Bullwhip and batching: An exploration

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Int. J. Production Economics 104 (2006) 408–418

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Bullwhip and batching: An exploration

Andrew Potter�, Stephen M. Disney

Logistics Systems Dynamics Group, Cardiff Business School, Cardiff University, Aberconway Building, Colum Drive,

Cardiff, CF10 3EU, Wales, UK

Received 16 April 2004; accepted 21 October 2004

Available online 26 February 2005

Abstract

The rounding of orders to achieve a batch size is recognised as a source of the bullwhip problem within supply chains.

While it is often advocated that batch sizes should be reduced as much as possible, there has been limited investigation

into the impact of batching on bullwhip. Here we consider scenarios where orders are placed only in multiples of a fixed

batch size, for both deterministic and stochastic demand rates. We derive a closed form expression for bullwhip when

demand is deterministic. This is validated through a simple model of a production control system. An expression for

bullwhip in a ‘‘pass on orders’’ scenario with stochastic demand is also derived and validated. Using simulation, we

show the impact of changing batch size on bullwhip in a production control system. Our results show that a manager

may achieve economies through batching while minimising the impact on bullwhip through the careful selection of the

batch size.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Bullwhip; Batching; Supply chains

1. Introduction

One of the main causes of additional costswithin supply chains is the bullwhip effect (Leeet al., 1997). This occurs when the variance oforders placed is distorted along the supply chain.Often, but not always, this distortion increases the

e front matter r 2005 Elsevier B.V. All rights reserve

e.2004.10.018

ng author. Tel.: +440 29 2087 6083;

87 4301.

ss: [email protected] (A. Potter).

variance. It has been recognised that there are fourmain causes of bullwhip:

�

d.

Demand signal processing—where amplificationis introduced as a result of companies res-ponding to feedback loops and time delays(Forrester, 1961).
� Order batching—where it is more economic for
demand to be aggregated to obtain economies ineither the production or transportation system(Burbidge, 1981).
� Gaming—this arises at times when there is a
shortage in supply or deliveries are missed, and

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A. Potter, S.M. Disney / Int. J. Production Economics 104 (2006) 408–418 409

was first studied by Houlihan (1987). In order toensure that they can satisfy the incomingdemand, an echelon will over-order if it isperceived that supply will be restricted.
� Pricing—where a company varies the price on a
product in order to stimulate demand. Butman(2003) provides an illuminating discussion onthe role of price-induced dynamics in supplychains.

This paper focuses on the order batching effectwithin a supply chain.

There are two main types of batching that canoccur. Time-based (or periodic) batching occurswhen orders are placed less frequently than theyare received. There is often a rational argument forhaving this type of batching. For instance, in thegrocery sector, demand is placed upon super-markets constantly due to customer purchases.However, for replenishment to be cost effective,orders are placed upon distribution centres onlyonce or twice a day. This ensures that the best useis made of both the warehouse and the transportfleet, with only a limited impact upon bullwhipand availability. Conversely, Metters (1997) de-scribes a scenario from the automotive sectorwhere a wire harness supplier orders every fourweeks as a result of using an MRP system. Thisresults in a significant amplification effect beingpassed along the supply chain.

Fig. 1. Production and packaging order

The other form of batching is related to orderquantities. Often, this form of batching resultsfrom the use of an Economic Order Quantity(EOQ) to minimise the ordering and inventoryholding costs of the company. Alternatively, amanufacturer may request that orders placed havea minimum quantity to justify a production batch.Packaging constraints may also play a role, withgoods being shipped in certain quantities. Thisavoids the need to break up packaged consign-ments, a process that may result in productbecoming damaged or lost. Finally, ordersmay be placed in multiples of full vehicle loads(for instance, a road trailer or container) inorder to obtain economies in transportation costs(Hall, 1987).Both of these batching effects can be seen in

Fig. 1, which is based on data collected from amanufacturer of soft drinks in the UK. The graphshows how much of a particular juice product wasproduced every day for four weeks and the ordersplaced by the company on their packagingsupplier. The packaging orders are subject to bothtime and order quantity batching. As can be seen,production takes place five days a week whileorders are placed on the packaging company onlyon a Tuesday. Therefore, production figures areaggregated into weekly totals. Further, the packa-ging can only be ordered in multiples of a fixedquantity.

s from a soft drinks manufacturer.

