中央财经大学学报2007年增刊
工业中存货控制的基本方法
BasicMethodsofInventoryControlinIndustry
顾常超高诚秀
GUChang.chsoGAOCbe“S-Xiu
(武汉大学数理统计学院武汉430072)
[摘要】存货控制一直是供应链管理中的重要部分。本论文主要介绍工业中存货控制的基本方法。关键有两部分。循环存货和安全存货,它们虽然简单,但在应用中起了关键作用。对于循环存货,我们首想到经典的EOQ公式,然后我们提出完全单位和边际单位数量折扣政策。对于安全存货,我们分析改进循环服务水平的重要性并且提供相应的连续评论和周期评论政策的解央方案。
Abstract:Inventorycontrolisalways∞importantpartinsupplychainmmlagemenLThispapermainlyintro-ducesSOIUebasicmethodsofinventmycontrolinindustry.Twoparts,cycleinventoryandsafeinventory,arefo-cusedsincetheyplaypivotalmlasinpracticethoughsimple.Forthecycleinventory,wefirstrecalltheclassicalEOQformula,andthenwepresentall—unitandmarginal—unitquantitydiscountPolicie8_Forthesafeinventory,weanalyzetheimportanceofimprovingcycleservicelevelandoffercorrespondingsolutionforbothcontinuousTe—viewandperiodicreviewPolicies.
[关键词】供应链循环存货安垒存货
Keywords:SupplychainCycleinventorySafeinventory
1.CycleInventory
a)EconomicOrderQuantity(EoQ)
Cycleinventoryexistsbenanseproducingorpurchasingin婚lotsalfowsastageofthesupplychaintoex-pfoiteconomiesofscaleandlowercesLThepresenceoffixedenstsassociatedwithorderingandtransportation,quantitydiscountsinproductpricingandshort—termdiscountsortradepromotionsencouragesdifferentstagesofasupplychaintoexploiteconomiesofscaleandorderinlargelots.Hereweltvlewthedeterminateinventoryabouthoweachofthesefactorsaffectsthelotsizeandcycleinventorieswithinthesupplychainandaboutmanagerialle-v∞thatreducecycleinventoryinthesupp|ywithoutraisingcosto”.
Firstlywedefinesomeno.finn:
Q:lotorbatchsizeofandorder;R:denmndperunittime.
c:materialcostwhichistherdeepaidp目unit;S:fixedcostincurredperorder.
h:holdingcostperyearagefractionofproductCOSL
Herewei印m≈theimpactofdemandvariabilitybecauseithaslittleeffectonlargesizeandcycleinventory.
Cyclei—lven呻=lotgiz以=孚,Numberofordersperyear=吾,
Annualordercmt=(吾)s,Annualholdingcost=(孚)^c
ForasizeofQ,thetotalannualcost(TC)isgivenbythefollowingformula‘”:
o作者简介:顾常超.男.武攫大学硬士研究生。102
中央财经大学学报2007年增刊
TC=(号)s+(罟)k+龌
TblIlinilni聪tlIetotaleotlt,wec蛐take山e6戚defiv“vewitll阳8pect
df7℃1J醇^ctoQ.Thenwehave:
i尹一矿+虿’
se岫llgtlIefirstderivativet0be扯m,wee蛆学ett11eeeonollIieorderqIl蚰dty(EOQ):
Q’=
b)Eeonomi嚣ofscme
hpmcdce,thetoExploitQ啪6.yDiseo嘶偿dependiIlg帆山eo【dersize.Twoeommollly
aIldmarginalumtquantitydi驼ounLHereWe
COSLpricepaidperunitiBnot㈣tartLItmaychangeusedsi∞一baseddi∞ount∞heme:aUunitsquanfitydimoutwiU锄一8Iy∞Ⅱle胡bt8oftlIe辩di8co皿tschem鹪OntlIe训annual
i.AU—unitquantitydj∞∞nts:
IntlIe暑e∞heme.t}lepricing∞hedulecontainsspecifiedbreal【points口o,旱l,q2,…q,,whereqo=0.Ⅱ衄。卜
ataveragederisplaeedtllatisatleast鹊large鹊吼butsmallertIl粕g“I,t11曲eachumtisobmined
tllec∞t0fc。.U∞EoQ叫icywidId珊erentQ;,Wecanfind血reeca∞8:
Case1:吼≤Q,<q…
酬一d
Cm2:口。‘吼
一cmtTCt=(参)s+(譬)k+qR%试annudcostTC。=(景)s+(詈)k+c。R
Case3:Qi’q…
n试…a-oostTC。=(击)s+(警)‰¨“
ii.Marginalunitquantitydiscount:
