaed o he demand closely. In his paper, he producion rae is assumed k = rf() [38], where r(>1) is a consan. In he producion period, when he producs have been produced, hey will be delivered o he reailer insananeously. ha is, he replenishmen rae of he reailer is k. (6) he reailer would sele he accoun a =and pay for he ineres charges on iems in sock a he rae I c over he inerval [, ] as.alernaively, he reailer seles he accoun a = and is no (1)
4 Discree Dynamics in Naure and Sociey Invenory level S Hence, he variaion of he invenory level I() wih respec o he ime can be described by he following differenial equaions. During he growh sage in he inerval [, ], he demand rae is D and he replenishmen rae is rd. herefore, he invenory level a ime is governed by di 1 () = (r 1) D d,, (2) wih he boundary condiions I 1 () =. Similarly, during he mauriy sage [, 1 ], he demand rae is D and he replenishmen rae is rd. herefore, he invenory level a ime is governed by 1 Figure 1: he invenory model for 1. di 2 () = (r 1) D d, 1, (3) wih he boundary condiions I 2 ( 1 )=S. Finally, a he mauriy sage [ 1,], he demand rae is D and he producion sops. herefore, he invenory level a ime is governed by Invenory level S 1 di 3 () = D d, 1, (4) wih he boundary condiions I 3 ( 1 )=Sand I 3 () =. Solving (2) (4), he values of I i () (i = 1, 2, 3) are as follows: Figure 2: he invenory model for 1. required o pay any ineres charges for iems for he sock during he whole cycle as.beforehe selemen of he accoun, he reailer can use sales revenueoearnheineresupoheendofheperiod a he rae I e. (7) he break poin of he demand funcion is less han. (8) For he invenory sysem, we analyze i in he Secion 3 for wo siuaions: 1 and 1. Given he above, i is possible o build he mahemaical invenory EOQ model wih he rade credi financing. 4. he Invenory Level 4.1. Invenory Level for 1. Basedonheabovenoaions and assumpions, he invenory sysem can be considered in hefollowing.ahebeginning,hesockleveliszero.he producion sars wih zero sock level a ime =,and he producion sops a ime = 1. hen he invenory level gradually diminishes due o demand, ill i becomes zero a =.hewholeprocessisrepeaedandhebehaviourof he invenory sysem is described in Figure 1. Based on he Figure 1, wecanknowhaheinvenorycycleherehashe hree sages. I 1 () = (r 1) 1 2 D 2,, I 2 () = (r 1) D (r 1) 1 2 D 2, 1, I 3 () =D ( ), 1. In addiion, using he boundary condiion I 2 ( 1 ) = I 3 ( 1 )=S,weobain (5) (r 1) D 1 (r 1) 1 2 D 2 =D ( 1 ). (6) Solving (6), we can obain 1 = +(1/2)(r 1). (7) r 4.2. Invenory Level for 1. he invenory sysem for 1 considered in he following is he same as ha in he siuaions of 1. he behaviour of he invenory sysem is described in Figure 2. Based on he Figure 2, we can know ha he invenory cycle here sill has he hree sages. he variaion of he invenory level I() wih respec o he ime can be described by he following differenial equaions. During he growh sage in he inerval [, 1 ], he demand rae is D and he replenishmen rae is rd. herefore, he invenory level a ime is governed by di 1 () = (r 1) D d, 1, (8) wih he boundary condiions I 1 () = and I 1 ( 1 )=S.
Discree Dynamics in Naure and Sociey 5 Similarly, during he mauriy sage [ 1,], he demand rae is D and he replenishmen rae is zero. herefore, he invenory level a ime is governed by di 2 () = D d, 1, (9) wih he boundary condiions I 2 ( 1 )=S. Finally, a he mauriy sage [, ], hedemandraeis D andhereisnoproducion.herefore,heinvenorylevel a ime is governed by di 3 () = D d,, (1) wih he boundary condiions I 3 () =. Solving (2) (4), he values of I i () (i = 1, 2, 3) are as follows: I 1 () = (r 1) 1 2 D 2, 1, I 2 () = 1 2 D 2 + 1 2 D 2 +D ( ), I 3 () =D ( ),. 1, (11) In addiion, using he boundary condiion I 1 ( 1 ) = I 2 ( 1 )=S,weobain (r 1) 1 2 D 2 1 = 1 2 D 2 1 + 1 2 D 2 +D ( ). (12) For 1, he annual sock holding cos is c h [ 1 I 1 () d + I 2 () d + 1 = c h [1 6 rd 3 1 1 6 D 3 I 3 () d] + 1 2 D 2 1 D 1 + 1 2 D 2 ]. (15) (3) here are eigh cases o occur in ineres earned and ineres charged for he iems kep in sock per year. From he value of, wehavewopossiblesiuaions: and. he differen mahemaical formulaions are discussed in he following according o he wo possible siuaions including eigh cases. 