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A. Potter, S.M. Disney / Int. J. Production Economics 104 (2006) 408–418410

In order to mitigate the negative impacts ofbatching, it is advocated that batch sizes should bereduced as much as possible. However, there hasbeen little academic study into how varying thebatch size can affect the level of bullwhipgenerated by a replenishment system. The aim ofthis paper is to contribute to this endeavour. Theinsights provided will enable managers to reducethe negative supply chain effects of batching.

Our paper proceeds by first outlining theliterature that has been published on the relation-ship between batching and bullwhip before de-scribing the research approach taken herein. Wethen derive the variance of output orders (whichwe use as a proxy for bullwhip) with both steady-state and stochastic demand patterns. These arevalidated through a simulation model beforemanagerial insights are discussed and conclusionsdrawn.

Table 1

Variance of orders when changing the time between orders or

batch size (adapted from Cachon, 1999)

Batch size ¼ 1 Demand variability

Time between

orders

Low Medium High

1 0.21 1.00 1.41

2 0.30 1.42 2.00

4 0.44 2.00 2.84

8 0.64 2.80 4.00

16 0.80 4.00 5.60

2. Impact of batching on bullwhip

Most of the literature studying the impact ofbatching on bullwhip has advocated the minimisa-tion of batch sizes, aiming for the ultimate goal ofa ‘‘Batch of One’’ (Burbidge, 1981). There may becircumstances where this is not possible eithertechnically, economically or socially. Therefore, itis important to understand the impact of varyingthe batch size on bullwhip in order to enablemanagers to make informed decisions. However,the amount of literature on this particular area islimited.

In Riddalls and Bennett (2001), a qualitativeassessment of the impact of batch production costson bullwhip has been undertaken. The model usedincorporates both a fixed batch production costand a variable unit cost. The aim is to minimise thetotal cost. There is no constraint on the amountthat can be produced in any one time period,resulting in a variable order quantity. They findthat bullwhip levels are related to the remainder ofthe ratio between the batch size and averagedemand.

Cachon (1999) considers a scenario with bothtime and quantity batching. Multiple customersplace orders on a supplier both at fixed time

intervals and in integer multiples of a fixed orderquantity. This order quantity is also a multiple ofaverage demand. He found three approaches toreducing bullwhip within the supply chain:

�
Balancing orders—instead of all the customersplacing orders at the same time, each customerorders on a different day. Therefore, thesupplier receives the same number of ordersevery day. � Reducing the size of the fixed order quantity. � Increasing the time between customer orders.
This last result appears counter-intuitive—forinstance, Towill (1997) clearly shows the benefitsof time compression in reducing bullwhip. How-ever, Cachon used the coefficient of variation as ameasure of bullwhip, whereas we prefer to use thevariance ratio as a measure. Adapting the resultsfrom Cachon (1999), Table 1 shows that the ordervariance does increase with the time betweenorders when the batch size is 1 for low, mediumand high demand variability.Pujawan (2004) also considers the impact of

changing the time between orders on bullwhip. Heinvestigated the impact of two different lot sizingrules with non-integer values for the time betweenorders. For both rules, a cyclical order patternemerged, although the minima appear to occur atdifferent times. In one case, they occur when theremainder for the time between orders was 0.5 (forinstance, at 2.5 or 3.5), and in the other, theminima are achieved on integer values of timebetween orders.

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Holland and Sodhi (2003) aimed to quantify thelevel of bullwhip induced in a two-echelon supplychain as a result of batching in the ordering rules.The retailer is only permitted to order in multiplesof a fixed batch size. Simulation models were runfor five different batch sizes and a statisticalanalysis of the results was carried out. They foundthat the level of bullwhip across one echelon wasproportional to the square of the batch size. Partof our paper looks to extend the findings of thiswork by considering a full range of batch sizes.

