The impact of lead time forecasting on the bullwhip effect

The impact of lead time forecasting on the bullwhip effect arxiv:1309.7374v3 [math.pr] 29 Jul 2015 ZbigniewMichna 1 andpeternielsen 2 1 DepartmentofMathematicsandCybernetics Wrocław University of Economics 2 DepartmentofMechanicalandManufacturingEngineering Aalborg University Abstract In this article we quantify the bullwhip effect(the variance amplification in replenishment orders) when demands and lead times are predicted in a simple two-stage supply chain with one supplier and one retailer. In recent research the impact of stochastic order lead timeonthebullwhipeffectisinvestigated,buttheeffectofneedingto predict/estimatetheleadtimeisnotconsideredinthesupplychain models. Under uncertainty conditions it is necessary to estimate the leadtimeforamemberofthesupplychaintoplaceanorder.wefind anewcauseofthebullwhipeffectintheformofleadtimeforecasting andwegiveanexactformofthebullwhipeffectmeasure(theratio of variances) when demands and lead times are predicted by moving averages. In the bullwhip effect measure we discover two terms amplifying the effect which are the result of lead time estimation. Correspondingauthor Email: zbigniew.michna@ue.wroc.pl Tel/fax: +48713680335 1

Keywords: supply chain, bullwhip effect, order-up-to level policy, stochastic lead time, lead time forecasting, demand forecasting, lead time demand forecasting 1 Introduction ThebullwhipeffectwasrecognizedbyForrester[10]inthemiddleofthetwentiethcenturyandwascoinedasatermbyProcter&Gamblemanagement. This phenomenon appears in supply chains as the variance amplification in replenishmentordersifonemovesupinasupplychain(seedisneyandtowill [8]andGearyetal.[11]forthedefinitionandhistoricalreview).Itisconsidered harmful because of its consequences which are(see e.g. Buchmeister et al.[4]): excessive inventory investment, poor customer service level, lost revenue, reduced productivity, more difficult decision-making, sub-optimal transportation, sub-optimal production etc. This makes it critical to find therootcausesofthebullwhipeffectandtoquantifytheincreaseindemandsvariabilityateachstageofthesupplychainasthisisdirectlylinked tocosts. Inthecurrentstateofresearchtypicallyfivemaincausesofthe bullwhip effect are considered(see e.g. Lee et al. [12] and[13]): demand forecasting, non-zero lead time, supply shortage, order batching and price fluctuation. To decrease the variance amplification in a supply chain(i.e. to reduce the bullwhip effect) we need to identify all factors causing the bullwhip effect and to quantify their impact on the effect. In this research we will investigate another cause of the bullwhip effect that is lead time forecasting. It is well known from inventory theory that themeanandvariabilityofleadtimeofasupplieraffectstheinventoryand order decisions of its customer. Although lead times are typically considered deterministic, they are actually not in many supply chains(see Chatfield et al.[5]). The impact of stochastic lead times in inventory systems has been intensively studied in the literature see e.g. Bagchi et al.[3], Hariharan and Zipkin[17], Mohebbi and Posner[19], Song[21] and[22], Song and Zipkin [23] and[24] and Zipkin[25]. Recent research investigates the influence of stochasticleadtimesonthebullwhipeffect(seee.g.soandzheng[20],duc etal.[9]orkimetal.[18]andreferencestherein)butnoneofthecurrent 2