5.1. odel 1: he Invenory odels for he Case 5.1.1. Case 1 Case 1.1 ( 1, shown in Figure 3). he annual ineres charged will be paid for he invenory no being sold afer he due dae. Duringheime[, ], heinvenory level is I 2 (); forheime[, ], heinvenorylevelisi 3 (). herefore, he annual ineres charged for he invenory no being sold afer he due dae is given by c p I c [ I 2 () d + I 3 () d] Solving (12), we can obain = c pi c [ 1 6 D 3 + 1 6 D 3 (16) 1 = 2 + 2. (13) r 5. ahemaical odels and Soluions he annual oal relevan cos consiss of he following elemens: ordering cos, holding cos, ineres payable, and ineres earned. he componens are evaluaed as in he following. (1) Annual ordering cos: A/. (2) Annual sock holding cos (excluding he ineres charges). For 1, he annual sock holding cos is c h [ 1 I 1 () d + I 2 () d + I 3 () d] 1 = c h [(r 1) 1 6 D 3 + 1 2 rd 2 1 (r 1) 1 2 D 2 1 + 1 2 D 2 D 1 ]. (14) + 1 2 D 2 D + 1 2 D 2 ]. During he ime [, ], he reailer can use he sales o accumulae he ineres. herefore, he annual ineres earned during he rade credi period is shown in he following: si e [ ( ) D d] = si e 6 D 3. (17) Case 1.2 ( 1, shown in Figure 4). he annual ineres charged will be paid for he invenory no being sold afer he due dae. Duringheime[, 1 ],heinvenory level is I 1 (); forheime[ 1,],heinvenorylevelisI 2 (); for he ime [, ],heinvenorylevelisi 3 (). herefore, he annual ineres charged for he invenory no being sold afer he due dae is given by c p I c [ 1 I 1 () d + I 2 () d + I 3 () d] 1 = c pi c [1 6 rd 3 1 1 6 D 3 + 1 2 D 2 1 D 1 + 1 2 D 2 (r 1) 1 6 D 3 ]. (18)
6 Discree Dynamics in Naure and Sociey S Invenory level Revenue 1 1 (a) Ineres charged for 1 (b) Ineres earned for 1 Figure 3: Ineres charged and ineres earned for 1. S Invenory level Revenue 1 1 (a) Ineres charged for 1 (b) Ineres earned for 1 Figure 4: Ineres charged and ineres earned for 1. During he ime [, ], he reailer can use he sales o accumulae he ineres. he annual ineres earned during he rade credi period is presened in he following: si e [ ( ) D d] = si e 6 D 3. (19) Case 1.3 ( 1, shown in Figure 5). he annual ineres charged will be paid for he invenory no being sold afer he due dae. Duringheime[, ], heinvenory level is I 1 ();forheime[, 1 ],heinvenorylevelisi 2 ();for he ime [ 1,],heinvenorylevelisI 3 (). he annual ineres charged for he invenory no being sold afer he due dae is shown by c p I c [ 1 I 1 () d + I 2 () d + I 3 () d] 1 = c pi c [(r 1) 1 6 D 3 + 1 2 D 2 1 + 1 2 D 2 D 1 1 6 (r 1) D 3 ]. (2) For he inerval [, ], he reailer can use he sales o accumulae he ineres. he annual ineres earned during he rade credi period is presened in he following: si e [ ( ) D d] = si e 6 D 3. (21) 5.2. he Opimal Replenishmen Policy for he odel 1. he resuls in previous Secion4.1 lead o he following oal cos funcions. (1) If ( 2 r+ 2 )/2,hais,/ r RC 1 () RC 11, 1, ha is 2 r+ 2 2 { RC 12, 1, = ha is 2 r+ 2 1 (r+1) 2 2 { RC { 13, 1, ha is 1 2 (r+1). (22)
Discree Dynamics in Naure and Sociey 7 S Invenory level Revenue 1 1 (a) Ineres charged for 1 (b) Ineres earned for 1 Figure 5: Ineres charged and ineres earned for 1. (2) If ( 2 r+ 2 )/2,hais / r RC 1 () { RC 12, 1, ha is 1 (r+1) = 2 { RC { 13, 1, ha is 1 2 (r+1), (23) where RC 11 = A + c h (1 6 rd 3 1 1 6 D 3 + 1 2 D 2 1 D 1 + 1 2 D 2 ) + c pi c ( 1 6 D 3 + 1 6 D 3 + 1 2 D 2 D + 1 2 D 2 ) si e 6 D 3 ; RC 12 = A + c h (1 6 rd 3 1 1 6 D 3 + 1 2 D 2 1 RC 13 D 1 + 1 2 D 2 ) + c pi c [1 6 rd 3 1 1 6 D 3 + 1 2 D 2 1 D 1 + 1 2 D 2 (r 1) 1 6 D 3 ] si e 6 D 3 ; = A + c h [(r 1) 1 6 D 3 + 1 2 D 2 1 + 1 2 D 2 D 1 ] + c pi c [(r 1) 1 6 D 3 + 1 2 D 2 1 + 1 2 D 2 D 1 1 6 (r 1) D 3 ] si e 6 D 3. (24) he problem is min RC 1 (). In order o obain he opimal soluions, we should sudy each of he branches and hen combine he resuls. heorem 1. (1) If α <, RC 13 () is a convex funcion of for = (, ). heopimalvalueof 13 corresponds o min{rc 13 ( 13 ), RC 13 ((1/2)(r + 1))}. (2) If α, RC 13 () is an increasing funcion wih for = (, ). For [(1/2)(r + 1), ), heopimalvalue 13 is obained when 13 = (1/2)(r + 1). I is similar o analyze he opimal soluions for oher branches as heorem 1. herefore, we can obain he soluion procedure for deermining he opimal replenishmen cycle and he order quaniy of he model 1 in he following algorihm. Algorihm 1. Sep 1. Inpu he values for all parameers wih. Sep 2.Compare/ r and.if / r,go o Algorihm 1.1; if / r, hen go o Algorihm 1.2. Algorihm 1.1 Sep 1. FindheglobalminimumofRC 11 (), says 11,as Sep 1.1.Compue 11,if 11 ( 2 r+ 2 )/2 and f 11 ( 11 ) >, find he min{rc 11 ( 11 ), RC 11 (), RC 11 (( 2 r +