Int}le鸵scheme.ⅡlepficingscheddecontainsspecifiedbreaI【pointsgo,口1,92…吼.go=0.Ⅱ粕orderofsizeqisplaeed,the6f8t吼一宁ollll妇arepricedatc0,tllenextqz—gl
have:un池arepricedat。l’粕d∞oILForeachValueofi,l乱Vibetllec似oford商Ilg91吼i协.We
K=C;(9l—qo)+Cr(啦一91)+…C‘一1(qi一吼.1)
TotalannualcostTC=护R+[K+(QⅦ)c。]耽+争K+(口-口1)c】
priceCi:1h∞wec∞gettlleoptimallotsizefor
1mere
Caseare√——1F一厨耶再可=再万a180twocaBe8:1:吼≤口。<口i+l
Total蛐ual
Case
T0诅lcostTC。=s+[K+(Q。一玑)q]7∥2+鲁[K+(Q一口{)c。]2:Qi<吼orQ.>q…nnucost他。=min{(鲁)s+K^/2+景K,‘i鲁)s+K。/2+旦qi+lPi+-)
forecastisneveF2.Safehlv蚰torydem肌dIn8bsolutelyfight.Andduetotlleeffectofleaddine,safeinventoryexisL
103Asknown
中央财经大学学报2007年增刊thispart,wewillreviewhowsafeinventorycanhelpasupplychainimproveproductavailabilitywithdemandvaria-bility.1heappropriatelevelofsafeinventoryisdeterminedbythefollowingtwofactors:theuncertaintyofdemandofsupplyandthedesiredlevelofproductavailability【3J.Firstlywedefinesomenotation:7,
R:averagedemandperunittime;叮R:Standarddeviationofdemandperperiod
L:Leadtimewhichistheintervalbetweenwhenanorderisplacedandwhenitisreceived.
Assumethatdemandforeachperiodi。isnormallydistributedwithamean磁anddeviationNiTotaldemandduringkperiodsisnormallydistributed访tlIameanofPandstandarddeviationofQ,where:
Productfillrate(矗):theP=甄iffil=勰趿a以=序丽=fractionofproductdemandthatissatisfiedformproductininventory.
Cycleservicelevel(CSL):thefactionofreplenishmentcyclesendwithallthecustomerdemandbeingmet.Therearetworeplenishmentpolicies.Theoneiscontinuousreviewinwhichinventoryiscontinuouslytracker,andanorderforlotsizeQisplacedwhentheinventorydeclinestothereorderpoint(BOP).Theotherisperiodicreviewinwhichinventorystatusischeckedatregularperiodicintervalsandanordercanbeplacedonlywhentheinventoryischecked.
a)ContinuousReviewPolicy
Inthispolicy,theexpecteddemandduringleadtime=RL,Safetyinventory.黯=ROP二RLBecausethedemandduringleadtimeisnormallydistributedwithameanOfR,andastandarddeviationof矿c,whereR‘=RLand口L=0kR.wecangetthecycleservicelevel:
CSL=F(ROP,RI,矿‘)=l火ROP,Rt,盯‘)dX,
Wedenoteexpectedshortageperreplenishmentcycle∞ESCwhichistheaverageunitsofdemandthat棚陀notsatisfiedfrominventoryinstockperreplenishmentcycle.Thenwecangettheproductfillrate
夕=1一ESC/Q=(Q—ESC)/Q
Letf(x)bethedensityfunctionofdemanddistributionduringtheleadtime.WecangetESCasfollow:
耶c=.;J(菇一RDP状菇)J;如=一5[1ROP一只(盖)]+盯z(盖)V工VL
WhereFsisthestandardnormalcumulativedistributionfunctionandfsisthestandardnormaldensityfunc。tiorL
Givendesiredcycleservicelevel,wecanevaluaterequiredsafetyinventory.
Assumewehavethefollowinginput:
Desiredcycleservicelevel=CSL,Meandemandduringleadtime=R£,standarddeviationofdemanddur-ingleadtime=o-L.BecauseROP=R£+∞,CSLcanbeidentifiedbysafetyinventoryss.Probability(demand≤RL+55)=CSL.Giventhatdemandisnormallydistribution.WehaveF(RL一55,RL,o-L)=CSLwhichisequalto吃+船=F“(CSL,R£,叽).
