MATHEMATICAL ECONOMICS 10(17)

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1 MATHEMATICA ECONOMICS 10(17) The Publishig House of Wrocław Uiversiy of Ecoomics Wrocław 014

2 Reviewers cooperaig wih he joural Maria Balcerowicz-Szkuik, Alessadro Barbiero, Fracesco Bosello, ucio e Capiai, Mirko Cesarii, Jacek Cukrowski, Igor N. ubia, Pier Alda Ferrari, Vladimir Gazda, Alessadro e Gregorio, Fracesca Greseli, Giacarlo Mazi, Maria Małoka, Jerzy Mika, Aaol Pilawski, Agiola Pollasri, Fracesco Porro, Giuseppe Porro, Emauela Raffiei, Nadia Solaro, Tadeusz Saisz, Achille Verizzi Copy-ediig Elżbiea Macauley, Tim Macauley Proof readig Barbara Cibis Typeseig Elżbiea Szlachcic Cover desig Rober Mazurczyk Fro cover paiig: Paris Pei Po (privae collecio) Iformaio of submiig ad reviewig paper is available o he Publishig House s websies The publicaio is disribued uder he Creaive Commos Aribuio 3.0 Aribuio-NoCommercial-Noerivs CC BY-NC-N The joural is idexed a: BazEko CEJSH EBSCO This publicaio is available a: www. joural.ue.wroc.pl, Copyrigh by Wrocław Uiversiy of Ecoomics Wrocław 014 ISSN e-issn The origial versio: pried

3 TABE OF CONTENTS Sabia ekowska Corollig he effec of muliple esig i Big aa... 5 Saisław Heilper Milivariae measures of depedece based o copulas Jausz Łyko, Aoi Smoluk O maximal social preferece Zbigiew Micha The bullwhip effec i he AR(1) demad srucure Paweł Siarka Exposure a defaul. The problem of is esimaio i he coex of capial requiremes Grzegorz Siek Modelig icome o he basis of disribuio mixure Malgorzaa Szerszuowicz O he search for codiios for producio process opimizaio Adrzej Wilkowski Iere securiy

4 M A T H E M A T I C A E C O N O M I C S No. 10(17) 014 THE BUWHIP EFFECT IN THE AR(1) EMAN STRUCTURE Zbigiew Micha Absrac. I his oe we derive he famous formula of F. Che, Z. rezer, J.K. Rya ad. Simchi-evi [000a] for he bullwhip effec measure i a simple wo-sage supply chai uder he assumpio ha demads cosiue auoregressive srucure of order 1. Our approach is a lile differe ha i Che e. al [000a] ad herefore we obai he formula as a equaliy ulike Che e. al [000a], where hey have i as a lower boud. Moreover, we aalyze he bullwhip effec measure formula ad i some cases we have differe coclusios ha i Che e. al [000a]. Keywords: supply chai, bullwhip effec, order-up-o level policy, demad forecasig, AR(1) srucure. JE Classificaio: F15, F3,. OI: /me Iroducio The bullwhip effec (he Forreser effec) was recogized by Forreser [1958] i he middle of he weieh ceury. The erm bullwhip effec was coied by Procer & Gamble s maageme. This pheomeo appears i supply chais as he variace amplificaio i repleishme orders if oe moves up i a supply chai. I is cosidered harmful because of is cosequeces which are (see e.g. [Buchmeiser e al. 008]): excessive iveory ivesme, poor cusomer service level, los reveue, reduced produciviy, more difficul decisio-makig, sub-opimal rasporaio, sub-opimal producio ec. I he curre sae of research, ypical five mai causes of he bullwhip effec are cosidered (see e.g. [ee, Padmaabha, Whag 1997a; 1997b]): demad forecasig, o-zero lead ime, supply shorage, order bachig ad price flucuaio. To decrease he variace amplificaio i a supply chai (i.e. o reduce he Forreser effec) we eed o ideify all Zbigiew Micha eparme of Mahemaics ad Cybereics, Wrocław Uiversiy of Ecoomics, Komadorska Sree 118/10, Wrocław, Polad. zbigiew.micha@ue.wroc.pl

