BImpact of Supply Chain Coordination for Deteriorating Goods with Stock-Dependent Demand Rate

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1 J. Sev. Sci. & Manageent, 8, : 3-7 Published Online August 8 in SciRes ( BIpact of Supply Chain Coodination fo Deteioating Goods with Stock-Dependent Deand Rate Chuanxu Wang PSchool of Econoy and Manageent,Shanghai Maitie Univesity Shanghai,China Eail: cxwang@shtu.edu.cn P ABSRAC o analyze effects of supply chain coodination fo deteioating goods with stock-dependent deand ate, this pape pesents decision odels fo ode quantity and odeing cycle unde two scenaios( decentalized supply chain, centalized supply chain. Nueical study is caied out to deonstate the effectiveness of the poposed odels, and to analyze the ipact of supply chain coodination on supply chain pofit. Sensitivity analysis is pefoed to study the ipact of diffeent paaetes associated with the odel, such as the ate of deteioation, the etaile s puchase cost, the anufactue s poduction cost, the etaile s and anufactue s holding cost on the supply chain pofit incease pecentages geneated by the supply chain coodination. Keywods: supply chain coodination, deteioating goods, odeing cycle, stock-dependent deand. Intoduction In eal life, any inventoy goods, such as agicultual poducts, fashion goods, dugs and high-tech poducts, ae subject to depletion though spoilage, shinkage, decay and obsolescene []. he deteioation is quite pevalent and should not be disegaded. Inventoy anageent fo deteioation goods has eceived any attentions fo eseaches and pactitiones. Most of the existing eseaches focus on the EOQ-based inventoy decision odels. Ghae and Schade [] pesented the EOQ odel by consideing the cobined effects of deand, usage and linea decay. Covet and Philip [3]used the vaiable deteioation ate of the two-paaete Weibull distibution, to foulate the inventoy decision odel unde the assuptions of a constant deand ate, with no shotages allowed. Philip [4] odified this odel by using the deteioation ate of the thee-paaete Weibull distibution. adikaalla [5] adopted gaa distibuted deteioation unde constant deand ove tie, without shotages. Moon and Lee [6] pesented the EOQ odel with a noally distibuted deteioation ate. Othe deteioation inventoy odels have extended pio eseach by consideing a tie-vaying deand function, with o without shotages. Dave and Patel [7] poposed an EOQ odel unde tie-popotional deand, with no shotages allowed. Sachan [8] extended thei odel by consideing shotages. Bahai-Kashani [9] genealized the poble by peitting vaiations in both eplenishent cycle length and ode quantity. Bose et al []developed an EOQ odel fo deteioation ites incopoating the effects of inflation, tie value of oney, a linealy tie-dependent deand ate and shotages. Replenishent decision odels unde tie-popotional deand and exponentially decaying deteioation ate was developed in []. It is obseved that lage quantities of consue goods displayed in a supeaket geneate highe deands. Silve and Peteson [] noted that the sales at the etail level tend to be popotional to the inventoy displayed. Gupta and Vat [], Mandal and Phaujda [3], Bake and Uban [4], Datta and Pal [5], etc developed the EOQ odels with stock-dependent deand ate. Mandal and Phaujda [6], Pal et al. [7] developed the inventoy odels fo deteioating ites with stock-dependent deand ate. In this pape, we have extended these woks, on deteioating inventoy eseach, by consideing deteioating goods with stock-dependent deand in a two-echelon supply chain consisting one anufactue and one etaile, the objective is to investigate the effects of supply chain coodination on pofit incease in the supply chain, and study the ipact of diffeent paaetes associated with the odel, such as the ate of deteioation, the etaile s puchase cost, the anufactue s poduction cost, the etaile s and anufactue s holding cost on the supply chain pofit incease pecentages geneated by the supply chain coodination.. Assuptions and Notations.. Assuptions ( he etaile eplenishes the stocks fo the exclusive souce on an EOQ basis. Replenishents ae instantaneous. ( Lead tie is assued to be zeo fo the sake of siplicity. Copyight 8 SciRes