research addresses the consequences of inherent need to estimate/ forecast lead time when it is stochastic. These works investigate the impact of random stochastic lead times on the bullwhip effect through the characteristics of their distribution e.g. mean value or variance. In the paper So and Zheng [20] supplier s delivery lead time depends on the existing order backlog at thesupplierwhichmeansthatitisnotdeterministicbutitdependsonthe retailer s order quantities. They solve this problem numerically. Inthetypicalapproachifoneassumesthatacertainfeatureisrandom thereisaneedtopredictitsvalueforthenextperiods. Inoursituation the relationship between supplier s lead time and its customer(a retailer) order quantities is very strong, especially when the supplier operates at tight capacity and has difficulties to adjust capacity and maintain constant deliveryleadtimetoitscustomers.weshouldalsonoticethattheretailerorder quantities can in turn determine the delivery time performance of the supplier. If a retailer observes uncertainty in demands and lead times(i.e. they arerandom)andhewantstoplaceanordertoasupplierduetoacertain stockpolicytofulfillcustomerordersinatimelymanner,heneedstopredict future customer s demands and future supplier s lead times. In other words the retailer needs to project his costumer s future demands over his supplier s lead time to determine the appropriate order quantity to this supplier. This is done by the so-called lead time demand forecasting to have the necessary required inventory to meet customer demands over the lead time. Lead time demand forecasting can be executed by demand forecasting and lead time forecasting. Thuswecannotavoidthatthevalueofafutureleadtimeis necessary to determine an order quantity to the supplier. It yields a need to predict lead times based on their previous values. Practically a retailer needstoestimate(toforecast)thevalueofthenextleadtimetomakean ordertoasupplier.thisneedforleadtimeestimationhasnotbeennoticed inpreviousworksontheimpactofastochasticleadtimeonthebullwhip effect. Inthispaperwefindanewcauseofthebullwhipeffectwhichisleadtime estimation through forecasting and we quantify its impact on the variance amplification in replenishment orders. Many papers assuming a deterministic lead time have studied the influence of different methods of demand fore- 3

casting on the bullwhip effect such as simple moving average, exponential smoothing, and minimum-mean-squared-error forecasts when demands are independent identically distributed or constitute integrated moving-average, autoregressive process or autoregressive-moving average(see Graves[16], Lee etal.[14],chenetal.[6]and[7],alwanetal.[2],zhang[26]andducetal. [9]).Alsotherecentworksontheimpactofleadtimesonthebullwhipeffect byagrawaletal.[1]andliandliu[15]shouldbenoted.however,ofthese papers the first one does not consider stochastic lead times and the second one investigates a transition state model with uncertainties in demands, production process, supply chain structure, inventory policy implementation and especially vendor order placement lead time delays. They find a maximally allowable vendor order placement lead time delay such that the supply chain system is exponentially stabilizable. This approach uses dynamical control systems theory and is not probabilistic(for similar models see the references inliandliu[15]). Inthispaperweconsidermovingaveragesasmethodsofdemandandlead timeforecastingandwefindanexactformofthebullwhipeffectmeasure related to the prediction of lead times and demands. More precisely we investigate a model where: a)asupplychaincontainstwostagesandconsistsofaretailerwhoreceives client demands and a supplier(customers retailer supplier (manufacturer)); b) customer demands constitute an iid sequence; c) lead times between the supplier and the retailer constitute an iid sequence; d)theretailerusestheorder-up-tolevelpolicytomakeanordertothe supplier; e) the retailer predicts the future values of demands and the future value ofleadtimesbasedonthesimplemovingaveragemethodusingpast observations that is we propose the following lead time demand forecast D L t = 4 L t 1 i=0 D t+i,

where L t istheforecastforanextleadtimeoftheordermadeatthe beginningofaperiod tand D t+i denotestheforecastforademand fortheperiod t+iatthebeginningofaperiod t. Thecrucialpointofourapproachisthelastsubpointe)anddiffersfrom thepreviousapproaches.namelyintheworkofducetal.[9]theleadtime demand forecast is defined as follows L Dt L t 1 = D t+i, i=0 where L t isthenextleadtimeatthebeginningofatime t. Thevalue of L t theretailerdoesnotknowatthebeginningofthetime t whenhe makesanordertothesupplier.thismeansthatthelastleadtimedemand forecastingisnotfeasibleinpractice. ThepaperofKimetal. [18]also investigates a stochastic lead time in supply chains and proposes lead time demand forecasting. More precisely the simply moving average method for leadtimedemandisproposedthatis D L t = 1 p p Dt j L, (1) j=1 where pisthedelayparameterofthepredictionand D L t j istheprevious knownleadtimedemandoftheordermadeatthebeginningofthetime t j.thisapproachispracticallyfeasible.letusnoticethat L t j 1 Dt j L = i=0 D t j+i, (2) where L t j isaleadtimeofanordermadeatthebeginningofthetime t j and D t j+i isthedemandfromtheperiod t j+i.combining(1)and(2) wegetadoublesumandwecannotexchangethesumsbecause L t j are different(compare it with Kim et al.[18]). In our approach we show that the bullwhip effect measure contains two summands depending on lead time forecasting. These terms amplify the valueofthebullwhipeffectmeasureandaretheevidencethatleadtime estimation in itself is another cause of the bullwhip effect. 5