8 Discree Dynamics in Naure and Sociey 2 )/2)}. And accordingly se 11. Compue Q 11 RC 11 ( 11 ).Else,gooSep1.2. and Sep 1.2.Findhemin{RC 11 (), RC 11 (( 2 r+ 2 )/2)} and accordingly se 11.CompueQ 11 and RC 11 ( 11 ). Sep 2. FindheglobalminimumofRC 12 (), says 12,as Sep 2.1. Compue 12,if( 2 r+ 2 )/2 12 (r + 1)/2 and f 12 ( 12 ) >,findhemin{rc 12 ( 12 ), RC 12 ((r + 1)/2), RC 12 (( 2 r + 2 )/2)} and accordingly se 12. Compue Q 12 and RC 12 ( 12 ). Else, go o Sep 2.2. Sep 2.2. Findhemin{RC 12 ((r + 1)/2), RC 12 (( 2 r+ 2 )/2)} and accordingly se 12. Compue Q 12 and RC 12 ( 12 ). Sep 3. FindheglobalminimumofRC 13 (), says 13,as Sep 3.1. Compue 13,if(r + 1)/2 13,andhense 13 = 13.CompueQ 13 and RC 13 ( 13 ).Else,gooSep3.2. Sep 3.2.Se 13 = (r + 1)/2.CompueQ 13 and RC 13 ( 13 ). Sep 4. Findhemin{RC 11 ( 11 ), RC 12 ( 12 ), RC 13 ( 13 )} and accordingly selec he opimal value for 1.Sop. Algorihm 1.2 Sep 1. FindheglobalminimumofRC 12 (), says 12,as Sep 1.1.Compue 12,if 12 (r+1)/2 and f 12 ( 12 )>, find he min{rc 12 ( 12 ), RC 12 ((r + 1)/2), RC 12 ()} and accordingly se 12.CompueQ 12 and RC 12 ( 12 ).Else,goo Sep 2.2. Sep 1.2. Findhemin{RC 12 ((r + 1)/2), RC 12 ()} and accordingly se 12.CompueQ 12 and RC 12 ( 12 ). Sep 2. FindheglobalminimumofRC 13 (), says 13,as Sep 2.1.Compue 13,if(r + 1)/2 13,andhense 13 = 13.CompueQ 13 and RC 13 ( 13 ). Else, go o Sep 2.2. Sep 2.2.Se 13 = (r + 1)/2.CompueQ 13 and RC 13 ( 13 ). Sep 3. Findhemin{RC 12 ( 12 ), RC 13 ( 13 )} and accordingly selec he opimal value for 1.Sop. 5.3. odel 2: he Invenory odels for he Case 5.3.1. Case 2 ( ) Case 2.1 ( 1 ). SimilaroCase1,heannual ineres charged for he invenory no being sold afer he due dae is given by c p I c I 3 () d = c pi c (1 2 D 2 D + 1 2 D 2 ). (25) he annual ineres earned during he rade credi period is shown in he following: si e [ ( ) D d+ ( ) D d] = si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ). (26) Case 2.2 ( 1 ). he annual ineres charged for he invenory no being sold afer he due dae is given by c p I c [ I 3 () d] = c pi c (1 2 D 2 D + 1 2 D 2 ). (27) he annual ineres earned during he rade credi period is shown in he following: si e [ ( ) D d+ ( ) D d] = si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ). (28) Case 2.3 ( 1 ). he annual ineres charged for he invenory no being sold afer he due dae is given by c p I c [ 1 I 2 () d + I 3 () d] 1 = c pi c ( 1 3 D 3 1 + 1 3 D 3 1 2 D 2 D +D 2 +D 2 1 2 D 2 ). (29) he annual ineres earned during he rade credi period is shown in he following: si e [ ( ) D d+ ( ) D d] = si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ). 5.3.2. Case 3: he Invenory odels for he Case < (3) Case 3.1 ( 1 <). he annual ineres charged for he invenory no being sold afer he due dae is zero. he annual ineres earned during he rade credi period is shown in he following: si e [ 1 ( ) D d+ K ()( ) d] = si e [1 2 D 2 1 3 D 3 +rd ( ) 1 2 2 1 ]. (31)
Discree Dynamics in Naure and Sociey 9 Case 3.2 ( 1 <). heannualinereschargedfor he invenory no being sold afer he due dae is zero. he annual ineres earned during he rade credi period is shown in he following: where RC 21 si e [ 1 ( ) D d+ K ()( ) d] = si e [1 2 D 2 1 3 D 3 +rd ( ) 1 2 2 +rd ( ) ( 1 )]. (32) = A + c h (1 6 rd 3 1 1 6 D 3 + 1 2 D 2 1 D 1 + 1 2 D 2 ) + c pi c (1 2 D 2 D + 1 2 D 2 ) 5.4. he Opimal Replenishmen Policy for he odel 2. Combine he invenory models of Cases 2 and 3, he resuls in previous subsecions lead o he following oal cos funcion. (1) If (1/2)(r + 1) RC 22 si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ); = A + c h [(r 1) 1 6 D 3 + 1 2 D 2 1 RC 2 () { = RC 22, { { RC 31, RC 21, RC 23, 1, ha is 1, ha