2 4 Chuanxu Wang (3 No backodes ae allowed. (4 Deand ate is dependent on the instantaneous inventoy level. he deand ate d( I of the ite, when the inventoy is I, is consideed in the fo di ( = α I β, whee α > and < β < ae scale and shape paaetes ( Bake and Uban 988. (5 he anufactue s poduction ate is geate than o equal to the deand ate facing the etaile. (6he anufactue is a ake-to-ode poduce; it has a lot-fo-lot poduction policy in esponse to the etaile s deand. In this paticula case, the length of the anufactue s poduction cycle is equal to the length of the etaile s eplenishent cycle... Notations p the sale pice fo the etaile; S the ode cost pe ode fo the etaile; M the setup cost pe lot fo the anufactue; c ( c pocueent (anufactuing cost pe unit fo etaile (anufactue; d the deand ate of the ite in the aketplace; q the poduction ate; h ( h the inventoy holding cost as a faction of the inventoy cost fo the etaile(anufactue; ( t the deteioation ate facing both the etaile and anufactue, ; I (, t I ( t inventoy level at tie t fo the etaile ( anufactue; Q the ode quantity fo the etaile; the eplenishent cycle ( o poduction cycle fo the etaile( anufactue; 3. Basic Model In this section, we fist deive the pofit odel fo the decentalized supply chain. Late, we pesent the pofit odel deived by consideing the centalized supply chain. 3.. he Decentalized Supply Chain In the decentalized supply chain, each entity within the supply chain ais to axiize its own pofit functions, with no consideation given to its countepat s eaction o pofit. he etaile akes a eplenishent decision based on an EOQ policy that includes inventoy holding cost and odeing cost. Duing the eplenishent cycle, the change in etaile s inventoy level depends on deand and deteioation and is given by []: di ( t β + I ( t = α[ I (] t t, ( As shown in Pal et al. (993 [7], equation ( can be ewitten as β α[ I ( t] β I ( t + α[ I ( t] By integating, we get β [ I ( t] ln + α[ I ( t] β = ln e whee C is integation constant. It can be eaanged as I ( β t = C e β di ( t = ( β ϑt ( + ln C, (3 (4 α + [ I ( t] By using the bounday condition on inventoy ( = Q,we can get C = (5 β α + Q Substituting (5 in (4, the etaile s inventoy level at tie t ( t can be expessed as I ( t [( Q e ] / he inventoy holding cost in a cycle fo the etaile is HC = h c I ( t (7 he etaile s total nube of deteioated goods in a cycle is given by D, β Q = QI ( α[ I ( t] β ( β /( β /( β = Q [( α + Q e α] / β ( β t /( β /( β = α + α (6 β ( β t β/( β α[( α + Q e α] (8 β/( β he pofit pe unit tie fo the etaile can be expessed as S = + β ( β t β/( β β/( β ( p c α[( α Q e α] / β ( β t /( β /( β hc α[( α+ Q e α] / D, cq (9 he pofit function (9 is highly nonlinea and cannot be solved by analytical ethods. We solve it by using optiization technology on copute and get the following optial values: Since the anufactue is a ake to-ode poduce, it has a lot-fo-lot poduction policy in esponse to the etaile s deand. Duing the poduction cycle, the change in anufactue s inventoy level is due to the cobined effect of poduction and deteioation: Copyight 8 SciRes

3 Ipact of Supply Chain Coodination fo Deteioating Goods 5 with Stock-Dependent Deand Rate di ( t = q I ( t t t, ( whee t and ae the stating and stopping poduction ties, espectively. With a ake-to-ode policy, the poduction quantity of the anufactue is equal to the deand quantity of the etaile. heefoe, we get: I ( t = I ( t ( t Solving the equation (6 by using the ethod poposed by Spiegel (96[8], we can get: q t I ( t = + Ce, ( whee C is constant. By using the bounday condition on inventoy I ( t =,we can obtain: q t C e = heefoe, the anufactue s inventoy level at tie t ( t t can be expessed as ( tt ( t tt ( tt q qe I ( t = e e q = (3 t he inventoy holding cost in a cycle fo the anufactue is q ( tt HC = h c I ( t = h c e t t q = + + ( t hc [ e ( t] q (4 he anufactue s total nube of deteioated goods in a cycle is given by = ( ( (5 D, Q q t I Q he pofit pe unit tie fo the anufactue is D, ( c c Q M HC c Q = ( t cq M hcq[ + e + ( t ] = ( t c q( t c q c qe + (6 Based on the optial ode quantity and odeing cycle as well as equation (, we can obtain optial pofit fo the anufactue using ( he Centalized Supply Chain In the centalized supply chain, the ode quantity and eplenishent cycle ae deteined by consideing the total pofit incued by both the etaile and the anufactue, so that the oveall pofit is axiized. he centalized supply chain equies infoation shaing between anufactue and etaile. Sequential o concuent engineeing will be beneficial to the infoation shaing. In this case, the total pofit pe unit tie fo the supply chain is the su of Equation (9 and (6. sc = + = β ( β t β/( β p α[( α + Q e α] S β/( β β ( β t /( β α[( α + α] /( β hc Q e β ( β /( β c [( α + Q e α] + /( β ( t M h c q[ + e + ( t ] ( t c q( t c q c qe + (7 Siilaly, since supply chain pofit function (7 is highly nonlinea, we solve it by using optiization technology on copute and get the optial values. 