2 Supply chain model Wewillmodelasupplychainwithtwostagesthatisoneretailerandonesupplier.Inourapproachtotheproblemofleadtimeforecastingweassumethat theretailerobservesdemands D t ofhiscostumers(usually tdenotesatime periodand D t isademandduringaperiodofthesamelength).morepreciselywewillassumethat {D t } t= constitutesasequenceofindependent identicallydistributedrandomvariableswith IED t = µ D and VarD t = σd 2 andagenericrandomvariablefordemandswillbedenotedby D.Similarly leadtimesareintroducedthatis L t istheleadtimeforanorderplacedby theretailertothesupplieratthebeginningoftheperiod t.randomvariablesofleadtimes {L t } t= areindependentidenticallydistributedwith IEL t = µ L and VarL t = σl 2 andagenericrandomvariableforleadtimes wewilldenoteby L.Letusnotethatwedonotimposeanyassumptionson thedistributionsof Dand L.Weassumeonlythattheirsecondmoments arefinite.thesequences {D t } t= and {L t} t= areindependentofeach other. Theleadtimedemandatthebeginningofaperiod tisdefinedas follows D L t = D t +D t+1 +...+D t+lt 1 = L t 1 i=0 D t+i. (3) Thisvalueisnotknownfortheretaileratthebeginningofaperiod tbut heneedstoforecastitsvaluetomakeanordertothesupplier.thenatural waytodothisistopredictdemandsandleadtimes. If D t+i denotesthe forecastforademandfortheperiod t + i atthebeginningofaperiod t (thatisafter i+1periods, i = 0,1,...)and F D t 1 = σ(d t 1,D t 2,...)is thesigmaalgebrageneratedbythedemandsuptoatime t 1then D t+i F D t 1, whichmeansthattheforecastatthebeginningoftheperiod tforaperiod t+iisafunctionofthepreviousknowndemands {D t 1,D t 2,...}.Similarly sinceleadtimesarerandomtheretailerneedstopredicttheirvaluesforthe nextperiodstomakeanorder.let F L t 1 = σ(l t 1,L t 2,...)bethesigma algebrageneratedbyleadtimesuptoatime t 1.Thusif L t istheforecast foranextleadtimeatthebeginningofaperiod tthengenerally L t F L t 1, 6

whichmeansthattheforecastforaperiod tisafunctionoftheprevious knownleadtimes {L t 1,L t 2,...}.Thustheretailermakinganordertoa supplier puts the following forecast for a lead time demand as follows D L t = L t 1 i=0 D t+i. (4) Employingthemovingaverageforecastmethodwiththelength n 1for demand forecasting we get D t+j = 1 n D t i, (5) n where j = 0,1,...and D t i i = 1,2,...,naredemandswhichhavebeen observedbytheretailertillthebeginningofaperiod t.similarly,theretailer predicts a lead time. Precisely, using the moving average forecast method withthelength m 1forleadtimeforecastingweobtain L t = 1 m L t i, (6) m where L t i i = 1,2,...,mareleadtimeswhichhavebeenobservedbythe retailertillthebeginningofaperiod t. Ifwewanttobemoreprecisewe needtoassumethatthedistributionofleadissuchthat L t M where M > 0 thatisleadtimesareboundedby M. Thisweassumeto avoidthesituationthatforexampletheleadtime L t 1 isnotknownatthe beginningofthetime twhenwemakeanorder.thenleadtimeforecasting is the following L t = 1 m L t M i (7) m thatiswegetbackatleast M periods. Forsimplicitywewilluseinour calculation the lead time forecast given in(6) because one can see slightly modifyingtheproofofth. 1thatthebullwhipeffectmeasureisthesame under assumption that lead times are bounded and applying the lead time 7