is 1 (r+1) 2 1, ha is 1 2 (r+1) r 1 (r 1) 2 1, ha is r 1 2 (r 1). (33) RC 23 + 1 2 D 2 D 1 ] + c pi c (1 2 D 2 D + 1 2 D 2 ) si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ); = A + c h [(r 1) 1 6 D 3 + 1 2 D 2 1 + 1 2 D 2 D 1 ] (2) If (1/2)(r + 1) + c pi c ( 1 3 D 3 1 + 1 3 D 3 + 1 2 D 2 D + D 2 1 2 D 2 ) RC 2 () { = RC 22, { { RC 31, RC 32, RC 23, 1, ha is 1 (r+1) 2 1, ha is 1 (r+1) 2 1, ha is r 1 (r 1) 2 1, ha is r 1 2 (r 1), (34) RC 31 si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ), = A + c h (1 6 rd 3 1 1 6 D 3 + 1 2 D 2 1 RC 32 D 1 + 1 2 D 2 ) si e (1 2 D 2 1 3 D 3 +rd ( ) 1 2 2 1 ); = A + c h [(r 1) 1 6 D 3 + 1 2 D 2 1
1 Discree Dynamics in Naure and Sociey + 1 2 D 2 D 1 ] si e [1 2 D 2 1 3 D 3 +rd ( ) 1 2 2 +rd ( ) ( 1 )]. (35) 1)/2), RC 22 (r (r + 1)/2)} and accordingly se 22. Compue Q 22 and RC 22 ( 22 ).Else,gooSep3.2. Sep 3.2. Find he min{rc 22 ((r + 1)/2), RC 22 (r (r + 1)/2)} and accordingly se 22.CompueQ 22 and RC 22 ( 22 ). Sep 4. FindheglobalminimumofRC 23 (), says 23,as Sep 4.1.Compue 23,if(r + 1)/2 23,andhense 23 = 23.CompueQ 23 and RC 23 ( 23 ).Else,gooSep4.2. In order o obain he opimal replenishmen decisions, weshouldsudyeachbranchofhecosfuncions. heorem 2. (1) If β <, RC 23 () is a convex funcion of for = (, ). heopimalvalueof 23 corresponds o min{rc 23 ( 23 ), RC 23 (r (1/2)(r 1))}. (2) If β, RC 23 () is an increasing funcion wih for = (, ). For [r (1/2)(r 1), ), heopimal value 23 is obained by 23 = r (1/2)(r 1). I is similar o analyze he opimal soluions for oher branches as heorem 2. Basedonhediscussion, hesoluion procedure for deermining he opimal replenishmen cycle and he order quaniy in he model 2 wih can be summarized ino he following algorihm. Algorihm 2. Sep 1. Inpu he values for all parameers wih. Sep 2.Compare(r + 1)/2 and.if(r + 1)/2,go o Algorihm 2.1; if (r + 1)/2, hen go o Algorihm 2.2. Algorihm 2.1 Sep 1. FindheglobalminimumofRC 31 (), says 31,as following. Sep 1.1.Compue 31,if 31 and f 31 ( 31 )>,find he min{rc 31 ( 31 ), RC 31 (), RC 31 ()} and accordingly se 31.CompueQ 31 and RC 31 ( 31 ).Else,gooSep1.2. Sep 1.2.Findhemin{RC 31 (), RC 31 ()} and accordingly se 31.CompueQ 31 and RC 31 ( 31 ). Sep 2. Find he global minimum of RC 21 (), says 21,as Sep 2.1.Compue 21,if 21 (r + 1)/2 and f 21 ( 21 )>, findhemin{rc 21 ( 21 ), RC 21 (), RC 21 ((r + 1)/2)} and accordingly se 21.CompueQ 21 and RC 21 ( 21 ).Else, go o Sep 2.2. Sep 2.2. Findhemin{RC 21 (), RC 21 ((r + 1)/2)} and accordingly se 21.CompueQ 21 and RC 21 ( 21 ). Sep 3. FindheglobalminimumofRC 22 (), says 22,as Sep 3.1. Compue 22,if(r + 1)/2 22 r (r + 1)/2 and f 22 ( 22 )>,findhemin{rc 22 ( 22 ), RC 22 ((r+ Sep 4.2.Se 23 = (r + 1)/2.CompueQ 23 and RC 23 ( 23 ). Sep 5. Findhemin{RC 31 ( 21 ), RC 21 ( 21 ), RC 22 ( 22 ), RC 23 ( 23 )} and accordingly selec he opimal value for 2. Sop. Algorihm 2.2 Sep 1. FindheglobalminimumofRC 31 (), says 31,as Sep 1.1.Compue 31,if 31 (r+1)/2 and f 31 ( 31 )>, find he min{rc 31 ( 31 ), RC 31 (), RC 31 ((r + 1)/2)} and accordingly se 31.CompueQ 31 and RC 31 ( 31 ).Else,goo Sep 1.2. Sep 1.2. Findhemin{RC 31 (), RC 31 ((r + 1)/2)} and accordingly se 31.CompueQ 31 and RC 31 ( 31 ). Sep 2. FindheglobalminimumofRC 32 (), says 32,as Sep 2.1.Compue 32,if(r + 1)/2 32 and f 32 ( 32 )>, findhemin{rc 32 ( 32 ), RC 32 ((r + 1)/2), RC 32 ()} and accordingly se 32.CompueQ 32 and RC 32 ( 32 ).Else, go o Sep 2.2. Sep 2.2. Findhemin{RC 32 ((r + 1)/2), RC 32 ()} and accordingly se 32.CompueQ 32 and RC 32 ( 32 ). Sep 3. FindheglobalminimumofRC 22 (), says 22,as Sep 3.1. Compue 22, if 22 r (r + 1)/2 and f 22 ( 22 ) >, find he min{rc 22 ( 22 ), RC 22 (), RC 22 (r (r + 1)/2)} and accordingly se 22. Compue Q 22 and RC 22 ( 22 ).Else,gooSep3.2. Sep 3.2.Findhemin{RC 22 (), RC 22 (r (r + 1)/2)} and accordingly se 22.CompueQ 22 and RC 22 ( 22 ). Sep 4. FindheglobalminimumofRC 23 (), says 23,as Sep 4.1.Compue 23,if(r + 1)/2 23,andhense 23 = 23.CompueQ 23 and RC 23 ( 23 ).Else,gooSep4.2. Sep 4.2.Se 23 = (r + 1)/2.CompueQ 23 and RC 23 ( 23 ). Sep 5. Findhemin{RC 31 ( 21 ), RC 32 ( 32 ), RC 22 ( 22 ), RC 23 ( 23 )} and accordingly selec he opimal value for 2. Sop.