4. Nueical Study o illustate the effect of the odels, we give the following nueical exaple: α =.5 units pe tie peiod, β =.4, =., S = $, M = $, p = $ pe unit, q = units, h =.35, h =.5, c = 3.5, c =.. he odels wee ipleented on a pesonal copute using Matheatica vesion 5.. he esults ae outlined in able, evealing that the pofit fo centalized supply chain is geate than that fo decentalized supply chain. Based on the nueical exaple consideed above, we now pefo the sensitivity analysis on the effects of changes in the odel paaetes such as the ate of deteioation, the etaile s puchase cost, the anufactue s poduction cost, the etaile s and anufactue s holding cost on the optial ode quantity fo the etaile, the optial eplenishent cycle ( o poduction cycle fo the etaile( anufactue, the optial supply chain pofit, and supply chain pofit incease pecentages geneated by the centalized policy. he effects of changes in the pa aetes values ae shown in able. he sensitivity analysis is pefoed by changing each of the paaetes by 5%, -%, +3% and +5% and keeping the othe paaetes unchanged. he esults ae deonstated in able. he Sollution Results Supply chain Q sc Decentalized Centalized Copyight 8 SciRes

4 6 Chuanxu Wang Paaetes Change (% c c h h able. Effect of Changes in the Model Paaetes Decentalized Supply Chain Q SC Results of Optiization Pocedue Centalized Supply Chain SC Pofit Incease (% Q table. he following obsevations can be ade fo ( Whethe the supply chain is centalized o decentalized, the optial ode quantity, the optial eplenishent cycle (o poduction cycle, the optial supply chain pofit ae deceasing in the deteioation ate. Meanwhile the pecentages of supply chain pofit incease ealized by eploying the centalized policy ae inceasing in the deteioation ates. ( Whethe the supply chain is centalized o decentalized, the optial ode quantity and eplenishent cycle (o poduction cycle decease, while the optial supply chain pofit inceases in etaile s unit pocueent cost. (3 Whethe the supply chain is centalized o decentalized, the optial ode quantity, the optial eplenishent cycle (o poduction cycle and supply chain pofit ae deceasing in anufactue s unit anufactuing cost. (4 Whethe the supply chain is centalized o decentalized, the optial ode quantity, the optial eplenishent cycle (o poduction cycle and supply chain pofit ae deceasing in etaile s and anufactue s inventoy holding cost ate. (5 he pecentages of supply chain pofit incease ealized by eploying the centalized policy ae inceasing in the deteioation ates, but ae deceasing in the etaile s puchase cost, the anufactue s poduction cost, the etaile s and anufactue s holding cost. 5. Conclusions his pape has investigated the effect of supply chain coodination fo deteioating goods with stock-dependent deand ate. wo pofit odels ae developed with soe assuptions based on exponentially decaying deteioation ates. he nueical study is conducted to deonstate the effectiveness of the poposed odels, and to analyze the ipact of supply chain coodination on supply chain pofit. Sensitivity analysis is pefoed to study the ipact of diffeent paaetes associated with the odel, such as the ate of deteioation, the etaile s puchase cost, the anufactue s poduction cost, the etaile s and anufactue s holding cost on the optial ode quantity fo the etaile, the optial eplenishent cycle ( o po- Copyight 8 SciRes