forecastgivenin(7).thusbyeq.(4),(5)and(6)wegettheforecastfora lead time demand as follows Dt L = L t D t = 1 mn m n L t i D t i. (8) Wehavetoindicatethatasimilarideato(8)appearedinChatfieldatel.[5] but there the bullwhip effect measure is simulated without showing the realtion between lead time forecasting and the bullwhip effect. We can employ theleadtimeforecast(7)to(8)butaswementionedthisdoesnotaffect the bullwhip effect measure. Moreover in our model the retailer applies a basestockpolicythatisasimpleorder-up-tolevelinventorypolicy.let S t betheinventorypositionatthebeginningofaperiod t(lateranorderis placed).iftheorder-up-tolevelpolicyisemployedthen S t isdeterminedin the following way S t = D L t +z σ t, (9) where σ t 2 = Var(D L t D L t ) isthevarianceoftheforecasterrorfortheleadtimedemandand z isthe normal z-score that specifies the probability that demand is fulfilled by the on-handinventoryanditcanbefoundbasedonagivenservicelevel. In somearticles σ t 2 isdefinedmorepracticallythatisinsteadofvarianceitis takentheempiricalvarianceof D L t D L t.thiscomplicatescalculationsvery muchbutwemustmentionthattheestimationof σ t 2 increasesthesizeof the bullwhip effect. These two approaches coincide if z=0. Thus the order quantity q t placedatthebeginningofaperiod tis q t = S t S t 1 +D t 1. (10) Ourmainpurposeistofind Varq t andthentocalculatethefollowingbullwhip effect measure BM = Varq t VarD t. 8

Proposition 1 The variance of the forecast error for the lead time demand doesnotdependon tandisasfollows σ t 2 = Var(D L t D L t ) = µ L σ 2 D + σ2 L µ2 D (m+1) m + µ2 L σ2 D n + σ2 L σ2 D mn. Proof: Bytheeq. (3)and(8)andassumingindependence,wegetthat IED L t = IE D L t = µ L µ D and Var(Dt L D t L ) = IE(D t L D t L ) 2 ( Lt 1 ) 2 = IE D t+i + 1 ( m ) 2 i=0 m 2 n 2IE L t i IE ( Lt 1 ) ( 1 m 2IE D t+i i=0 m IE ( n L t i ) 1 n IE ( n ) 2 D t i ) D t i = IELIED 2 +IE(L(L 1))(IED) 2 + 1 m 2 n 2[mIEL2 +m(m 1)(IEL) 2 ][nied 2 +n(n 1)(IED) 2 ] 2(IELIED) 2 = µ L (σ 2 D +µ 2 D)+(σ 2 L +µ 2 L µ L )µ 2 D + 1 mn (σ2 L +mµ2 L )(σ2 D +nµ2 D ) 2µ2 L µ2 D = µ L σ 2 D + σ2 L µ2 D (m+1) m + µ2 L σ2 D n + σ2 L σ2 D mn which finishes the proof. Sincethevarianceoftheforecasterrorfortheleadtimedemandisindependentof twehavefromtheeq.(9)and(10) q t = D L t D L t 1 +D t 1 whichpermitstocalculatethevarianceof q t. Proposition2Thevarianceofanorderquantityinaperiod tisgivenas Varq t = 2σ2 L σ2 D (m+n 1) m 2 n 2 + 2σ2 L µ2 D m 2 + 2µ2 L σ2 D n 2 + 2µ Lσ 2 D n +σ 2 D. 9

Proof: Letusnotethat Dt 1 L = 1 mn m n L t 1 i D t 1 i m+1 n+1 L t i = 1 D t i mn i=2 i=2 = 1 ( m )( n ) L t i +L t m 1 L t 1 D t i +D t n 1 D t 1 mn = D t L + 1 m mn (D t n 1 D t 1 ) L t i + 1 n mn (L t m 1 L t 1 ) D t i + 1 mn (L t m 1 L t 1 )(D t n 1 D t 1 ). Thusweget q t = 1 m mn (D t n 1 D t 1 ) L t i 1 n mn (L t m 1 L t 1 ) D t i 1 mn (L t m 1 L t 1 )(D t n 1 D t 1 )+D t 1. Byindependenceitiseasytonoticethat IEq t = µ D.Soletuscomputethe secondmomentof q t IEq t 2 = = 1 m m 2 n 2IE(D t n 1 D t 1 ) 2 IE( L t i ) 2 + 1 m 2 n 2IE(L t m 1 L t 1 ) 2 IE( n D t i ) 2 + 1 m 2 n 2IE(L t m 1 L t 1 ) 2 IE(D t n 1 D t 1 ) 2 +IED t 1 2 + 2 m n m 2 n 2IE[(L t m 1 L t 1 ) L t i ]IE[(D t n 1 D t 1 ) D t i ] + 2 m m 2 n 2IE(D t n 1 D t 1 ) 2 IE[(L t m 1 L t 1 ) L t i ] 2 m mn IE[D t 1(D t n 1 D t 1 )]IE( L t i ) 10