Discree Dynamics in Naure and Sociey 11 6. Numerical Examples and Sensiiviy Analysis In his secion, we carry ou some numerical examples o illusrae he algorihms obained in he previous secions. Addiionally, we also provide a sensiiviy analysis of he values of mos imporan parameers on he opimal decisions of he reailer and he oal cos. 6.1. Numerical Examples Example 1 (/ r ). he inpu parameers are A= $5 per order, D = 5 unis, r=2, s=$1 per uni, c h = $5 per uni per year, c p = $4 per uni, I e =.15, I c =.8, =.3 year, and =.4 year. Using Algorihm 1, wecancalculaerc 11 ( 11 ) = 351.8 wih 11 =.4, RC 12 ( 12 ) = 393.2 wih 12 =.425, and RC 13 ( 13 ) = 142.6 wih 13 =.6. herefore, he opimal order cycle for he reailer is 1 = 13 =.6 year and he opimal order quaniy is Q 1 = 8 unis. he minimum of he oal cos is RC 1 = $142.6. Example 2 ( / r). he inpu parameers are A= $5 per order, D = 5 unis, r=2, s=$1 per uni, c h = $5 per uni per year, c p = $4 per uni, I e =.15, I c =.8, =.2 year, and =.4 year. Using Algorihm 1, wecancalculaerc 12 ( 12 ) = 441.6 wih 12 =.4, andrc 13 ( 13 ) = 19.7 wih 13 =.6. herefore, he opimal order cycle for he reailer is 1 = 13 =.6 year and he opimal order quaniy is Q 1 = 8 unis. he minimum of he oal cos is RC 1 = $19.7. Example 3 ((r + 1)/2 ). he inpu parameers are A=$5 per order, D = 5 unis, r=2, s=$4 per uni, c h = $5 per uni per year, c p = $2 per uni, I e =.15, I c =.8, =.6 year, and =.4 year. Using Algorihm 2, wecancalculaerc 31 ( 31 ) = 92.3 wih 31 =.4, RC 21 ( 21 ) = 347. wih 21 =.6, RC 22 ( 22 ) = 17.2 wih 22 =.8, andrc 23 ( 23 )= 533.2 wih 23 = 1.4. herefore, he opimal order cycle for he reailer is 1 = 31 =.4 year and he opimal order quaniy is Q 1 =4unis. he minimum of he oal cos is RC 2 = $92.3. Example 4 ((r + 1)/2 ). he inpu parameers are A= $5 per order, D = 5 unis, r=2, s=$4 per uni, c h = $5 per uni per year, c p = $2 per uni, I e =.15, I c =.8, =.9 year, and =.4 year. Using Algorihm 2,wecancalculaeRC 31 ( 31 ) = 576.1 wih 31 =.6367, RC 32 ( 32 ) = 261.7 wih 32 =.8, RC 22 ( 22 ) = 416.3 wih 22 =.9, andrc 23 ( 23 )= 114.1 wih 23 = 2.3. herefore, he opimal order cycle for he reailer is 1 = 31 =.4 year and he opimal order quaniy is Q 1 = 12 unis. he minimum of he oal cos is RC 2 = $261.7. 6.2. Sensiiviy Analysis. In he model, here are many parameers influencing he decisions of he members. Bu according o he pas research, he effecs of some parameers on he decisions are analyzed in many lieraures such as he ordering cos per order. herefore, we choose hree parameers, which are direcly relaed o our innovaions, o conduc he sensiiviy analysis for obaining ineresing managemen insighs. In his subsecion, we presen he sensiiviy analysis ofhemodelsmenionedabovewihrespecohehree parameers of r,, and. 6.2.1. he Impac of he Changes of he Parameer r. Basic parameers values for he case are A =$5 per order, D = 5 unis, s=$1 per uni, c h = $5 per uni per year, c p = $4 per uni, I e =.15, I c =.8, =.3 year, and =.6 year.heimpacsofhechangesofheparameerr on he reailer decisions and cos are shown in Figure 6. Basic parameers values for he case are A=$25 per order, D = 5 unis, s=$4 per uni, c h = $5 per uni per year, c p = $2 per uni, I e =.15, I c =.8, =.7 year, and =.3 year.heimpacsofhechangesofheparameer r on he reailer decisions and is cos are shown in Figure 7. Based on Figures 6 and 7, wecanseehaifhedemand dependen producion rae is lower, as he increasing of he parameer r, he opimal order cycle and he opimal order quaniy are increasing. I means ha if he value of producion rae is small, he reailer will lower he frequency of puing he orders o cu down he order cos. If he demand dependen rae is higher, he demand dependen producion rae has no effec on he opimal order cycle and he order quaniy. For he limi case, when he producion rae is infinie, as mos papers presened, he producion rae has no influence on he members decisions. For he oal cos, as he increase of he parameer r, he oal cos is firs increasing and hen decreasing wih his parameer. I demonsraed ha when he value of producion rae is small, he on-hand invenory level is low, which resuls in low oal cos. When he demand dependen producion rae is much higher, he reailer s profi is decreasing wih his rae, because he reailer can ge he producs as soon as possible o beer mee he demand of he marke; on he oher hand, in general, here is no shorage o occur for he reailer, which will save he shorage cos for he reailer. 6.2.2. he Impac of he Changes of he Parameer. Basic parameers values are A=$25 per order, D = 5 unis, s=$4 per uni, c h = $5 per uni per year, c p = $2 per uni, I e =.15, I c =.8, r=2,and=.5year. Based on Figure 8,wecanseehaashedelayedpaymen ime offered by he suppliers is increasing ill =, he opimal order cycle and he opimal order quaniy are nonincreasing, which demonsrae ha when he delayed paymen ime is less han he growh sage ime of he