Disney et al. (2003a) consider the impact oftransportation batching within a Vendor ManagedInventory (VMI) supply chain. Because inventoryis managed as a whole across the two supply chainechelons, they show it is possible to achievesimultaneously both full transport batches and areduction in bullwhip. The scenario consideredhere differs in that each echelon is responsible formanaging their inventory by placing replenishmentorders.

3. Methodology

In order to analyse the impact of batching onbullwhip, this paper primarily draws on themanipulation of mathematical equations. A simu-lation model, used to both explore behaviourinitially and to validate our analytical findings, hassupplemented the mathematical investigation. Themodel comprises a single echelon within a supplychain and is based on the Inventory and Order-

DelayTi

1

Tw

1

Forecastdemand

Target Stock Level

Target WIPLevel

Order

Ta

Fig. 2. The automated variable pipeline IOPBCS

Based Production Control System (IOBPCS)model proposed in Towill (1982). This model hasa structure similar to those found in industry(Coyle, 1977). The actual model also incorporatespipeline feedback and is illustrated in Fig. 2. Ta;Tp; T i and Tw are parameters in the model relatingto forecasting, production delay, inventory andwork in progress (WIP) respectively.The block diagram has been translated into a

spreadsheet using difference equations. This waschosen due to its simplicity and the ability tomanipulate the results into appropriate formats.Batching is introduced to the model through theorder rule, using the ROUND function in thespreadsheet program. This rounds orders eitherup or down to the nearest factor of the batch size.The initial parameter values were T a ¼ 8; Tp ¼ 2;T i ¼ 4 and Tw ¼ 8:In measuring bullwhip, there are a number

of different approaches that can be adopted.El-Beheiry et al. (2004) provide a detailed reviewof the main measures. At a simple level, a proxyfor the level of bullwhip is to consider themaximum order placed during a simulation run,as used by Riddalls and Bennett (2001). While thisprovides a qualitative idea as to the behaviour ofbullwhip, it is not necessarily suitable for findingan analytic solution. Another approach is to dividethe coefficient of variation for orders placed by thecoefficient of variation for orders received (Fran-soo and Wouters, 2000). In calculating thevariability of orders, both Cachon (1999) andPujawan (2004) have adopted this coefficient of

Production

Forecast

WIP

StockLevel

Demand

∫

∫

Tp

(APVIOBPCS) model (John et al., 1994).

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variation approach. We prefer to consider bull-whip as the quotient of the variance of ordersplaced and the variance of orders received:

Bullwhip ¼s2orders

s2demand

. (1)

In this paper, analysis of only the output variancestakes place. This is because it is assumed that thedemand signal is independent of the batch size,resulting in the demand variance being constantthroughout. As an alternative, the standarddeviation could have been used. While this wouldhave reduced the magnitude of the findings, theunderlying trends and managerial insights wouldremain unchanged.

4. Steady-state demand

Let us start by assuming demand is a constant.Numerical investigations revealed that the systemstabilises around two batch sizes—the multiplesabove and below the demand level. A typicaltime series output from the model is illustrated inFig. 3. As can be seen, the orders form a repeatingcycle with the average order level being equal tothe level of demand. The variance of orders placedon suppliers is equivalent to that for one of thecycles.