+ 2 m 2 n 2IE[(L t m 1 L t 1 ) 2 IE[(D t n 1 D t 1 ) n D t i ) 2 mn IE(L t m 1 L t 1 )IE[D t 1 (D t n 1 D t 1 )] 2 mn IE(L t m 1 L t 1 )IE(D t 1 n D t i ] = 2σ2 D m 2 n 2[m(σ2 L +µ 2 L)+m(m 1)µ 2 L]+ 2σ2 L m 2 n 2[n(σ2 D +µ 2 D)+n(n 1)µ 2 D] + 4σ2 L σ2 D m 2 n 2 +σ2 D +µ 2 D + 2 m 2 n 2(µ2 L σ 2 L µ 2 L)(µ 2 D σ 2 D µ 2 D) + 4σ2 D m 2 n 2(µ2 L σ2 L µ2 L ) 2 mn (µ2 D σ2 D µ2 D )mµ L + 4σ2 L m 2 n 2(µ2 D σ2 D µ2 D ) 0 0 = 2σ2 L σ2 D (m+n 1) + 2σ2 L µ2 D + 2µ2 L σ2 D + 2µ LσD 2 +σ m 2 n 2 m 2 n 2 D 2 n +µ2 D which gives the thesis. Thuswecanderivetheexactformofthebullwhipeffectmeasure. Theorem1Themeasureofthebullwhipeffecthasthefollowingform BM = Varq t = 2σ2 L (m+n 1) + 2σ2 L µ2 D VarD t m 2 n 2 m 2 σd 2 + 2µ2 L n 2 + 2µ L n +1. Remark1Wegetthesameformulaifweemploytheleadtimeforecast(7) under assumption that lead times are bounded. Letusanalyzetheformula.Thefirstsummandintheformulaincludesthe impact of the forecast of lead times and demands. The second summand showstheinfluenceofthepredictionofleadtimes. Thethirdandfourth ones give the amplification of the variance by demand forecasting. The effect isverylarge(seethenextsectionandthetablesbelow)ifwetake m = 1 thatisinthecaseiftheforecastofanextleadtimeisbasedononelast observationoftheleadtimethenweget BM = Varq t VarD t = 2σ2 L n + 2σ2 L µ2 D σ 2 D = 2σ2 L µ2 D σ 2 D 11 + 2µ2 L n 2 + 2µ L n +1 + 2µ2 L n + 2(µ L +σl 2) +1. 2 n

Ifleadtimesaredeterministicthatis L t = L = const.thenthebullwhip effect is described by BM = Varq t VarD t = 2L2 n 2 + 2L n +1, whichisconsistentwiththeresultofchenetal.[6].weshouldnoticethat Ducetal.[9]alsoobtainedtheresultofChenetal.[6]inaspecialcaseand asanexactvalueofthebullwhipeffect(notalowerbound).chenetal.[6] getthisasalowerboundbecausetheydefinetheerror σ t astheempirical varianceof D L t D L t. Now we investigate what happens if the number of past observations of leadtimesordemandsarelargethatisif m or n. Soifthe number of past lead times included in the forecast(the delay parameter of forecasting) goes to infinity we get lim BM = 2µ2 L m n + 2µ L 2 n +1. Thisshowsthattheimpactofthepredictionofleadtimesdisappearsifthe number of previous lead times included in the forecast is very large. Similarly ifthenumberofdemandsusedinthepredictionisgrowingtoinfinitythen lim BM = 2σ2 L µ2 D n m 2 σd 2 +1. Theeffecthasnotdisappearedanditremainsconstantiftheratio µ 2 D /σ2 D doesnotchangeanditislinearwithrespectto σ 2 L.Moreoverthistermcan beveryharmfulif missmall(seethenextsectionandthetablesbelow). 3 Numericalexamples We will numerically investigate the measure of the bullwhip effect. Especially wewillconsidereveryterminformulaofth.1.thusletusput BM 1 = 2σ2 L(m+n 1) m 2 n 2, BM 2 = 2σ2 Lµ 2 D m 2 σ 2 D, BM 3 = 2µ2 L n 2 + 2µ L n. (11) 12