5 54 Zbigiew Micha he facors causig he bullwhip effec ad o quaify heir impac o he effec. I he rece paper of Micha ad Nielse [013], aoher cause of bullwhip is idicaed, ha is lead ime forecasig i he case of sochasic lead imes. Moreover Micha ad Nielse [013] claim ha demad sigal processig ad lead ime sigal processig are equally impora causes of he bullwhip effec. May papers have sudied he ifluece of differe mehods of demad forecasig o he Forreser effec such as simple movig average, expoeial smoohig, ad miimum-mea-squared-error forecass whe demads are idepede ideically disribued or cosiue iegraed movig-average, auoregressive process or auoregressivemovig average (see: [Graves 1999; ee, So, Tag 000; Che e. al 000a; 000b]). Few aricles have ivesigaed he impac of sochasic lead imes o he bullwhip effec (see [uc, uog, Kim 008; Kim e al. 006; o e al. 014; Micha, Nielse 013; Micha, Nielse, Nielse 013; 014; Nielse, Micha, o 014). Moreover, Micha ad Nielse [013] is he oly paper ha cosiders lead ime forecasig, ad is impac o bullwhip is expressed quaiaively. I his aricle we assume ha demads cosiue auoregressive srucure of order 1 ad lead ime is cosa. Addiioally, demads are prediced usig he simple movig average mehod. Fially we rediscover he famous formula of F. Che, Z. rezer, J.K. Rya ad. Simchi- evi [000a]. I our approach we obai i as a equaliy, ulike he resul of Che e. al [000a] where hey obai a lower boud. We discuss his differece ad aalyze he bullwhip effec measure formula.. Supply chai model wih movig average forecasig mehod We will model a supply chai wih wo sages, ha is oe reailer ad oe supplier. We will assume ha he reailer observes demads (usually deoes a umber of a ime period ha is a day or a week ad is a demad durig a period of he same legh). We will assume ha cosiues a saioary sequece of radom variables wih E = µ ad Var, ad a geeric radom variable for demads will be deoed by. More precisely, demads cosiue he firs-order auoregressive process where µ > 0 ad ρ < 1 ad 1, (1) is a sequece of idepede ideically disribued radom variables such ha E = 0 ad Var = σ. I is easy o oice ha

6 The bullwhip effec i he AR(1) demad srucure 55 E = µ = ad Var 1 1. Moreover we assume ha he lead imes are deermiisic ad are equal o. This is raher a urealisic assumpio i may supply chais bu i is accepable i he firs approach o he bullwhip problem. Thus he lead ime demad a he begiig of period is defied as follows i i0.... () The lead ime demad is he quaiy which is ecessary o place a order. Usig eq. (1) oe ca show k 1 k k i k i i0 1. (3) If i deoes he forecas for a demad durig he period + i a he begiig of period (ha is afer i periods), he employig he movig average forecasig mehod wih he predicio horizo we ge 1 i k 1 (4) where k k = 1,,... are demads which have bee observed by he reailer uil he begiig of period. Thus i is aural o defie he forecas for a lead ime demad as follows k 1 1 i k i0 k1. (5) Moreover, i our model he reailer applies a base sock policy ha is a simple order-up-o-level iveory policy. e S be he iveory posiio a he begiig of period (laer a order is placed). If he order-up-o-level policy is employed he S is deermied i he followig way where S z, (6) Var (7) is he variace of he forecas error for he lead ime demad ad z is he ormal z-score ha specifies he probabiliy ha demad is fulfilled by he o-had iveory ad i ca be foud based o a give service level. I