5 Ipact of Supply Chain Coodination fo Deteioating Goods 7 with Stock-Dependent Deand Rate duction cycle fo the etaile( anufactue, the optial supply chain pofit, and supply chain pofit incease pecentages geneated by the supply chain coodination. he following obsevations can be obtained fo nueical analysis: ( A centalized policy is found to be always supeio to a decentalized policy in tes of pofit incease, especially when the deteioation ates ae high. (Whethe the supply chain is centalized o decentalized, the optial ode quantity, the optial eplenishent cycle (o poduction cycle, the optial supply chain pofit ae deceasing in the deteioation ate, etaile s unit pocueent cost, anufactue s unit anufactuing cost, as well as etaile s and anufactue s inventoy holding cost ate. he optial supply chain pofit ae deceasing in the deteioation ate, anufactue s unit anufactuing cost, as well as etaile s and anufactue s inventoy holding cost ate. Meanwhile it is inceasing in etaile s unit pocueent cost. (3he pecentages of supply chain pofit incease ealized by eploying the centalized policy ae inceasing in the deteioation ates. he poposed odels can be used to analyze gicultual poducts, fashion goods, dugs and high-tech poducts supply chain. It is obseved that lage quantities of these deteioationg goods displayed in a supeaket tend to geneate highe deands. Howeve, the odels consideed in this pape ae soewhat idealized. In eality, when supply chain is coodinated, thee ae likely to be soe costs incued fo infoation shaing schee. he infoation shaing could also affect poduction, inventoy, and othe opeations. he futue eseach will futhe conside these factos. hese odels can be extended in the futue eseach to conside oe geneal deteioation ates. Anothe extension possibility would be to use othe eplenishent policies. In addition to pofit odels, the cost odels can also be applied to evaluate the extent of coodination. 6. Acknowledgeent his pape is patially suppoted by Shanghai Shuguang Poga unde gant no. 6SG48. REFERENCES [] Chen, J. M., Chen. H. Effects of joint eplenishent and channel coodination fo anaging ultiple deteioating poducts in a supply chain, Jounal of the opeational eseach society, 56,5, pp [] Ghae P. N., Schade GF. A odel fo exponentially decaying inventoies, Jounal of Industy Engineeing, 5, 963, pp [3] Covet R. P., Philip GC. An EOQ odel fo ites with Weibull distibution deteioation, AIIE ansactions, 5, 973, pp [4] Philip G. C. A genealized EOQ odel fo ites with Weibull distibution deteioation, AIIE ansactions, 6, 974, pp [5] adikaalla,p. R. An EOQ inventoy odel fo ites with Gaa distibuted deteioation. AIIE ansactions,, 978, pp.78. [6] Moon I., Lee S. he effects of inflation and tie value of oney on an econoic ode quantity odel with a ando poduct life cycle, Euopean Jounal of Opeational Reseach, 5,, pp [7] Dave, U., Patel L. K. (, S i policy inventoy odel fo deteioation ites with tie popotional deand. Jounal of the opeational eseach society, 3, 98, pp [8] Sachan, R. S.On (, S i policy inventoy odel deteioation ites with tie popotional deand. Jounal of the opeational eseach society, 35, 984, pp.3-9. [9] Bahai-Kashani H., Replenishent schedule fo deteioating ites with tie-popotional deand. Jounal of Opeational Reseach Society, 4, 989, pp [] Bose, S., Goswai, A., Chaudhui,K.S., An EOQ odel fo deteioating ites with linea-dependent deand ate and shotages unde inflation and tie discounting. Jounal of the Opeational Reseach Society, 46, 995, pp [] Silve,E. A., Peteson,R., Decision systes fo inventoy anageent and poduction planning, nd edition, Wiley, New Yok, 985. [] Gupta, R., Vat, P., Inventoy odel fo stock-dependent consuption ate. Opseach, 3, 986, pp.9-4. [3] Mandal, B. N., Phaujda, S., A note on an inventoy odel with stock-dependent consuption ate. Opseach, 6, 989a, pp [4] Bake, R. C. and Uban,. L., A deteinistic inventoy syste with an inventoy-level-dependent deand ate. Jounal of Opeational Reseach Society, 39(9, 988, pp [5] Datta,. K., Pal, A. K., Effects of inflation and tie-value of oney on an inventoy odel with linea tie-dependent deand ate and shotages. Euopean Jounal of Opeational Reseach, 5, 99, pp.-8. [6] Mandal, B. N., Phaujda, S., An inventoy odel fo deteioating ites and stock-dependent consuption ate. Jounal of Opeational Reseach Society, 4(5, 989b., pp [7] Pal, S., Goswai, A., Chaudhui, K.S., A deteinistics inventoy odel fo deteioating ites with stock-dependent deand ate. Intenational Jounal of Poduction Econoics, 3, 993, pp [8] Spiegel, M. R., Applied Diffeential Equations. Pentice-Hall: Englewood Cliffs, NJ, 96. Copyight 8 SciRes