12 Discree Dynamics in Naure and Sociey 1.2 1.1 28 26 24 he opimal order cycle 1.9.8.7 he opimal order quaniy 22 2 18 16 14.6.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Demand dependen producion rae: r (a) he opimal order cycle he oal cos 7 65 6 55 5 45 4 35 3 12 1 8 2 2.5 3 3.5 4 4.5 5 5.5 6 Demand dependen producion rae: r (b) he opimal order quaniy 25 2 2.5 3 3.5 4 4.5 5 5.5 6 Demand dependen producion rae: r (c) he oal cos for he reailer Figure 6: he impac of he changes of he parameer r when. new producs inroducion, he reailer will shoren he replenishmen cycle o ake advanage of he rade credi more frequenly for accumulaing he ineres; on he oher hand, he shor order cycle can make he reailer adjus is order decisions more quickly for meeing he changeable demandwihinhegrowhsageofheproducs. When >, ha is, he delayed paymen ime is larger han he breakpoin of he demand, during he mauriy sage of he producs, he delayed paymen ime has no effec on he opimal order cycle and he opimal order quaniy. herefore, during he mauriy sage of he new producs, he reailer s replenishmen policies are no influenced by he delayed paymen ime offered by he supplier. However, he oal cos for he reailer is decreasing wih he delayed paymen ime for he produc life cycle. herefore, he reailer hopes he supplier can offer hem he delayed paymen ime as long as possible o obain more profis. 6.2.3. he Impac of he Changes of he Parameer. Basic parameers values are A=$25 per order, D = 5 unis, s=$4 per uni, c h = $5 per uni per year, c p = $2 per uni, I e =.15, I c =.8, r=2,and =.5 year. Based on Figure 9, we can see ha as he demand breakpoin is increasing; he opimal order cycle and he opimal order quaniy are decreasing a shor ime firsly and hen increasing for he las ime. ha is, when he breakpoin is large, he new producs moving o he mauriy period a slow speed, he demand is changed slowly. herefore, in
Discree Dynamics in Naure and Sociey 13.9 12 he opimal order cycle.85.8.75.7.65.6.55.5 2 3 4 5 6 7 8 9 Demand dependen producion rae: r (a) he opimal order cycle 35 he opimal order quaniy 11 1 9 8 7 6 5 2 3 4 5 6 7 8 9 Demand dependen producion rae: r (b) he opimal order quaniy 3 25 he oal cos 2 15 1 5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 Demand dependen producion rae: r (c) he oal cos for he reailer Figure 7: he impac of he changes of he parameer r when. he relaive sable marke, he reailer can increase he order quaniy and replenishmen cycle o lower he cos. he oal cos for he reailer is increasing wih he demand breakpoin. Hence, for he producs, if he growh sageofheproduclifecycleislonger,hecosofhereailer will be higher. herefore, he reailer should make some measures, such as adverisemen, o induce he producs demandmovingohemauriysageassoonaspossible. 7. Conclusions os of he exising invenory models under rade credi financing are assumed ha he demand is a consan and he replenishmen rae is infinie or a consan. However, in pracice, he demand rae is he ramp ype funcion of ime for some cases, such as he new producs and he holiday relaed producs. On he oher hand, he producion rae is relaed o he marke demand. When he marke is beer, he producion rae will be improved. When he reailer makes an order, i can be me faser. ha is, he replenishmen of he reailer is relaed o he marke demand. herefore, in his paper, we developed an EOQ model under rade credi financing wih ramp ype demand and he demand dependen producion rae. Subsequenly, he algorihmsareproposedodecideheopimalreplenishmen cycle and he opimal order quaniy for he reailer. Finally, he numerical analysis is demonsraed o illusrae he models and he sensiively analysis is carried ou o give some managemen insighs. Basedonhesudy,wemainlyfoundhefollowing. (1) If he value of producion rae is small, he reailer will lower he frequency of puing he orders o cu down