7 56 Zbigiew Micha pracice he erm z has o proec from large deviaios of demads abou. The defiiio of i his paper differs from he defiiio i Che e. al [000a]. They use more pracical ad feasible defiiio of lead ime deviaio, ha is i i C,,, (8) 1 k1 which is a esimaio of (7) ad C(, ρ, ) is a cosa depedig o, ρ, (see [Che e. al 000a]). I pracice we do o avoid he esimaio of (7) bu for heoreical reasos we use defied i (7) because i simplifies calculaios ad is equivale o he assumpio z = 0, ha is he safey loadig is zero. Therefore we ge a equaliy ulike Che e. al [000a] (see Theorem 1 below) where hey obai i as a lower boud (for z = 0 heir boud is igh). This meas ha he esimaio of lead ime demad deviaio icreases he order variace. Thus usig he order-up-o-level policy, he order quaiy q placed a he begiig of period is q = S S (9) Our mai purpose is o fid Varq ad he o calculae he followig bullwhip effec measure q / q BM Var E. Var / E Proposiio 1. The variace of he forecas error for he lead ime demad defied i (7) does o deped o. This easily follows from he saioariy of he sequece. Sice he variace of he forecas error for he lead ime demad is idepede of we have from equaios (6) ad (9) ha q 1 1 k 1k k1 k1 1 k 1 k 0 1k , where i he las equaliy we used formula (3). (10)

8 The bullwhip effec i he AR(1) demad srucure 57 I is easy o oice ha E = Eq, hus we ca derive he exac form of he bullwhip effec measure. Theorem 1. The measure of he bullwhip effec has he followig form Varq BM 1 1. (11) Var Proof. By formula (10) we ge Varq 1 k Var k0 1 Var Var 1 1 Var, which gives he hesis of he heorem. Formula (11) is he well-kow resul of Che e. al [000a]. To our kowledge i is he firs resul which quaifies he bullwhip effec as he quoie of variaces. Var 3. Aalysis of he bullwhip effec measure ad coclusios We ge he famous resul of Che e al. [000a] bu i heir paper he above formula is a lower boud for he bullwhip effec measure. This is caused by he esimaio (8) of ad cofirms ha he forecasig (esimaio) of ay parameer whe placig a order is he mai cause of he variace amplificaio. We eed o explai ha he lack of forecasig, ha is wihou usig ay mehod (hi or miss), is also a forecas bu he wors oe ad his should amplify he order variace mos of all. Thus we ca claim ha he esimaio of he safey loadig is also a cause of he bullwhip effec ad he he effec is a leas as i (11). Ulike he aalysis i Che e al. [000a], we have differe coclusios for eve ad odd i he formula. More precisely, if is large he erm ρ does o coribue much i he bullwhip effec measure eve for ρ close o oe. If is small ad ρ is close o egaive oe i does maer if is

9 58 Zbigiew Micha odd or eve. We ca explai his i he followig way, ha is a eve umber of demads employed i he predicio o place a order dampes order variabiliy because hey are egaively correlaed ad for small eve hey reduce pairwise. Thus we ca coclude ha for small ad egaively correlaed demads, a eve umber of demads ake i he simple movig average predicio reduces he bullwhip effec, ha is he effec has o be smaller ha i he case of a odd umber of demads ake i he predicio of demad. Alhough his reducio of bullwhip ca cause he icrease of he sock level variabiliy which yields a poor cusomer service level. I seems ha he bes circumsaces o maage he bullwhip problem is he case whe demads are posiively correlaed ad he correlaio is close o oe ρ 1 (bu ρ < 1 ), he he effec will be egligible if we simply ake he las demad o predic demads ( = 1). Similarly, we ca proceed if demads are egaively correlaed ad he correlaio is close o egaive oe ρ 1 (ρ > 1). The he bes choice for he predicio horizo of demads is =, bu i his siuaio as we oiced before we ca observe aoher isufficiecy i supply chais ha is he icrease of he sock level variabiliy. Aoher ieresig case we ca observe whe is large, which meas ha he umber of pas demad observaios ake io he simple average mehod coverges o ifiiy. I his case he effec disappears which meas ha he bullwhip effec measure goes o oe. I he work of Micha ad Nielse [013], sochasic lead imes are cosidered which are prediced usig he simple movig average mehod wih he delay parameer m of predicio. The large does o elimiae he effec because lead ime forecasig sill causes he effec (see Theorem 1 of Micha ad Nielse [013]). The bullwhip effec is also elimiaed if = 0, ha is he lead ime is zero (see eq. (11)). However his is obvious ad iuiive because he orders are realized immediaely ad demads are fulfilled o-lie ad o sorehouse is ecessary. Obviously his is o possible i pracice. I he aricle of Micha ad Nielse (013), he demads are idepede ad ideically disribued ad lead imes are also idepede ad ideically disribued. The formulas i Micha ad Nielse [013] ad (11), coicide if he lead ime is deermiisic (i [Micha, Nielse 013] ad ρ = 0 i (11).