Hence,wecanfindsafeinventoryssusingthefollowingformula[.】:ss=F’1(CSL)盯L
Inpractice,theleadtimemaynotbeconstant.Weassumethatitfollowsanormaldistribution.AndwesetStbetllestandarddeviationofleadtime.Thenthemeanandstandarddeviationofthedemandduringtheleadtimea_re:Rt=RL,吼=√k;+萨五
b)PeriodicReviewPolicy
Inperiodicreviewpolicies,wesetTbethereviewinterval。therealleadtimehereisT+Lfollowtheanalysis104
中央财经大学学报2007年增刊abovewehave:Rr+‘已R(T+L),矿r+£=/r+厶旷R.So,wecanfindthesafeinventory鼹usingthefollowingfor-mula:豁=F。1(CSL)o-mc
3.Conclusion
ThispapermainlyintroducessomebasicmethodsofinventorycontrolinindustryandfocusontWOparts,cycleinventoryandsafeinventory,sincetlleyareeasilyimplementedinpractice.Forthecycleinventory,wepresentall—unitandmarginal—unitquantitydiscountpolicies.Forthesafeinventory,weanalyzetheimportanceofimpro-vmgcycleservicelevelandoffercorrespondingsolutionforbothcontinuousreviewandperiodicreviewpolicies.Wehopethisresearchcanofferaguidelinetosomepracticers.
References
[1】Dye,cY.,oIly蛐g.LY.,2005.AnEOQmodelforpeaishableitemsunderstock—dopendcntsellingrateandtime—dependentp矗rti矗lbscklogsin昏EuropeanJournal0fOperationalResearch.163.776-783.
mh[2]Padmanabhan,G.,Vrat,P.,1995.EOQmodelsforperishableitemsunderstockdependentsellingrate.EuropeanJournalofOperationalRe-
86,281.
[3】c凹al,&k,Gurmsekamn,九,1995.Anintegratedproductioninventory—marketingmodelfordeterioratingitems.Computers&IndustrialEngineering28(4),755—762-
【4]Yah,}L,Chug,T.eE,1998.Op血I讨productionstoppingandrestartingtimesforallEOQmodelwithdeterioratingitems.Joumal0fOper-ationslResearchSociety49,1288—1295.105
中央财经大学学报2007年增刊
工业中存货控制的基本方法
BasicMethodsofInventoryControlinIndustry
顾常超高诚秀
GUChang.chsoGAOCbe“S-Xiu
(武汉大学数理统计学院武汉430072)
[摘要】存货控制一直是供应链管理中的重要部分。本论文主要介绍工业中存货控制的基本方法。关键有两部分。循环存货和安全存货,它们虽然简单,但在应用中起了关键作用。对于循环存货,我们首想到经典的EOQ公式,然后我们提出完全单位和边际单位数量折扣政策。对于安全存货,我们分析改进循环服务水平的重要性并且提供相应的连续评论和周期评论政策的解央方案。
Abstract:Inventorycontrolisalways∞importantpartinsupplychainmmlagemenLThispapermainlyintro-ducesSOIUebasicmethodsofinventmycontrolinindustry.Twoparts,cycleinventoryandsafeinventory,arefo-cusedsincetheyplaypivotalmlasinpracticethoughsimple.Forthecycleinventory,wefirstrecalltheclassicalEOQformula,andthenwepresentall—unitandmarginal—unitquantitydiscountPolicie8_Forthesafeinventory,weanalyzetheimportanceofimprovingcycleservicelevelandoffercorrespondingsolutionforbothcontinuousTe—viewandperiodicreviewPolicies.
[关键词】供应链循环存货安垒存货
Keywords:SupplychainCycleinventorySafeinventory
1.CycleInventory
a)EconomicOrderQuantity(EoQ)
Cycleinventoryexistsbenanseproducingorpurchasingin婚lotsalfowsastageofthesupplychaintoex-pfoiteconomiesofscaleandlowercesLThepresenceoffixedenstsassociatedwithorderingandtransportation,quantitydiscountsinproductpricingandshort—termdiscountsortradepromotionsencouragesdifferentstagesofasupplychaintoexploiteconomiesofscaleandorderinlargelots.Hereweltvlewthedeterminateinventoryabouthoweachofthesefactorsaffectsthelotsizeandcycleinventorieswithinthesupplychainandaboutmanagerialle-v∞thatreducecycleinventoryinthesupp|ywithoutraisingcosto”.