14 Discree Dynamics in Naure and Sociey 15.86.84 145 he opimal order cycle.82.8.78.76.74.72 he opimal order cycle 14 135 13 125.7.1.2.3.4.5.6.7.8 he delayed paymen ime: 12.1.2.3.4.5.6.7.8 he delayed paymen ime: (a) he opimal order cycle (b) he opimal order quaniy 5 45 4 he oal cos 35 3 25 2 15 1.1.2.3.4.5.6.7.8 he delayed paymen ime: (c) he oal cos for he reailer Figure 8: he impac of he changes of he parameer. he order cos. If he demand dependen rae is higher, he demand dependen producion rae has no effec on he opimal order cycle and he order quaniy. For he limi case, when he producion rae is infinie, as mos papers presened, he producion rae has no influence on he members decisions. (2) When he delayed paymen ime is less han he growh sage ime of he new producs inroducion, he reailer will shoren he replenishmen cycle o ake advanage of he rade credi more frequenly for accumulaing he ineres; on he oher hand, he shor order cycle can make he reailer adjus is order decisions more quickly for meeing he changeable demand wihin he growh sage of he producs. When he delayed paymen ime is larger han he breakpoin of he demand, wihin he mauriy sage of he producs, he delayed paymen ime has no effec on he opimal order cycle and he opimal order quaniy. he oal cos for he reailer is decreasing wih he delayed paymen ime. herefore, he reailer hopes he supplier can offer hem he delayed paymen ime as long as possible. (3) As he demand breakpoin is increasing, he opimal order cycle and he opimal order quaniy are decreasing firs and hen increasing. he oal cos for he reailer is increasing wih he demand breakpoin. Hence,forheproducs,ifhegrowhsageofhe produc life cycle is longer, he cos of he reailer
Discree Dynamics in Naure and Sociey 15 3 45 he opimal order cycle 2.5 2 1.5 1 he opimal order quaniy 4 35 3 25 2 15 1.5.1.2.3.4.5.6.7.8.9 Demand breakpoin: (a) he opimal order cycle 7 5.1.2.3.4.5.6.7.8.9 Demand breakpoin: (b) he opimal order quaniy 6 5 he oal cos 4 3 2 1.1.2.3.4.5.6.7.8.9 Demand breakpoin: (c) he oal cos for he reailer Figure 9: he impac of he changes of he parameer. will be higher. herefore, he reailer should make some measures, such as adverisemen, o induce he producs demand moving o he mauriy sage as soon as possible. he research presened in his paper can be exended in several ways. For example, oher demand funcions can be furher discussed considering he demand dependen producion rae in he EOQ invenory model wih he rade credi financing. Addiionally, he models can be generalized o consider he shorage or he parial backlogging. Furhermore, he influence of he poor qualiy producs on he EOQ model can be discussed o obain some managemen insighs. Appendices A. Proof of heorem 1 he firs derivaive for a minimum of RC 13 is wih RC 13 f 13 () =A+(c h +c p I c ) = 1 2 f 13 () + 1 f 13 () (A.1) [(r 1) 1 6 D 3 + 1 2 D 2 1 + 1 2 D 2 D 1 ]
16 Discree Dynamics in Naure and Sociey 1 6 c pi c (r 1) D 3 si e 6 D 3, f 13 () =(c h +c p I c ) [ 1 2r (r 1) D 2 +D ( r 1 )]; r 13 () =(c pi c +c h )D r 1. r (A.2) f Based on (A.1), we have If α,hen RC 13 / >. RC 13 is an increasing funcion wih for = (, ). For [(1/2)(r + 1), ), he opimal value 13 is obained when = (1/2)(r + 1). he opimal value of he order level is Q 13 = D d + 13 D d. Based on he analysis above, i is easy o obain heorem 1. B. Proof of heorem 2 Calculae he firs derivaives of he RC 23 wih respec o : RC 13 = 1 2 { A (c h +c p I c ) RC 23 = 1 2 f () + 1 f () ; (B.1) [(r 1) 1 6 D 3 1 8r D (r 1) 2 3 r 1 2r D 2 ] + 1 6 c pi c (r 1) D 3 + si e 6 D 3 }. (A.3) wih f 23 () =A+c h [(r 1) 1 6 D 3 + 1 2 D 2 1 + 1 2 D 2 D 1 ] he second derivaives for RC 13 wih respec o is 2 RC 13 2 +c p I c ( 1 3 D 3 1 + 1 3 D 3 1 2 D 2 D + D 2 = 2 3 f () 2 2 f () + 1 f () = 2 3 { A (c h +c p I c ) (A.4) +D 2 1 2 D 2 ) si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ), (B.2) [(r 1) 1 6 D 3 1 8r D (r 1) 2 3 ] + 1 6 c pi c (r 1) D 3 + si e 6 D 3 }. We can know ha lim ( RC 13 / ) >. Le us se α= A (c h +c p I c )D 3 (r 1) ( 1 24 + 1 8r ) + 1 6 D 3 (c p I c (r 1) +si e ). (A.5) f 23 () =c h [ 1 2r (r 1) D 2 +D ( r 1 )] r +c p I c ( 2D 1 1 r 2 +2D ) ; f 23 () =c hd r 1 1 2c r p I c r 3 D. herefore, we have If α<,henlim ( RC 13 / ) <. herefore, he inermediae value heorem yields ha here exiss 13 as he roo of RC 13 / = for = (, ). Andwhenα<, we have 2 RC 13 / 2 >.his 13 no only exis bu also is unique. If 13 is feasible, ha is (1/2)(r+1) 13 for (22)and (23), he opimal replenishmen cycle is 13 = 13.Oherwise, if 13 is infeasible, he opimal replenishmen cycle is 13 = (1/2)(r + 1). he opimal value of he order level is Q 13 = D d + 13 D d. RC 23 = 1 2 { A c h [(r 1) 1 6 D 3 1 8r D (r 1) 2 3 r 1 2r D 2 ]