In the tables below we have investigated the dependence of the bullwhip effectonthevaluesof m(thenumberofpastleadtimesusedinforecasting) foragivenvalueof n = 5,10,20,30 (thenumberofpastdemandsused inforecasting), σ D /µ D = 0.5 (coefficientofdemandvariation), µ L = 3 (expectedvalueofleadtimes)and σ L = 2(standarddeviationofleadtimes). The tables show the impact of lead time forecasting on the bullwhip effect. Itisevidentthatforsmall m(e.g. m = 3or5)theterms BM 1 and BM 2 contributeverymuchtothebullwhipeffectandwhen mislargetheimpact of lead time forecasting on the bullwhip effect almost disappears but the effect remains by demand forecasting. For example for n = 5 if m changes from3to50thebullwhipeffectmeasurevariesfrom6.72444to2.93971(see Tab.1).Thismeansthatleadtimeforecastingcanreducetheeffectmore thantwiceasmuch.similarlye.g.intab.4if mchangesfrom3to50the bullwhip effect measure varies from 4.80716 to 1.23308 which indicates that thereductionintheeffectcanbealmostfourtimesasmuch.moreoverthe effectisverybigif m = 1. Thisfollowsfromthefactthattheforecastis basedonthelastknownvalueoftheleadtimeandtheenvironmentis very random because we assume that lead times are mutually independent. Wehavealsovisualizedthebullwhipeffectmeasureasafunctionoftwo variables.infig.1themeasureofthebullwhipeffectasafunctionof mand σ L hasbeenplottedwhere mchangesbetween20and40, σ L [0.5,6]and n = 10, σ D /µ D = 0.5and µ L = 3.Themeasureofthebullwhipeffectasa functionof m = 5,6,...,40and µ L [1,10]for n = 10, σ D /µ D = 0.5and σ L = 3isshowninFig.2.Fig.3showsthemeasureofthebullwhipeffect dependingon n = 20,21,...,40and σ L [0.5,6]for m = 10, σ D /µ D = 0.5 and µ L = 3.SimilarlyFig.4presentsthebullwhipeffectasafunctionof mand nwheretheirvalueschangebetween20and40for σ D /µ D = 0.5, µ L = 3 and σ L = 3. InFig. 5wevisualizethemeasureofthebullwhip effectdependingon n = 20,21,...40 and µ L [1,10] where m = 10, σ D /µ D = 0.5and σ L = 3. 13

4 Conclusions and further research opportunities Inthispaperwehaveinvestigatedtheimpactofleadtimeforecastingonthe variance amplification in a simple two-stage supply chain with one supplier and one retailer, who employs the base stock policy for replenishment and the moving averages method for lead time and demand forecasting. The exact form of the bullwhip effect measure indicates that lead time forecasting is a crucially contributing factor to the effect. The forecast of lead times gives two newsummands BM 1 and BM 2 (seeeq.(11))inthebullwhipeffectmeasure which substantially increase the value of the effect. These two summands are linearasafunctionofleadtimevariance σ 2 L andintensifytheeffectbythe increaseof σ 2 L.Thenextfactorcausedbyleadtimeforecastingisthelength ofthesampleofleadtimesusedintheforecastthatisthevalue m(seeth. 1). If this value increases and goes to infinity then the variance amplification decreases and the impact of lead time forecasting disappears. We should also notethattheterm BM 1 isoforder 1/mforlarge mthatis O(1/m)and theterm BM 2 isoforder 1/m 2 forlarge mthatis O(1/m 2 )whichmeans thatthesummand BM 1 hasabiggerinfluenceontheeffectforlarge m.it isinterestingthattheterm BM 1 canbeneglectedifthelengthofdemand observations applied in demand forecasting that is n will be large because thesummand BM 1 isalso O(1/n). Summarizingweoughttostatethat lead time forecasting is in fact a critically contributing factor to the bullwhip effectanditsimpactcannotbeomittedinthedesignandmanagementof supplychains. Itisalsoworthnotingthattheeffectstemsfromtheneed toestimatetheleadtime,anddoesnotonlydependontheexpectationand variance of the lead time. The future research opportunities are widespread and necessary for the development of the supply chain management. In the further approaches to lead time forecasting problem we need to investigate other structures than iid of lead times and demands. Even if we consider more complicated structure of demands for example autoregressive-moving average leaving iid structure of lead times then this will complicate derivations of the bullwhip effect measure to a significant degree. Other opportunities lie in different forecasting 14