Firstlywedefinesomeno.finn:
Q:lotorbatchsizeofandorder;R:denmndperunittime.
c:materialcostwhichistherdeepaidp目unit;S:fixedcostincurredperorder.
h:holdingcostperyearagefractionofproductCOSL
Herewei印m≈theimpactofdemandvariabilitybecauseithaslittleeffectonlargesizeandcycleinventory.
Cyclei—lven呻=lotgiz以=孚,Numberofordersperyear=吾,
Annualordercmt=(吾)s,Annualholdingcost=(孚)^c
ForasizeofQ,thetotalannualcost(TC)isgivenbythefollowingformula‘”:
o作者简介:顾常超.男.武攫大学硬士研究生。102
中央财经大学学报2007年增刊
TC=(号)s+(罟)k+龌
TblIlinilni聪tlIetotaleotlt,wec蛐take山e6戚defiv“vewitll阳8pect
df7℃1J醇^ctoQ.Thenwehave:
i尹一矿+虿’
se岫llgtlIefirstderivativet0be扯m,wee蛆学ett11eeeonollIieorderqIl蚰dty(EOQ):
Q’=
b)Eeonomi嚣ofscme
hpmcdce,thetoExploitQ啪6.yDiseo嘶偿dependiIlg帆山eo【dersize.Twoeommollly
aIldmarginalumtquantitydi驼ounLHereWe
COSLpricepaidperunitiBnot㈣tartLItmaychangeusedsi∞一baseddi∞ount∞heme:aUunitsquanfitydimoutwiU锄一8Iy∞Ⅱle胡bt8oftlIe辩di8co皿tschem鹪OntlIe训annual
i.AU—unitquantitydj∞∞nts:
IntlIe暑e∞heme.t}lepricing∞hedulecontainsspecifiedbreal【points口o,旱l,q2,…q,,whereqo=0.Ⅱ衄。卜
ataveragederisplaeedtllatisatleast鹊large鹊吼butsmallertIl粕g“I,t11曲eachumtisobmined
tllec∞t0fc。.U∞EoQ叫icywidId珊erentQ;,Wecanfind血reeca∞8:
Case1:吼≤Q,<q…
酬一d
Cm2:口。‘吼
一cmtTCt=(参)s+(譬)k+qR%试annudcostTC。=(景)s+(詈)k+c。R
Case3:Qi’q…
n试…a-oostTC。=(击)s+(警)‰¨“
ii.Marginalunitquantitydiscount:
Int}le鸵scheme.ⅡlepficingscheddecontainsspecifiedbreaI【pointsgo,口1,92…吼.go=0.Ⅱ粕orderofsizeqisplaeed,the6f8t吼一宁ollll妇arepricedatc0,tllenextqz—gl
have:un池arepricedat。l’粕d∞oILForeachValueofi,l乱Vibetllec似oford商Ilg91吼i协.We
K=C;(9l—qo)+Cr(啦一91)+…C‘一1(qi一吼.1)
TotalannualcostTC=护R+[K+(QⅦ)c。]耽+争K+(口-口1)c】
priceCi:1h∞wec∞gettlleoptimallotsizefor
1mere
Caseare√——1F一厨耶再可=再万a180twocaBe8:1:吼≤口。<口i+l
Total蛐ual
Case
T0诅lcostTC。=s+[K+(Q。一玑)q]7∥2+鲁[K+(Q一口{)c。]2:Qi<吼orQ.>q…nnucost他。=min{(鲁)s+K^/2+景K,‘i鲁)s+K。/2+旦qi+lPi+-)
forecastisneveF2.Safehlv蚰torydem肌dIn8bsolutelyfight.Andduetotlleeffectofleaddine,safeinventoryexisL
103Asknown
中央财经大学学报2007年增刊thispart,wewillreviewhowsafeinventorycanhelpasupplychainimproveproductavailabilitywithdemandvaria-bility.1heappropriatelevelofsafeinventoryisdeterminedbythefollowingtwofactors:theuncertaintyofdemandofsupplyandthedesiredlevelofproductavailability【3J.Firstlywedefinesomenotation:7,
R:averagedemandperunittime;叮R:Standarddeviationofdemandperperiod
L:Leadtimewhichistheintervalbetweenwhenanorderisplacedandwhenitisreceived.
Assumethatdemandforeachperiodi。isnormallydistributedwithamean磁anddeviationNiTotaldemandduringkperiodsisnormallydistributed访tlIameanofPandstandarddeviationofQ,where:
Productfillrate(矗):theP=甄iffil=勰趿a以=序丽=fractionofproductdemandthatissatisfiedformproductininventory.