Let us quantify the variance across one of thecycles. There are c occurrences of order size m andd occurrences of order size n: The variance of theorders placed is calculated using Eq. (2). Details ofits derivation can be found in Appendix A. Notice

Fig. 3. Orders placed under steady-state demand with batching.

that we consider a single cycle as population, not asample as it is clear that as the sample sizeincreases to infinity the variance approaches thepopulation variance given by Eq. (2).

s2 ¼

PN

i¼1ðm� xiÞ2

N¼

b2cd

ðcþ dÞ2, (2)

where b is the batch size.From Eq. (2), it is possible to calculate the

variance of orders placed if the frequency withwhich the two batch sizes occur is also known.However, further investigations reveal that thefrequency of the different order sizes occurringdoes not need to be known explicitly in order todetermine order variance. This is achieved byexploiting the properties of congruences to give

s2 ¼ ðb�mod½m; b�Þmod½m; b�. (3)

We have plotted Eq. (3) in Fig. 4 to demonstratethe impact of the batch size on the outputvariance. When the batch size is less than or equalto average demand, bullwhip is minimised whenmod m; b½ � ¼ 0 (that is when the quotient of batchsize and average demand is an integer). Numericalinvestigations reveal that between these points, awaveform emerges with peaks halfway between theminima. The height of these peaks increases withbatch size. When the batch size is greater thanaverage demand, the level of bullwhip presentcontinues to rise linearly. By varying the levelof average demand, it can be shown thatthe amplitude of the peaks is also determined bythis variable.

Fig. 4. Impact of batch size on bullwhip for steady demand.

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5. Variable (stochastic) demand

While Section 4 enables the quantification of theimpact of batching on bullwhip, the results areonly valid when the demand rate is constant. Inreality, demand can be extremely variable. There-fore, we will now extend our approach tosituations where demand is variable and morethan two batch sizes may be ordered.

Let us assume that demand is a normallydistributed, stationary stochastic variable witha known mean, m; and variance, s2

D: We assumem� 4s2

D so that the probability of negativedemand is negligible. Furthermore the orderingdecision used by the supply chain player toschedule production is to simply round demandto the nearest multiple of the batch size. In such ascenario the probability of a replenishment orderof bðsþ xÞ being placed, p½bðsþ xÞ�; is given by

p½bðsþ xÞ� ¼

Z bðsþ0:5þxÞ

bðs�0:5þxÞ

e�ðt�mÞ2=2s2

Dffiffiffiffiffiffi2pp

sD

dt. (4)

Appendix A details our derivation of the variancesof the orders placed where K ¼ H is the numberbatch sizes realized:

s2 ¼b2PH

K¼2 p½bðK þ s� 1Þ�PK�1

n¼1 ðK � nÞ2p½bðnþ s� 1Þ�PK¼1

n¼0 p½bðnþ sÞ�� 2

.

(5)

It could be argued that, by using this probabilitydensity function approach, there is an infinitenumber of order sizes. However, the probability ofmany of the larger and smaller orders being placedis extremely small and they can be excluded fromthe variance calculation. Hence, only a finite

Table 2

Comparison of theoretical and simulated variance values

No. of batch

sizes

Average

demand

Standard deviation

of demand

Batch size

2 1000 300 1750

3 1000 300 900

4 1000 300 620

5 1000 300 500

6 1000 300 400

7 1000 300 350

number of multiples of the batch size need to beconsidered and Eq. (5) expands out to a closed-form expression. The first seven of these aredocumented for interested readers in Table A.1in the appendix.To demonstrate the use of Eq. (5), consider a

scenario where demand follows a normal distribu-tion with a mean of 1000 units per period, astandard deviation of 300 and a batch size of 900.Orders are received every time period and im-mediately passed on to the supplier having beenrounded to the nearest batch size. No allowance ismade for any difference between target and actualinventory levels or forecasts of future demand.Consequently, the delay between ordering anddelivery does not need to be considered in thisinstance. This has been shown by Sterman (1989)to be an effective strategy to minimise the costs ina supply chain. From a simplified version of thesimulation model, it was found that such ademand pattern produces orders of only threedifferent batch sizes: 0, 900 and 1800 units. Usinga table of normal distribution probabilities (Silveret al., 1998) or Eq. (4), it can be found that thevalues for c; d and e are 0.03362, 0.84538 and0.121, respectively. Applying the equation forthree batch sizes as given in Table A.1, the outputvariance can be calculated as 119,057. Thiscompares well against the results from 10 runs ofthe simulation model, where an average outputvariance of 119,092 was obtained, a difference of�0.03% between the theoretical and simulationresults.The other equations in Table A.1 were also

validated in this manner and the results can befound in Table 2. In each case, the simulation