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Table1: Themeasureofthebullwhipeffectfor n = 5, σ D /µ D = 0.5, µ L = 3and σ L = 2(BM 3 = 1.920). m BM 1 BM 2 BM 1 1.60000 32.00000 36.52000 3 0.24888 3.55555 6.72444 5 0.11520 1.28000 4.31520 10 0.04480 0.32000 3.28480 15 0.02702 0.14222 3.08924 20 0.01920 0.08000 3.01920 25 0.01484 0.05120 2.98604 30 0.01208 0.03555 2.96764 35 0.01018 0.02612 2.95631 40 0.00880 0.02000 2.94880 45 0.00774 0.01580 2.94354 50 0.00691 0.01280 2.93971 Table2: Themeasureofthebullwhipeffectfor n = 10, σ D /µ D = 0.5, µ L = 3and σ L = 2(BM 3 = 0.780). m BM 1 BM 2 BM 1 0.80000 32.00000 34.58000 3 0.10666 3.55555 5.44222 5 0.04480 1.28000 3.10480 10 0.01520 0.32000 2.11520 15 0.00853 0.14222 1.93075 20 0.00580 0.08000 1.86580 25 0.00435 0.05120 1.83555 30 0.00346 0.03555 1.81902 35 0.00287 0.02612 1.80899 40 0.00245 0.02000 1.80245 45 0.00213 0.01580 1.79793 50 0.00188 0.01280 1.79468 18

Table3: Themeasureofthebullwhipeffectfor n = 20, σ D /µ D = 0.5, µ L = 3and σ L = 2(BM 3 = 0.345). m BM 1 BM 2 BM 1 0.40000 32.00000 33.74500 3 0.04888 3.55555 4.94944 5 0.01920 1.28000 2.64420 10 0.00580 0.32000 1.67080 15 0.00302 0.14222 1.49024 20 0.00195 0.08000 1.42695 25 0.00140 0.05120 1.39760 30 0.00108 0.03555 1.38164 35 0.00088 0.02612 1.37200 40 0.00073 0.02000 1.36573 45 0.00063 0.01580 1.36143 50 0.00055 0.01280 1.35835 Table4: Themeasureofthebullwhipeffectfor n = 30, σ D /µ D = 0.5, µ L = 3and σ L = 2(BM 3 = 0.220). m BM 1 BM 2 BM 1 0.26666 32.00000 33.48666 3 0.03160 3.55555 4.80716 5 0.01208 1.28000 2.51208 10 0.00346 0.32000 1.54346 15 0.00173 0.14222 1.36396 20 0.00108 0.08000 1.30108 25 0.00076 0.05120 1.27196 30 0.00058 0.03555 1.25613 35 0.00046 0.02612 1.24658 40 0.00038 0.02000 1.24038 45 0.00032 0.01580 1.23612 50 0.00028 0.01280 1.23308 19

2.6 2.4 2.2 BM 2 1.8 1.6 6 4 σ L 2 0 20 25 m 30 35 40 Figure1:Theplotofthebullwhipeffectmeasureasafunctionof mand σ L where n = 10, σ D /µ D = 0.5and µ L = 3. 20

BM 8 7 6 5 4 3 2 1 10 5 µ L 0 0 10 m 20 30 40 Figure2:Theplotofthebullwhipeffectmeasureasafunctionof mand µ L where n = 10, σ D /µ D = 0.5and σ L = 3. 21

4.5 4 3.5 BM 3 2.5 2 1.5 1 6 4 σ L 2 0 20 25 30 n 35 40 Figure3:Theplotofthebullwhipeffectmeasureasafunctionof nand σ L where m = 10, σ D /µ D = 0.5and µ L = 3. 22

1.6 1.5 BM 1.4 1.3 20 25 30 n 35 40 20 25 m 30 35 40 Figure4:Theplotofthebullwhipeffectmeasureasafunctionof mand n where σ D /µ D = 0.5, µ L = 3and σ L = 3. 23

3.5 3 BM 2.5 2 1.5 10 5 µ L 0 20 25 n 30 35 40 Figure5:Theplotofthebullwhipeffectmeasureasafunctionof nand µ L where m = 10, σ D /µ D = 0.5and σ L = 3. 24