Discree Dynamics in Naure and Sociey 17 +si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ) c p I c [ 1 3 D 3 1 2 D 2 +D 2 ]} +c p I c D. (B.3) he second derivaives for RC 23 wih respec o is 2 RC 23 2 = 2 3 { A c h [(r 1) 1 6 D 3 1 8r D (r 1) 2 3 ] +si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ) c p I c [ 1 3 D 3 1 2 D 2 +D 2 ]}. (B.4) We know ha lim ( RC 23 / ) >. Le us se β= A c h [(r 1) 1 6 D 3 1 8r D (r 1) 2 3 ] +si e ( 1 2 D 2 + 1 6 D 3 + 1 2 D 2 ) c p I c [ 1 3 D 3 1 2 D 2 +D 2 ]. (B.5) If β <, hen lim ( RC 23 / ) <. herefore, he inermediae value heorem yields ha here exiss RC 23 / = for = (, ). Andwhenβ<,wehave 2 RC 23 / 2 >.heopimalvalueof 23 canbeobained by solving he equaion RC 23 / =. 23 no only exis bu also is unique. If 23 is feasible, ha is, r (1/2)(r 1) 23 for (33)and(34), he opimal replenishmen cycle 23 = 23. Oherwise, he opimal replenishmen cycle is 23 = r (1/2)(r 1). he opimal order quaniy is Q 23 = D d+ 23 D d. If β, RC 23 / >, sorc 23 is an increasing funcion wih for = (, ).Forr (1/2)(r 1), he opimal value of 23 canbeobainedby = r (1/2)(r 1). he opimal order quaniy is Q 23 = D d+ 23 D d. Based on he analysis above, i is easy o obain heorem 2. Conflic of Ineress he auhors declare ha here is no conflic of ineress regarding he publicaion of his paper. Acknowledgmens he paper is funded by he Naional Naural Science Foundaion of China (7132115, 7117218), he inisry of educaion of Humaniies, and Social Sciences Projec (13YJC63121). References [1] K.-J. Chung and J.-J. Liao, he simplified soluion algorihm for an inegraed supplier-buyer-invenory model wih wopar rade credi in a supply chain sysem, European Journal of Operaional Research, vol. 213, no. 1, pp. 156 165, 211. [2] Y.-C. sao and G.-J. Sheen, Dynamic pricing, promoion and replenishmen policies for a deerioraing iem under permissible delay in paymens, Compuers & Operaions Research, vol. 35, no. 11, pp. 3562 358, 28. [3] H. Soni and N. H. Shah, Opimal ordering policy for sockdependen demand under progressive paymen scheme, European Journal of Operaional Research,vol.184,no.1,pp.91 1, 28. [4] J.-. eng, I.-P. Krommyda, K. Skouri, and K.-R. Lou, A comprehensive exension of opimal ordering policy for sockdependen demand under progressive paymen scheme, European Journal of Operaional Research,vol.215,no.1,pp.97 14, 211. [5] B. Sarkar, An EOQ model wih delay in paymens and ime varying deerioraion rae, ahemaical & Compuer odelling,vol.55,no.3-4,pp.367 377,212. [6] J.. eng, J. in, and Q. H. Pan, Economic order quaniy model wih rade credi financing for non-decreasing demand, Omega,vol.4,no.3,pp.328 335,212. [7] S. K. Goyal, Economic order quaniy under condiions of permissible delay in paymens, Journal of he Operaional Research Sociey,vol.36,no.4,pp.335 338,1985. [8] P. Chu, K.-J. Chung, and S.-P. Lan, Economic order quaniy of deerioraing iems under permissible delay in paymens, Compuers & Operaions Research, vol.25,no.1,pp.817 824, 1998. [9] S. P. Aggarwal and C. K. Jaggi, Ordering policies of deerioraing iems under permissible delay in paymens, Journal of he Operaional Research Sociey,vol.46,no.5,pp.658 662,1995. [1] A... Jamal, B. R. Sarker, and S. Wang, An ordering policy for deerioraing iems wih allowable shorage and permissible delay in paymen, JournalofheOperaionalResearchSociey, vol. 48, no. 8, pp. 826 833, 1997. [11] H.-J. Chang and C.-Y. Dye, An invenory model for deerioraing iems wih parial backlogging and permissible delay in paymens, Inernaional Journal of Sysems Science,vol.32,no. 3, pp. 345 352, 21. [12] J.-. eng, On he economic order quaniy under condiions of permissible delay in paymens, Journal of he Operaional Research Sociey,vol.53,no.8,pp.915 918,22. [13] K.-J. Chung and Y.-F. Huang, he opimal cycle ime for EPQ invenory model under permissible delay in paymens, Inernaional Journal of Producion Economics,vol.84,no.3,pp. 37 318, 23. [14] R. Uhayakumar and P. Parvahi, A wo-sage supply chain wih order cos reducion and credi period incenives for deerioraing iems, Inernaional Journal of Advanced anufacuring echnology,vol.56,no.5 8,pp.799 87,211.
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