Theoretical

variance

Average simulation

variance

Error (%)

684,457 686,182 �0.25

119,058 119,092 �0.03

125,202 124,096 0.89

110,965 110,314 0.59

103,581 103,035 0.53

100,095 100,897 �0.79

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results are the average of 10 runs of the model,each of 2500 time periods in length. As can beseen, there is only a very small percentage errorbetween the theoretical and actual values in eachcase. These differences can be largely attributed tothe imperfect stochastic process in the simulation,although some of the error is due to the non-negative demand assumption we have made. Morecareful construction of Eq. (4) may be able toeliminate this however.

This approach can also be used where the orderrule rounds up to the nearest batch size. All thatneeds to be changed is the upper and lower boundsof the integration in Eq. (5). In this particularinstance, they would become bðsþ xÞ andbðs� 1þ xÞ; respectively.

Obtaining solutions of a similar nature to (5) forthe full APVIOBPCS production control system inFig. 2, however, is more challenging. This isbecause the input demand signal is influenced byboth the parameter values and feedback loopswithin the model. While the mean for ordersplaced is the same as the input signal, the standarddeviation is different. The above method cantherefore only be applied if this impact is known.Currently, we do not understand the nature of thistransformation. Therefore, we have to resort tosimulation to investigate the impact of changingbatch size on bullwhip with a stochastic demandrate. As can be seen in Fig. 5, the trend is similar tothat in Fig. 4 with minima on multiples of batchsizes and peaks that increase in amplitude halfwaybetween. Also shown is the output variance fororders with no batching in the order rule. As canbe seen, the performance at the minima is

Fig. 5. Impact of batching on bullwhip with stochastic demand.

comparable (but not as good as) with the modelwith no rounding in it.

6. Sensitivity analysis

The above simulation results are based on theAPVIOBPCS model with one specific set ofparameters. It may be that results found areparticular to this combination of values. There-fore, a sensitivity analysis was conducted on thefour parameters associated with the model. In thecase of T a; T i and Tw; the values were increasedand decreased by 25% and 50% while Tp wasvaried between 1 and 5. The results of thesensitivity analysis are presented in Fig. 6.When varying T a and T i; the system behavesqualitatively in a similar manner to a normalAPVIOBPCS model, with the output variancedecreasing as the parameter increases because theyare smoothing the signals into the order calcula-tion. Equally, the results generated for Tp are notunexpected, with the output variance increasingwith the parameter value. In all three cases, whenthe batch size is greater than 720 and not close to afactor of average demand, there is little differencebetween the results. This is because the impact ofthe rounding coupled with the ratio of batch sizeto average demand negates any effect of theparameters. However, when T i ¼ 2 there is asignificant reduction in the performance of thissystem. Possibly, this is due to the system beingclose to the critical stability boundary and thereare significant oscillations in production orderrate. When varying Tw; there is a negligible impactupon the results.

7. Managerial implications

The findings of this research provide animportant insight for managers into how theycan minimise the on-costs to the supply chain dueto batching. By selecting a batch size that is amultiple of average demand, it is possible toreduce the level of bullwhip generated by such anordering policy. However, there is an incentive toopt for the smallest batch size possible. By doing

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1210864

1210864

Ta Ti23456

Tw Tp12345

170016001500140013001200110010009008007006005004003002001000

Batch Size

170016001500140013001200110010009008007006005004003002001000

Batch Size

170016001500140013001200110010009008007006005004003002001000

Batch Size

170016001500140013001200110010009008007006005004003002001000

Batch Size

28

24

20

16

12

8

4

0Out

put V

aria

nce

(in 1

00,0

00s)

28

24

20

16

12

8

4

0Out

put V

aria

nce

(in 1

00,0

00s)

28

24

20

16

12

8

4

0Out

put V

aria

nce

(in 1

00,0

00s)

28

24

20

16

12

8

4

0Out

put V

aria

nce

(in 1

00,0

00s)Ta Ti

TpTw

Fig. 6. Sensitivity analysis on the APVIOBPCS model with batching in the order rule.


so, the reactivity of the system to a long-termchange in average demand is reduced. Table 3shows the impact when average demand increasesbut the batch size remains the same. There is lessof an impact with a small batch size. This is alsohighlighted in Fig. 5 by the increasing amplitude ofthe peaks.