Cycleservicelevel(CSL):thefactionofreplenishmentcyclesendwithallthecustomerdemandbeingmet.Therearetworeplenishmentpolicies.Theoneiscontinuousreviewinwhichinventoryiscontinuouslytracker,andanorderforlotsizeQisplacedwhentheinventorydeclinestothereorderpoint(BOP).Theotherisperiodicreviewinwhichinventorystatusischeckedatregularperiodicintervalsandanordercanbeplacedonlywhentheinventoryischecked.
a)ContinuousReviewPolicy
Inthispolicy,theexpecteddemandduringleadtime=RL,Safetyinventory.黯=ROP二RLBecausethedemandduringleadtimeisnormallydistributedwithameanOfR,andastandarddeviationof矿c,whereR‘=RLand口L=0kR.wecangetthecycleservicelevel:
CSL=F(ROP,RI,矿‘)=l火ROP,Rt,盯‘)dX,
Wedenoteexpectedshortageperreplenishmentcycle∞ESCwhichistheaverageunitsofdemandthat棚陀notsatisfiedfrominventoryinstockperreplenishmentcycle.Thenwecangettheproductfillrate
夕=1一ESC/Q=(Q—ESC)/Q
Letf(x)bethedensityfunctionofdemanddistributionduringtheleadtime.WecangetESCasfollow:
耶c=.;J(菇一RDP状菇)J;如=一5[1ROP一只(盖)]+盯z(盖)V工VL
WhereFsisthestandardnormalcumulativedistributionfunctionandfsisthestandardnormaldensityfunc。tiorL
Givendesiredcycleservicelevel,wecanevaluaterequiredsafetyinventory.
Assumewehavethefollowinginput:
Desiredcycleservicelevel=CSL,Meandemandduringleadtime=R£,standarddeviationofdemanddur-ingleadtime=o-L.BecauseROP=R£+∞,CSLcanbeidentifiedbysafetyinventoryss.Probability(demand≤RL+55)=CSL.Giventhatdemandisnormallydistribution.WehaveF(RL一55,RL,o-L)=CSLwhichisequalto吃+船=F“(CSL,R£,叽).
Hence,wecanfindsafeinventoryssusingthefollowingformula[.】:ss=F’1(CSL)盯L
Inpractice,theleadtimemaynotbeconstant.Weassumethatitfollowsanormaldistribution.AndwesetStbetllestandarddeviationofleadtime.Thenthemeanandstandarddeviationofthedemandduringtheleadtimea_re:Rt=RL,吼=√k;+萨五
b)PeriodicReviewPolicy
Inperiodicreviewpolicies,wesetTbethereviewinterval。therealleadtimehereisT+Lfollowtheanalysis104
中央财经大学学报2007年增刊abovewehave:Rr+‘已R(T+L),矿r+£=/r+厶旷R.So,wecanfindthesafeinventory鼹usingthefollowingfor-mula:豁=F。1(CSL)o-mc
3.Conclusion
ThispapermainlyintroducessomebasicmethodsofinventorycontrolinindustryandfocusontWOparts,cycleinventoryandsafeinventory,sincetlleyareeasilyimplementedinpractice.Forthecycleinventory,wepresentall—unitandmarginal—unitquantitydiscountpolicies.Forthesafeinventory,weanalyzetheimportanceofimpro-vmgcycleservicelevelandoffercorrespondingsolutionforbothcontinuousreviewandperiodicreviewpolicies.Wehopethisresearchcanofferaguidelinetosomepracticers.
References
[1】Dye,cY.,oIly蛐g.LY.,2005.AnEOQmodelforpeaishableitemsunderstock—dopendcntsellingrateandtime—dependentp矗rti矗lbscklogsin昏EuropeanJournal0fOperationalResearch.163.776-783.
mh[2]Padmanabhan,G.,Vrat,P.,1995.EOQmodelsforperishableitemsunderstockdependentsellingrate.EuropeanJournalofOperationalRe-
86,281.
[3】c凹al,&k,Gurmsekamn,九,1995.Anintegratedproductioninventory—marketingmodelfordeterioratingitems.Computers&IndustrialEngineering28(4),755—762-
【4]Yah,}L,Chug,T.eE,1998.Op血I讨productionstoppingandrestartingtimesforallEOQmodelwithdeterioratingitems.Joumal0fOper-ationslResearchSociety49,1288—1295.105