10 The bullwhip effec i he AR(1) demad srucure 59 Ackowledgeme The auhor wishes o hak he Naioal Cere of Sciece for is suppor uder he gra 01/07/B/HS4/0070. Refereces Buchmeiser B., Pavlijek J., Palcic I., Polajar A. (008). Bullwhip effec problem i supply chais. Advaces i Producio Egieerig & Maageme 3(1). Pp Che F., rezer Z., Rya J.K., Simchi-evi. (000a). Quaifyig he bullwhip effec i a simple supply chai. Maageme Sciece 46(3). Pp Che F., rezer Z., Rya J.K., Simchi-evi. (000b). The impac of expoeial smoohig forecass o he bullwhip effec. Naval Research ogisics 47(4). Pp o N.A.., Nielse P., Micha Z., Nielse I.E. (014). Quaifyig he Bullwhip Effec of Muli-echelo Sysem wih Sochasic epede ead Time. Advaces i Producio Maageme Sysems: Iovaive ad Kowledge-Based Producio Maageme i a Global-ocal World. Pp uc T.T.H., uog H.T., Kim Y.. (008). A measure of he bullwhip effec i supply chais wih sochasic lead ime. The Ieraioal Joural of Advaced Maufacurig Techology 38(11-1). Pp Forreser J.W. (1958). Idusrial dyamics a major break-hrough for decisiomakig. Harvard Busiess Review 36(4). Pp Graves S.C. (1999). A sigle-iem iveory model for a o-saioary demad process. Maufacurig & Service Operaios Maageme 1(1). Pp Kim J.G., Chafield., Harriso T.P., Hayya J.C. (006). Quaifyig he bullwhip effec i a supply chai wih sochasic lead ime. Europea Joural of Operaioal Research 173(). Pp ee H.., Padmaabha V., Whag S. (1997a). Iformaio disorio i a supply chai: he bullwhip effec. Maageme Sciece 43(3). Pp ee H.., Padmaabha V., Whag S. (1997b). The bullwhip effec i supply chais. Sloa Maageme Review 38(3). Pp ee H.., So K.C., Tag C.S. (000). The value of iformaio sharig i a wolevel supply chai. Maageme Sciece 46(5). Pp Micha Z., Nielse P. (013). The impac of lead ime forecasig o he bullwhip effec. arxiv prepri arxiv: Micha, Z., Nielse, I.E., Nielse, P. (013). The bullwhip effec i supply chais wih sochasic lead imes. Mahemaical Ecoomics 9(16). Pp Micha Z., Nielse P., Nielse I.E. (014). The impac of lead ime forecasig o he bullwhip effec. arxiv prepri arxiv:

11 60 Zbigiew Micha Nielse P., Micha Z., o N.A.. (014). A Empirical Ivesigaio of ead Time isribuios. Advaces i Producio Maageme Sysems: Iovaive ad Kowledge-Based Producio Maageme i a Global-ocal World. Pp