However, if the average demand and itsvariability is known, it may be the case that amore simplified approach to replenishment may betaken whereby goods are supplied on a regular

basis (perhaps with procurement through a stand-ing order). This eliminates the need for aninventory control system, although care wouldhave to be taken to ensure inventory drift didnot occur.

8. Conclusions

Companies often undertake batching in theirreplenishment rules in order to realise economies

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Table 3

Reactivity of bullwhip to changes in average demand

Batch size Variance when average

demand ¼ 1440 units/period

Variance when average

demand ¼ 1600 units/period

% change (%)

1440 20,095 205,980 925

720 26,414 90,296 242

480 28,111 53,660 91

360 27,607 37,049 34

288 25,286 30,226 20

240 23,522 27,400 16

Fig. A.1. Illustration of a two-batch size scenario.


of scale in production, packaging or transporta-tion. This has implications for the whole supplychain through the introduction of the bullwhipproblem. While it has been recognised that the bestway of minimising this is to aim for a batch size ofone, there has been little study of the impact whenthe batch size is varied. This paper builds on thework of Holland and Sodhi (2003) by consideringa full range of batch sizes, both greater and lessthan average demand. Additionally, theoreticalequations for the derivation of the variance levelsfor orders placed are presented and are validatedthrough the use of a simulation model. It has beenshown that bullwhip levels from batching can bereduced if the batch size is a multiple of averagedemand. Between these minima, the level ofbullwhip rises and falls in a waveform, reachinga peak at the halfway point. Using this informa-tion, it is possible for managers to reduce the levelof bullwhip they introduce into the supply chain,even if they wish to exploit batching principles.

Future work will look at expanding the applic-ability of the equations from the current pass onorders scenario to a more complex production andinventory control model where the output signalincludes an allowance for forecast demand anderrors between target and actual inventory andWIP levels. This has been considered to someextent in Section 5, but we feel a more luminoussolution is required. In addition, investigations willbe carried out to increase our understanding of thecost implications of batching for both the indivi-dual echelon and the wider supply chain. Whilebatching should provide economies in transportcosts, it is important to understand the link with

inventory holding costs that inevitably occurswhen ordering in batch quantities.

Appendix A. Derivation of the variance ratios

First consider the case when there are two sizesof orders placed. This is representative of, forexample, the case of the constant demand scenarioin Section 4. Now we know the variance is given byEq. (A.1), where m ¼ mean, xi is the realisation ofthe order at time i; and N is the size of thepopulation.

s2 ¼

PN

i¼1ðm� xiÞ2

N(A.1)

Note that the variance expression is left unchangedwhen the order of the realisations of the xi

are rearranged in time. Thus we may sort fromlowest to highest batch quantities without con-sequences, see Fig. A.1. In the case of stochasticdemand, this property suggests that if demand is

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Table A.1

Variance expressions for increasing numbers of batch sizes

No. of batch sizes Variance equation

2 s2 ¼b2ðcdÞ

ðcþdÞ2

3 s2 ¼b2ðeðdþ4cÞþcdÞ

ðcþdþeÞ2

4 s2 ¼b2ðf ðeþ4dþ9cÞþeðdþ4cÞþcdÞ

ðcþdþeþf Þ2

5 s2 ¼b2ðgðfþ4eþ9dþ16cÞþf ðeþ4dþ9cÞþeðdþ4cÞþcdÞ

ðcþdþeþfþgÞ2

6 s2 ¼b2 hðgþ4fþ9eþ16dþ25cÞþgðfþ4eþ9dþ16cÞþf ðeþ4dþ9cÞþeðdþ4cÞþcdð Þ

ðcþdþeþfþgþhÞ2

7 s2 ¼b2 jðhþ4gþ9fþ16eþ25dþ36cÞþhðgþ4fþ9eþ16dþ25cÞþgðfþ4eþ9dþ16cÞþf ðeþ4dþ9cÞþeðdþ4cÞþcdð Þ

ðcþdþeþfþgþhþjÞ2

Notation c ¼ p½bs�; d ¼ p½bðsþ 1Þ�; e ¼ p½bðsþ 2Þ�; f ¼ p½bðsþ 3Þ�;

g ¼ p½bðsþ 4Þ�; h ¼ p½bðsþ 5Þ�; j ¼ p½bðsþ 6Þ�

Fig. A.2. Illustration of a three-batch size scenario.


auto-correlated, this auto-correlation will have noimpact on the order variance when batching isused. This is in contrast to the case when orderrates are allowed to roam freely, where thestructure of the stochastic demand process playsa very significant and complex role: for examplesee Disney et al. (2003b). However, to date, wehave yet to investigate this order rate closely.

Inspection will show that the mean demand isequal to

m ¼cmþ dn

cþ d¼

cbsþ dbðsþ 1Þ

cþ d

¼ c%bsþ d%bðsþ 1Þ, ðA:2Þ

where c is the number of occurrences that an orderquantity of m is realised and d is the number ofoccurrences that an order quantity of n is realised.Denote the batch size b then m ¼ bs and n ¼

bðsþ 1Þ: Note that we have also defined c and d asa percentage (C% and d% respectively). In this casethe denominator is unity and can be omitted.

Now the variance expression (A.1) may beexpressed in terms of b; c; d and s as follows.When an order of m ¼ bs is placed,

m� x ¼bd

cþ d, (A.3)

and when an order n ¼ bðsþ 1Þ is placed,

m� x ¼bd

cþ d� b. (A.4)

Using (A.2)–(A.4) in Eq. (A.1), we obtain

s2 ¼cðbd=ðcþ dÞÞ2 þ dðbd=ðcþ dÞ � bÞ2

cþ d

¼b2cd

ðcþ dÞ2. ðA:5Þ

Now let us consider a scenario with threeinstances of the order size (see Fig. A.2). Themean demand in this case is seen to be

m ¼cbsþ dbðsþ 1Þ þ ebðsþ 2Þ

cþ d þ e

¼ c%bsþ d%bðsþ 1Þ þ e%bðsþ 2Þ, ðA:6Þ

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and we may reconstruct the variance expressionusing the same procedure as before to produce:

s2 ¼cðbðd þ 2eÞ=ðcþ d þ eÞÞ2 þ dðbðd þ 2eÞ=ðcþ d þ eÞ � bÞ2 þ eðbðd þ 2eÞ=ðcþ d þ eÞ � 2bÞ2

cþ d þ e

¼b2ðeðd þ 4cÞ þ cdÞ

ðcþ d þ eÞ. ðA:7Þ

Notice there are now 3 rather than 2 terms in thenumerator and denominator. We may repeat thisprocess indefinitely, for increasing instances ofbatch sizes. Table A.1 shows the equations for upto seven batch sizes. Using this, we may thenobtain a general variance expression inductively asshown in Eq. (A.8):

s2 ¼b2PH

K¼2 p½bðK þ s� 1Þ�PK�1

n¼1 ðK � nÞ2p½bðnþ s� 1Þ�PK�1

n¼0 p½bðnþ sÞ�� 2

(A.8)

where p½:� is the number of occurrences (orpercentages) of the order quantity bðsþ xÞ beingplaced in the time series, and K ¼ H is the numberof batch sizes realised in the time series.

References

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Bullwhip and batching: An exploration

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